Database of Primitive Groups

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This database contains a list of primitive groups of degree up to 50. The list has been prepared by C. C. Sims and, while it is believed to be complete, users assuming the completeness of the list do so at their own risk.

Each group is labelled according to its degree. For example, the label of the 3rd primitive group of degree 24 is 3, 24. An example and a table of basic information about each group may be found at the end of this file.

The groups are accessed through the following functions only.

PrmGroup(d, n): This function returns the n-th group of degree d.
PrmInfo(d, n): This function returns a string describing the structure of the n-th group of degree d.
PrmNumberOfDegree(d): This function returns the number of groups of degree d which are stored in the library.
PrmGroupSatisfying(f): Given a boolean valued function f: GrpPerm --> BoolElt (which may either be an intrinsic function or a user defined function), return a group satisfying f(G). This function runs through all the stored groups, expanding each from the stored generators and applies the predicate f until it finds a suitable one. If no group is found, an error message is printed.
PrmGroupOfDegreeSatisfying(d, f): As PrmGroupSatisfying(f), except it only runs through the groups of degree d.
PrmGroupsSatisfying(f): As PrmGroupSatisfying(f), except a sequence of all such groups is returned.
PrmGroupsOfDegreeSatisfying(d, f): As PrmGroupOfDegreeSatisfying(d, f), except a sequence of all such groups is returned.
PrmProcess(): Return a "process" for looping over all the stored groups. Initially it points to the first group (of degree 2).
PrmProcessOfDegree(d): Return a "process" for looping over all the stored groups of degree d. Initially it points to the first group of degree d.
PrmProcessOfDegree(): Return a "process" for looping over all the stored groups of degree d where lo <= d <= hi. Initially it points to the first group of degree lo.
PrmProcessIsEmpty(P): Return whether the process P currently points to a group.
PrmProcessGroup(P): Given a process P which currently points to a group, return that group.
PrmProcessInfo(P): Given a process P which currently points to a group, return the string describing the structure of the group.
PrmProcessLabel(P): Given a process P which currently points to a group, return the label d, n of the group.
PrmProcessNext(~P): Given a process P which currently points to a group, modify it so that it points to the next group if there is one or make it empty if there is not.

In the following table, all the primitive groups of degree up to 50 are described. The meaning of the columns are partially enshrouded in the mists of time. An explanation follows with possible errors marked by (*).

Deg, No: The label of the particular group G = PrmGroup(Deg, No).
Order: The order of G
t: The transitivity t of G. A prefix "s" denotes that G is sharply t-transitive. A suffix "p" denotes that G is (t+1/2)-transitive (*). An empty field denotes that G is 1-transitive.
+/-: If G is odd, then the column is marked "-".
Fr: If G is Frobenius, then the column is marked "*".
N: A description of the minimal normal subgroup N of G (*). If G is simple, then the column is marked "-". If G has an EARNS, then is it marked "e.a.". Otherwise, the N is denoted dGn and refers to the n-th primitive group in this table of degree d.
G(n): A description of a one-point stabilizer of G, if it is isomorphic to another primitive group. (*) Denoted as N above.
G^(t) orbs: The orbits of a t-point stabilizer S of G. If S is the trivial subgroup, the column is left blank. Otherwise they are described by a comman separated list of terms l^n, where l is the orbit length and n is the number of orbits of that length. If n is 1, it is left out.
Comments: This column attempt to give a description of the group in a very small number of characters. As far as possible, the ATLAS notation is used for describing extensions. The constituent groups are described very tersely, e.g., D(5) denotes the dihedral group of order 2*5. The only abbreviations that should need special note are PYL and PZL (and similar). The Y is used to represent the Greek letter Gamma, Z for Sigma. So the group PYL(2,8) refers to the group which in Magma can be represented by asking for PGammaL(2, 8). On occasion, the group is described in a footnote.

===============================================================================
Deg |  No |   Order  |  t   |+/-| Fr. |  N   | G(n) | G^(t) orbs | Comments    
===============================================================================
  2 |   1 |        2 |   2  | - |     |  -   |      |            | S(2)=2
  3 |   1 |        3 |      |   |     |  -   |      |            | A(3)=3
    |   2 |        6 |  s3  | - |  *  |  3G1 |  2G1 |            | S(3)
  4 |   1 |       12 |  s2p |   |  *  | e.a. |  3G1 |            | A(4)
    |   2 |       24 |  s4  | - |     | e.a. |  3G2 |            | S(4)=AGL(2,2)
  5 |   1 |        5 |      |   |     |  -   |      |            | 5
    |   2 |       10 |      |   |  *  |  5G1 |      | 1,2^2      | 5:2=D(5)
    |   3 |       20 |  s2  | - |  *  |  5G1 |      |            | 5:4=AGL(1,5)
    |   4 |       60 |  s3p |   |     |  -   |  4G1 |            | A(5)
    |   5 |      120 |  s5  | - |     |  5G4 |  4G2 |            | S(5)
  6 |   1 |       60 |   2p |   |     |  -   |  5G2 | 1^2,2^2    | PSL(2,5)
    |   2 |      120 |  s3  | - |     |  6G1 |  5G3 |            | PGL(2,5)
    |   3 |      360 |  s4p |   |     |  -   |  5G4 |            | A(6)
    |   4 |      720 |  s6  | - |     |  6G3 |  5G5 |            | S(6)
  7 |   1 |        7 |      |   |     |  -   |      |            | 7
    |   2 |       14 |      | - |  *  |  7G1 |      | 1,2^3      | 7:2=D(7)
    |   3 |       21 |      |   |  *  |  7G1 |      | 1,3^2      | 7:3
    |   4 |       42 |  s2  | - |  *  |  7G1 |      |            | AGL(1,7)
    |   5 |      168 |   2  |   |     |  -   |      | 1^3,4      | PSL(3,2)
    |   6 |     2520 |  s5p |   |     |  -   |  6G3 |            | A(7)
    |   7 |     5040 |  s7  | - |     |  7G6 |  6G4 |            | S(7)
  8 |   1 |       56 |  s2p |   |  *  | e.a. |  7G1 |            | AGL(1,8)
    |   2 |      168 |   2p |   |     | e.a. |  7G3 | 1^2,3^2    | AYL(1,8)
    |   3 |      168 |   2p |   |     |  -   |  7G3 | 1^2,3^2    | PSL(2,7)
    |   4 |      336 |  s3  | - |     |  8G3 |  7G4 |            | PGL(2,7)
    |   5 |     1344 |   3  |   |     | e.a. |  7G5 | 1^4,4      | ASL(3,2)
    |   6 |    20160 |  s6p |   |     |  -   |  7G6 |            | A(8)
    |   7 |    40320 |  s8  | - |     |  8G6 |  7G7 |            | S(8)
  9 |   1 |       36 |      |   |  *  | e.a. |      | 1,4^2      | 3^2:4
    |   2 |       72 |      | - |     | e.a. |      | 1,4^2      | 3^2:D(4)
    |   3 |       72 |  s2  | _ |  *  | e.a. |      |            | footnote *1
    |   4 |       72 |  s2  |   |  *  | e.a. |      |            | 3^2:Q(8)=M(9)
    |   5 |      144 |   2  | - |     | e.a. |      | 1^3,2^3    | AYL(1,9)
    |   6 |      216 |   2  |   |     | e.a. |      | 1^3,3^2    | 3^2:(2'A(4))
    |   7 |      432 |   2  | - |     | e.a. |      | 1^3,6      | AGL(2,3)
    |   8 |      504 |  s3p |   |     |  -   |  8G1 |            | PSL(2,8)
    |   9 |     1512 |   3p |   |     |  9G8 |  8G2 | 1^3,3^2    | PYL(2,8)
    |  10 |     9!/2 |  s7p |   |     |  -   |  8G6 |            | A(9)
    |  11 |       9! |  s9  | - |     |  9G10|  8G7 |            | S(9)
 10 |   1 |       60 |      |   |     |  -   |      | 1,3,6      | A(5)
    |   2 |      120 |      | - |     | 10G1 |      | 1,3,6      | S(5)
    |   3 |      360 |   2p |   |     |  -   |  9G1 | 1^2,4^2    | PSL(2,9)
    |   4 |      720 |   2p | - |     | 10G3 |  9G2 | 1^2,4^2    | S(6)
    |   5 |      720 |  s3  | - |     | 10G3 |  9G3 |            | PGL(2,9)
    |   6 |      720 |  s3  |   |     | 10G3 |  9G4 |            | M(10)
    |   7 |     1440 |   3  | - |     | 10G3 |  9G5 | 1^4,2^3    | PYL(2,9)
    |   8 |    10!/2 |  s8p |   |     |  -   |  9G10|            | A(10)
    |   9 |      10! | s10  | - |     | 10G8 |  9G11|            | S(10)
 11 |   1 |       11 |      |   |     |  -   |      |            | 11
    |   2 |       22 |      | - |  *  | 11G1 |      | 1,2^5      | D(11)
    |   3 |       55 |      |   |  *  | 11G1 |      | 1,5^2      | 11:5
    |   4 |      110 |  s2  | - |  *  | 11G1 |      |            | AGL(1,11)
    |   5 |      660 |   2p |   |     |  -   | 10G1 | 1^2,3,6    | PSL(2,11)
    |   6 |     7920 |  s4  |   |     |  -   | 10G6 |            | M(11)
    |   7 |    11!/2 |  s9p |   |     |  -   | 10G8 |            | A(11)
    |   8 |      11! | s11  | - |     | 11G7 | 10G9 |            | S(11)
 12 |   1 |      660 |   2p |   |     |  -   | 11G3 | 1^2,5^2    | PSL(2,11)
    |   2 |     1320 |  s3  | - |     | 12G1 | 11G4 |            | PGL(2,11)
    |   3 |     7920 |   3p |   |     |  -   | 11G5 | 1^3,3,6    | M(11)
    |   4 |    95040 |  s5  |   |     |  -   | 11G6 |            | M(12)
    |   5 |    12!/2 | s10p |   |     |  -   | 11G7 |            | A(12)
    |   6 |      12! | s12  | - |     | 12G5 | 11G8 |            | S(12)
 13 |   1 |       13 |      |   |     |  -   |      |            | 13
    |   2 |       26 |      |   |  *  | 13G1 |      | 1,2^6      | D(13)
    |   3 |       39 |      |   |  *  | 13G1 |      | 1,3^4      | 13:3
    |   4 |       52 |      | - |  *  | 13G1 |      | 1,4^3      | 13:4
    |   5 |       78 |      |   |  *  | 13G1 |      | 1,6^2      | 13:6
    |   6 |      156 |  s2  | - |  *  | 13G1 |      |            | AGL(1,13)
    |   7 |     5616 |   2  |   |     |  -   |      | 1^2,2,9    | PSL(3,3)
    |   8 |    13!/2 | s11p |   |     |  -   | 12G5 |            | A(13)
    |   9 |      13! | s13  | - |     | 13G8 | 12G6 |            | S(13)
 14 |   1 |     1092 |   2p |   |     |  -   | 13G5 | 1^2,6^2    | PSL(2,13)
    |   2 |     2184 |  s3  | - |     | 14G1 | 13G6 |            | PGL(2,13)
    |   3 |    14!/2 | s12p |   |     |  -   | 13G8 |            | A(14)
    |   4 |      14! | s14  | - |     | 14G3 | 13G9 |            | S(14)
 15 |   1 |      360 |      |   |     |  -   |      | 1,6,8      | A(6)
    |   2 |      720 |      |   |     | 15G1 |      | 1,6,8      | S(6) 
    |   3 |     2520 |   2  |   |     |  -   |      | 1^3,12     | A(7)
    |   4 |    20160 |   2  |   |     |  -   |      | 1^3,12     | PSL(4,2)
    |   5 |    15!/2 | s13p |   |     |  -   | 14G3 |            | A(15)
    |   6 |      15! | s15  | - |     | 15G5 | 14G4 |            | S(15)
 16 |   1 |       80 |      |   |  *  | e.a. |      | 1,5^3      | 2^4:5
    |   2 |      160 |      |   |     | e.a. |      | 1,5^3      | 2^4:D(5)
    |   3 |      240 |  s2  |   |  *  | e.a. |      |            | AGL(1,16)
    |   4 |      288 |      |   |     | e.a. |      | 1,6,9      | (A(4)xA(4)):2
    |   5 |      320 |      |   |     | e.a. |      | 1,5,10     | (2^4:5).4
    |   6 |      480 |   2  |   |     | e.a. |      | 1^4,2^6    | AGL(1,16):2
    |   7 |      576 |      |   |     | e.a. |      | 1,6,9      | 2^4.3^2:4
    |   8 |      576 |      |   |     | e.a. |      | 1,6,9      | 2^4.S(3)xS(3)
    |   9 |      960 |      |   |     | e.a. |      | 1,5,10     | 2^4:A(5)
    |  10 |      960 |   2  |   |     | e.a  |      | 1^2,2,4^3  | AYL(1,16)
    |  11 |      960 |   2  |   |     | e.a. |      | 1^2,2,4^3  | ASL(2,4)
    |  12 |     1152 |      |   |     | e.a. |      | 1,6,9      | (S(4)xS(4)):2
    |  13 |     1920 |      |   |     | e.a. |      | 1,5,10     | 2^4:S(5)
    |  14 |     1920 |   2  |   |     | e.a. |      | 1^2,2,4,8  | ASL(2,4):2
    |  15 |     2880 |   2  |   |     | e.a. |      | 1^4,12     | AGL(2,4)
    |  16 |     5760 |   2  |   |     | e.a. |      | 1^2,2,12   | AYL(2,4)
    |  17 |     5760 |   2p |   |     | e.a. | 15G1 | 1^2,6,8    | 2^4.A(6)
    |  18 |    11520 |   2p |   |     | e.a. | 15G2 | 1^2,6,8    | 2^4.S(6)
    |  19 |    40320 |   3  |   |     | e.a. | 15G3 | 1^4,12     | 2^4.A(7)
    |  20 |   322560 |   3  |   |     | e.a. | 15G4 | 1^4,12     | 2^4.L(4,2)
    |  21 |    16!/2 | s14p |   |     |  -   | 15G5 |            | A(16)
    |  22 |      16! | s16  | - |     | 16G21| 15G6 |            | S(16)
 17 |   1 |       17 |      |   |     |  -   |      |            | 17
    |   2 |       34 |      |   |  *  | 17G1 |      | 1,2^8      | D(17)
    |   3 |       68 |      |   |  *  | 17G1 |      | 1,4^4      | 17:4
    |   4 |      136 |      |   |  *  | 17G1 |      | 1,8^2      | 17:8
    |   5 |      272 |  s2  | - |  *  | 17G1 |      |            | AGL(1,17)
    |   6 |     4080 |  s3  |   |     |  -   | 16G3 |            | PSL(2,16)
    |   7 |     8160 |   3  |   |     | 17G6 | 16G6 | 1^5,2^6    | PSL(2,16):2
    |   8 |    16320 |   3  |   |     | 17G6 | 16G10| 1^3,2,4^3  | PYL(2,16)
    |   9 |    17!/2 | s15p |   |     |  -   | 16G21|            | A(17)
    |  10 |      17! | s17  | - |     | 17G9 | 16G22|            | S(17)
 18 |   1 |     2448 |   2p |   |     |  -   | 17G4 | 1^2,8^2    | PSL(2,17)
    |   2 |     4896 |  s3  | - |     | 18G1 | 17G5 |            | PGL(2,17)
    |   3 |    18!/2 | s16p |   |     |  -   | 17G9 |            | A(18)
    |   4 |      18! | s18  | - |     | 18G3 | 17G10|            | S(18)
 19 |   1 |       19 |      |   |     |  -   |      |            | 19
    |   2 |       38 |      | - |  *  | 19G1 |      | 1,2^9      | D(19)
    |   3 |       57 |      |   |  *  | 19G1 |      | 1,3^6      | 19:3
    |   4 |      114 |      | - |  *  | 19G1 |      | 1,6^3      | 19:6
    |   5 |      171 |      |   |  *  | 19G1 |      | 1,9^2      | 19:9
    |   6 |      342 |  s2  | - |  *  | 19G1 |      |            | AGL(1,19)
    |   7 |    19!/2 | s17p |   |     |  -   | 18G3 |            | A(19)
    |   8 |      19! | s19  | - |     | 19G7 | 18G4 |            | S(19)
 20 |   1 |     3420 |   2p |   |     |  -   | 19G5 | 1^2,9^2    | PSL(2,19)
    |   2 |     6840 |  s3  | - |     | 20G1 | 19G6 |            | PGL(2,19)
    |   3 |    20!/2 | s18p |   |     |  -   | 19G7 |            | A(20)
    |   4 |      20! | s20  | - |     | 20G3 | 19G8 |            | S(20)
 21 |   1 |      336 |      | - |     |      |      | 1,4,8^2    | PGL(2,7)
    |   2 |     2520 |      |   |     |  -   |      | 1,10^2     | A(7)
    |   3 |     5040 |      | - |     | 21G2 |      | 1,10^2     | S(7)
    |   4 |    20160 |   2  |   |     |  -   |      | 1^2,3,16   | PSL(3,4)=M_21
    |   5 |    40320 |   2  | - |     | 21G4 |      | 1^2,3,16   | PZL(3,4)
    |   6 |    60480 |   2  |   |     | 21G4 |      | 1^2,3,16   | PGL(3,4)
    |   7 |   120960 |   2  | - |     | 21G4 |      | 1^2,3,16   | PYL(3,4)
    |   8 |    21!/2 | s19p |   |     |  -   | 20G3 |            | A(21)
    |   9 |      21! | s21  | - |     | 21G8 | 20G4 |            | S(21)
 22 |   1 |   443520 |   3  |   |     |  -   | 21G4 | 1^3,3,16   | M(22)
    |   2 |   887040 |   3  | - |     | 22G1 | 21G5 | 1^3,3,16   | M(22):2
    |   3 |    22!/2 | s20p |   |     |  -   | 21G8 |            | A(22)
    |   4 |      22! | s22  | - |     | 22G3 | 21G9 |            | S(22)
 23 |   1 |       23 |      |   |     |  -   |      |            | 23
    |   2 |       46 |      | - |  *  | 23G1 |      | 1,2^11     | D(23)
    |   3 |      253 |      |   |  *  | 23G1 |      | 1,11^2     | 23:11
    |   4 |      506 |  s2  | - |  *  | 23G1 |      |            | AGL(1,23)
    |   5 | 10200960 |   4  |   |     |  -   | 22G1 | 1^4,3,16   | M(23)
    |   6 |    23!/2 | s21p |   |     |  -   | 22G3 |            | A(23)     
    |   7 |      23! | s23  | - |     | 23G6 | 22G4 |            | S(23)
 24 |   1 |     6072 |   2p |   |     |  -   | 23G3 | 1^2,11^2   | PSL(2,23)
    |   2 |    12144 |  s3  | - |     | 24G1 | 23G4 |            | PGL(2,23)
    |   3 |244823040 |   5  |   |     |  -   | 23G5 | 1^5,3,16   | M(24)
    |   4 |    24!/2 | s22p |   |     |  -   | 23G6 |            | A(24)
    |   5 |      24! | s24  | - |     | 24G4 | 23G7 |            | S(24)
 25 |   1 |       75 |      |   |  *  | e.a. |      | 1,3^8      | 5^2:3
    |   2 |      150 |      |   |  *  | e.a. |      | 1,6^4      | 5^2:6
    |   3 |      150 |      |   |     | e.a. |      | 1,3^4,6^2  | 5^2:S(3)
    |   4 |      200 |      |   |     | e.a. |      | 1,4^4,8    | 5^2:D(4)
    |   5 |      200 |      | - |  *  | e.a. |      | 1,8^3      | 5^2:8
    |   6 |      200 |      |   |  *  | e.a. |      | 1,8^3      | 5^2:Q(8)
    |   7 |      300 |      |   |  *  | e.a. |      | 1,12^2     | 5^2:12
    |   8 |      300 |      |   |  *  | e.a. |      | 1,12^2     | 5^2:Q(12)
    |   9 |      300 |      |   |     | e.a. |      | 1,6^4      | 5^2:D(6)
    |  10 |      400 |      | - |     | e.a. |      | 1,8,16     | 5^2:8:2
    |  11 |      400 |      |   |     | e.a. |      | 1,8^3      | 5^2:D(4):2
    |  12 |      600 |      |   |     | e.a. |      | 1,12^2     | 5^2:4xD(3)
    |  13 |      600 |  s2  |   |  *  | e.a. |      |            | 5^2:(Q(8)x3)
    |  14 |      600 |  s2  | - |  *  | e.a. |      |            | AGL(1,25)
    |  15 |      600 |  s2  | - |  *  | e.a. |      |            | 5^2:3:8
    |  16 |      800 |      | - |     | e.a. |      | 1,8,16     | 5^2:O+(2,5)
    |  17 |     1200 |   2  |   |     | e.a. |      | 1^5,2^10   | footnote *2
    |  18 |     1200 |   2  | - |     | e.a. |      | 1^5,2^10   | AYL(1,25)
    |  19 |     2400 |   2  | - |     | e.a. |      | 1^5,4^5    | footnote *3
    |  20 |     3000 |   2  |   |     | e.a. |      | 1^5,5^4    | ASL(2,5)
    |  21 |     6000 |   2  |   |     | e.a. |      | 1^5,2^10   | ASL(2,5):2
    |  22 |     7200 |      |   |     |      |      | 1,8,16     | (A(5)xA(5)):2
    |  23 |    12000 |   2  | - |     | e.a. |      | 1^5,20     | AGL(2,5)
    |  24 |    14400 |      |   |     |      |      | 1,8,16     | (A_5xA_5):2^2
    |  25 |    14400 |      | - |     |      |      | 1,8,16     | (A(5)xA(5)):4
    |  26 |    28800 |      | - |     |      |      | 1,8,16     | (S(5)xS(5)):2
    |  27 |    25!/2 | s23p |   |     |  -   | 24G4 |            | A(25)
    |  28 |      25! | s25  | - |     | 25G27| 24G5 |            | S(25)
 26 |   1 |     7800 |   2p |   |     |  -   | 25G7 | 1^2,12^2   | PSL(2,25)
    |   2 |    15600 |   2p |   |     | 26G1 | 25G12| 1^2,12^2   | PZL(2,25) 
    |   3 |    15600 |  s3  | - |     | 26G1 | 25G14|            | PGL(2,25)
    |   4 |    15600 |  s3  | - |     | 26G1 | 25G15|            | PSL(2,25):2
    |   5 |    31200 |   3  | - |     | 26G1 | 25G18| 1^6,2^10   | PYL(2,25)
    |   6 |    26!/2 | s24p |   |     |  -   | 25G27|            | A(26)
    |   7 |      26! | s26  | - |     | 26G6 | 25G28|            | S(26)
 27 |   1 |      324 |      |   |     | e.a. |      | 1,4^2,6,12 | 3^3.A(4)
    |   2 |      351 |      |   |  *  | e.a. |      | 1,13^2     | 3^3:13
    |   3 |      648 |      |   |     | e.a. |      | 1,6,8,12   | 3^3(A(4)x2)
    |   4 |      648 |      | - |     | e.a. |      | 1,4^2,6,12 | 3^3.S(4)
    |   5 |      648 |      | - |     | e.a. |      | 1,6,8,12   | 3^3.2.A(4)
    |   6 |      702 |  s2  | - |  *  | e.a. |      |            | AGL(1,27)
    |   7 |     1053 |      |   |     | e.a. |      | 1,13^2     | 3^3.13.3
    |   8 |     1296 |      | - |     | e.a. |      | 1,6,8,12   | 3^3(S(4)x2)
    |   9 |     2106 |   2  | - |     | e.a. |      | 1^3,3^8    | AYL(1,27)
    |  10 |    25920 |      |   |     |  -   |      | 1,10,16    | PSp(4,3)
    |  11 |    51840 |      |   |     | 27G10|      | 1,10,16    | PSp(4,3):2
    |  12 |   151632 |   2  |   |     | e.a. |      | 1^3,24     | ASL(3,3)
    |  13 |   303264 |   2  | - |     | e.a. |      | 1^3,24     | AGL(3,3)
    |  14 |    27!/2 | s25p |   |     |  -   | 26G6 |            | A(27)
    |  15 |      27! | s27  | - |     | 27G14| 26G7 |            | S(27)
 28 |   1 |      336 |      |   |     |      |      | 1,3,6^2,12 | PGL(2,7)
    |   2 |      504 |      |   |     |  -   |      | 1,9^3      | PSL(2,8)
    |   3 |     1512 |   2  |   |     | 28G2 |      | 1^4,2^12   | PYL(2,8)
    |   4 |     6048 |   2  |   |     |  -   |      | 1^2,2,8^3  | PSU(3,3^2)
    |   5 |     9828 |   2p |   |     |  -   | 27G2 | 1^2,13^2   | PSL(2,27)
    |   6 |    12096 |   2  |   |     | 28G4 |      | 1^2,2,8,16 | PYU(3,3^2)
    |   7 |    19656 |  s3  | - |     | 28G5 | 27G6 |            | PGL(2,27)
    |   8 |    20160 |      |   |     |  -   |      | 1,12,15    | A(8)
    |   9 |    29484 |   2p |   |     | 28G5 | 27G7 | 1^2,13^2   | PSL(2,27):3
    |  10 |    40320 |      |   |     | 28G8 |      | 1,12,15    | S(8)
    |  11 |    58968 |   3  | - |     | 28G5 | 27G9 | 1^4,3^8    | PYL(2,27)
    |  12 |  1451520 |   2p |   |     |  -   | 27G11| 1^2,10,16  | PSp(6,2)
    |  13 |    28!/2 | s26p |   |     |  -   | 27G14|            | A(28)
    |  14 |      28! | s28  | - |     | 28G13| 27G15|            | S(28)
 29 |   1 |       29 |      |   |     |  -   |      |            | 29
    |   2 |       58 |      |   |  *  | 29G1 |      | 1,2^14     | D(29)
    |   3 |      116 |      | - |  *  | 29G1 |      | 1,4^7      | 29:4
    |   4 |      203 |      |   |  *  | 29G1 |      | 1,7^4      | 29:7
    |   5 |      406 |      |   |  *  | 29G1 |      | 1,14^2     | 29:14
    |   6 |      812 |  s2  | - |  *  | 29G1 |      |            | AGL(1,29)
    |   7 |    29!/2 | s27p |   |     |  -   | 28G13|            | A(29)
    |   8 |      29! | s29  | - |     | 29G7 | 28G14|            | S(29)
 30 |   1 |    12180 |   2p |   |     |  -   | 29G5 | 1^2,14^2   | PSL(2,29)
    |   2 |    24360 |  s3  | - |     | 30G1 | 29G6 |            | PGL(2,29)
    |   3 |    30!/2 | s28p |   |     |  -   | 29G7 |            | A(30)
    |   4 |      30! | s30  | - |     | 30G3 | 29G8 |            | S(30)
 31 |   1 |       31 |      |   |     |  -   |      |            | 31
    |   2 |       62 |      | - |  *  | 31G1 |      | 1,2^15     | D(31)
    |   3 |       93 |      |   |  *  | 31G1 |      | 1,3^10     | 31:3
    |   4 |      155 |      |   |  *  | 31G1 |      | 1,5^6      | 31:5
    |   5 |      186 |      | - |  *  | 31G1 |      | 1,6^5      | 31:6
    |   6 |      310 |      | - |  *  | 31G1 |      | 1,10^3     | 31:10
    |   7 |      465 |      |   |  *  | 31G1 |      | 1,15^2     | 31:15
    |   8 |      930 |  s2  | - |  *  | 31G1 |      |            | AGL(1,31)
    |   9 |   372000 |   2  |   |     |  -   |      | 1^2,4,25   | PSL(3,5)
    |  10 |  9999360 |   2  |   |     |  -   |      | 1^3,28     | PSL(5,2)
    |  11 |    31!/2 | s29p |   |     |  -   | 30G3 |            | A(31)
    |  12 |      31! | s31  | - |     | 31G11| 30G4 |            | S(31)
 32 |   1 |      992 |  s2p |   |  *  | e.a. | 31G1 |            | AGL(1,32)
    |   2 |     4960 |   2p |   |     | e.a. | 31G4 | 1^2,5^6    | AYL(1,32)
    |   3 |    14880 |   2p |   |     |  -   | 31G7 | 1^2,15^2   | PSL(2,31)
    |   4 |    29760 |  s3  | - |     | 32G3 | 31G8 |            | PGL(2,31)
    |   5 |319979520 |   3  |   |     | e.a. | 31G10| 1^4,28     | ASL(5,2)
    |   6 |    32!/2 | s30p |   |     |  -   | 31G11|            | A(32)
    |   7 |      32! | s32  | - |     | 32G6 | 31G12|            | S(32)
 33 |   1 |    32736 |  s3p |   |     |  -   | 32G1 |            | PSL(2,32)
    |   2 |   163680 |   3p |   |     | 33G1 | 32G2 | 1^3,5^6    | PYL(2,32)
    |   3 |    33!/2 | s31p |   |     |  -   | 32G6 |            | A(33)
    |   4 |      33! | s33  | - |     | 33G3 | 32G7 |            | S(33)
 34 |   1 |    34!/2 | s32p |   |     |  -   | 33G3 |            | A(34)
    |   2 |      34! | s34  | - |     | 34G1 | 33G4 |            | S(34)
 35 |   1 |     2520 |      |   |     |  -   |      | 1,4,12,18  | A(7)
    |   2 |     5040 |      |   |     | 35G1 |      | 1,4,12,18  | S(7)
    |   3 |    20160 |      |   |     |  -   |      | 1,16,18    | A(8)
    |   4 |    40320 |      |   |     | 35G3 |      | 1,16,18    | S(8)
    |   5 |    35!/2 | s33p |   |     |  -   | 34G1 |            | A(35)
    |   6 |      35! | s35  | - |     | 35G5 | 34G2 |            | S(35)
 36 |   1 |      504 |      |   |     |  -   |      | 1,7^3,14   | PSL(2,8)
    |   2 |      720 |      | - |     |      |      | 1,5,10^3   | PGL(2,9)
    |   3 |      720 |      | - |     |      |      | 1,5,10,20  | M(10)
    |   4 |     1440 |      | - |     |      |      | 1,5,10,20  | PYL(2,9)
    |   5 |     1512 |      |   |     | 36G1 |      | 1,14,21    | PYL(2,8)
    |   6 |     6048 |      |   |     |  -   |      | 1,7^2,21   | PSU(3,3^2)
    |   7 |     7200 |      | - |     |      |      | 1,10,25    | (A(5)xA(5)):2
    |   8 |    12096 |      |   |     | 36G6 |      | 1,14,21    | PYU(3,3^2)
    |   9 |    14400 |      | - |     |      |      | 1,10,25    | ((A5xA5):2)2
    |  10 |    14400 |      | - |     |      |      | 1,10,25    | (A(5)xA(5)).4
    |  11 |    25920 |      |   |     |  -   |      | 1,15,20    | PSp(4,3)
    |  12 |    28800 |      | - |     |      |      | 1,10,25    | (S(5)xS(5)):2
    |  13 |    51840 |      |   |     | 36G11|      | 1,15,20    | PSp(4,3):2
    |  14 |   181440 |      |   |     |  -   |      | 1,14,21    | A(9)
    |  15 |   259200 |      | - |     |      |      | 1,10,25    | (A(6)xA(6)):2
    |  16 |   362880 |      | - |     | 36G14|      | 1,14,21    | S(9)
    |  17 |   518400 |      | - |     |      |      | 1,10,25    | (A(6)xA(6)):4
    |  18 |   518400 |      | - |     |      |      | 1,10,25    | (A_6xA_6):2^2
    |  19 |  1036800 |      | - |     |      |      | 1,10,25    | (S(6)xS(6)):2
    |  20 |  1451520 |   2p |   |     |  -   | 35G4 | 1^2,16,18  | PSp(6,2)
    |  21 |    36!/2 | s34p |   |     |  -   | 35G5 |            | A(36)
    |  22 |      36! | s36  | - |     | 36G21| 35G6 |            | S(36)
 37 |   1 |       37 |      |   |     |  -   |      |            | 37
    |   2 |       74 |      |   |  *  | 37G1 |      | 1,2^18     | D(37)
    |   3 |      111 |      |   |  *  | 37G1 |      | 1,3^12     | 37:3
    |   4 |      148 |      | - |  *  | 37G1 |      | 1,4^9      | 37:4
    |   5 |      222 |      |   |  *  | 37G1 |      | 1,6^6      | 37:6
    |   6 |      333 |      |   |  *  | 37G1 |      | 1,9^4      | 37:9
    |   7 |      444 |      | - |  *  | 37G1 |      | 1,12^3     | 37:12
    |   8 |      666 |      |   |  *  | 37G1 |      | 1,18^2     | 37:18
    |   9 |     1332 |  s2  | - |  *  | 37G1 |      |            | AGL(1,37)
    |  10 |    37!/2 | s35p |   |     |  -   | 36G21|            | A(37)
    |  11 |      37! | s37  | - |     | 37G10| 36G22|            | S(37)
 38 |   1 |    25308 |   2p |   |     |  -   | 37G8 | 1^2,18^2   | PSL(2,37)
    |   2 |    50616 |  s3  | - |     | 38G1 | 37G9 |            | PGL(2,37)
    |   3 |    38!/2 | s36p |   |     |  -   | 37G10|            | A(38)
    |   4 |      38! | s38  | - |     | 38G3 | 37G11|            | S(38)
 39 |   1 |    39!/2 | s37p |   |     |  -   | 38G3 |            | A(39)
    |   2 |      39! | s39  | - |     | 39G1 | 38G4 |            | S(39)
 40 |   1 |    25920 |      |   |     |  -   |      | 1,12,27    | PSp(4,3)
    |   2 |    25920 |      |   |     |  -   |      | 1,12,27    | PSp(4,3)
    |   3 |    51840 |      |   |     | 40G1 |      | 1,12,27    | PSp(4,3):2
    |   4 |    51840 |      | - |     | 40G2 |      | 1,12,27    | PSp(4,3):2
    |   5 |  6065280 |   2  |   |     |  -   |      | 1^2,2,36   | PSL(4,3)
    |   6 | 12130560 |   2  | - |     | 40G5 |      | 1^2,2,36   | PGL(4,3)
    |   7 |    40!/2 | s38p |   |     |  -   | 39G1 |            | A(40)
    |   8 |      40! | s40  | - |     | 40G7 | 39G2 |            | S(40)
 41 |   1 |       41 |      |   |     |  -   |      |            | 41
    |   2 |       82 |      |   |  *  | 41G1 |      | 1,2^20     | D(41)
    |   3 |      164 |      |   |  *  | 41G1 |      | 1,4^10     | 41:4
    |   4 |      205 |      |   |  *  | 41G1 |      | 1,5^8      | 41:5
    |   5 |      328 |      | - |  *  | 41G1 |      | 1,8^5      | 41:8
    |   6 |      410 |      |   |  *  | 41G1 |      | 1,10^4     | 41:10
    |   7 |      820 |      |   |  *  | 41G1 |      | 1,20^2     | 41:20
    |   8 |     1640 |  s2  | - |  *  | 41G1 |      |            | AGL(1,41)
    |   9 |    41!/2 | s39p |   |     |  -   | 40G7 |            | A(41)
    |  10 |      41! | s41  | - |     | 41G9 | 40G8 |            | S(41)
 42 |   1 |    34440 |   2p |   |     |  -   | 41G7 | 1,20^2     | PSL(2,41)
    |   2 |    68880 |  s3  | - |     | 42G1 | 41G8 |            | PGL(2,41)
    |   3 |    42!/2 | s40p |   |     |  -   | 41G9 |            | A(42)
    |   4 |      42! | s42  | - |     | 42G3 | 41G10|            | S(42)
 43 |   1 |       43 |      |   |     |  -   |      |            | 43
    |   2 |       86 |      | - |  *  | 43G1 |      | 1,2^21     | D(43)
    |   3 |      129 |      |   |  *  | 43G1 |      | 1,3^14     | 43:3
    |   4 |      258 |      | - |  *  | 43G1 |      | 1,6^7      | 43:6
    |   5 |      301 |      |   |  *  | 43G1 |      | 1,7^6      | 43:7
    |   6 |      602 |      | - |  *  | 43G1 |      | 1,14^3     | 43:14
    |   7 |      903 |      |   |  *  | 43G1 |      | 1,21^2     | 43:21
    |   8 |     1806 |  s2  | - |  *  | 43G1 |      |            | AGL(1,43)
    |   9 |    43!/2 | s41p |   |     |  -   | 42G3 |            | A(43)
    |  10 |      43! | s43  | - |     | 43G9 | 42G4 |            | S(43)
 44 |   1 |    39732 |   2p |   |     |  -   | 43G7 | 1^2,21^2   | PSL(2,43)
    |   2 |    79464 |  s3  | - |     | 44G1 | 43G8 |            | PGL(2,43)
    |   3 |    44!/2 | s42p |   |     |  -   | 43G9 |            | A(44)
    |   4 |      44! | s44  | - |     | 44G3 | 43G10|            | S(44)
 45 |   1 |      720 |      |   |     |      |      | 1,4,8^3,16 | PGL(2,9)
    |   2 |      720 |      |   |     |      |      | 1,4,8,16^2 | M(10)
    |   3 |     1440 |      |   |     |      |      | 1,4,8,16^2 | PYL(2,9)
    |   4 |    25920 |      |   |     |  -   |      | 1,12,32    | PSp(4,3)
    |   5 |    51840 |      | - |     | 45G4 |      | 1,12,32    | PSp(4,3):2
    |   6 |  1814400 |      |   |     |  -   |      | 1,16,28    | A(10)
    |   7 |  3628800 |      |   |     | 45G6 |      | 1,16,28    | S(10)
    |   8 |    45!/2 | s43p |   |     |  -   | 44G3 |            | A(45)
    |   9 |      45! | s45  | - |     | 45G8 | 44G4 |            | S(45)
 46 |   1 |    46!/2 | s44p |   |     |  -   | 45G8 |            | A(46)
    |   2 |      46! | s46  | - |     | 46G1 | 45G9 |            | S(46)
 47 |   1 |       47 |      |   |     |  -   |      |            | 47
    |   2 |       94 |      | - |  *  | 47G1 |      | 1,2^23     | D(47)
    |   3 |     1081 |      |   |  *  | 47G1 |      | 1,23^2     | 47:23
    |   4 |     2162 |  s2  | - |  *  | 47G1 |      |            | AGL(1,47)
    |   5 |    47!/2 | s45p |   |     |  -   | 46G1 |            | A(47)
    |   6 |      47! | s47  | - |     | 47G5 | 46G2 |            | S(47)
 48 |   1 |    51888 |   2p |   |     |  -   | 47G3 | 1^2,23^2   | PSL(2,47)
    |   2 |   103776 |  s3  | - |     | 48G1 | 47G4 |            | PGL(2,47)
    |   3 |    48!/2 | s46p |   |     |  -   | 47G5 |            | A(48)
    |   4 |      48! | s48  | - |     | 48G3 | 47G6 |            | S(48)
 49 |   1 |      196 |      |   |  *  | e.a. |      | 1,4^12     | 7^2:4
    |   2 |      294 |      | - |     | e.a. |      | 1,3^6,6^5  | 7^2:S(3)
    |   3 |      392 |      |   |  *  | e.a. |      | 1,8^6      | 7^2:8
    |   4 |      392 |      |   |  *  | e.a. |      | 1,8^6      | 7^2:Q(8)
    |   5 |      392 |      | - |     | e.a. |      | 1,4^6,8^3  | 7^2:D(4)
    |   6 |      588 |      |   |  *  | e.a. |      | 1,12^4     | 7^2:Q(12)
    |   7 |      588 |      |   |  *  | e.a. |      | 1,12^4     | 7^2:12
    |   8 |      588 |      | - |     | e.a. |      | 1,6^7,12   | 7^2:D(6)
    |   9 |      784 |      |   |  *  | e.a. |      | 1,16^3     | 7^2:Q(16)
    |  10 |      784 |      | - |  *  | e.a. |      | 1,16^3     | 7^2:16
    |  11 |      784 |      | - |     | e.a. |      | 1,8^6      | 7^2:D(8)
    |  12 |      882 |      | - |     | e.a. |      |1,6^2,8^2,18| 7^2:3xD(3)
    |  13 |     1176 |      |   |  *  | e.a. |      | 1,24^2     | 7^2:24
    |  14 |     1176 |      |   |  *  | e.a. |      | 1,24^2     | 7^2:3xQ(8)
    |  15 |     1176 |      |   |  *  | e.a. |      | 1,24^2     | 7^2:Q(8):3
    |  16 |     1176 |      |   |     | e.a. |      | 1,8^3,24   | 7^2:Q(8):3
    |  17 |     1176 |      | - |     | e.a. |      | 1,12^4     | 7^2:3:D(4)
    |  18 |     1176 |      | - |     | e.a. |      | 1,12^2,24  | 7^2:3xD(4)
    |  19 |     1568 |      | - |     | e.a. |      | 1,16^3     | 7^2:Q(16):2
    |  20 |     1764 |      |   |     | e.a. |      | 1,12,36    | 7^2:3xQ(12)
    |  21 |     1764 |      | - |     | e.a. |      | 1,12,18^2  | 7^2:3xD(6)
    |  22 |     2352 |  s2  |   |  *  | e.a. |      |            | footnote *4
    |  23 |     2352 |  s2  |   |  *  | e.a. |      |            | 7^2:(3xQ(16))
    |  24 |     2352 |  s2  | - |  *  | e.a. |      |            | footnote *5
    |  25 |     2352 |      | - |     | e.a. |      | 1,24^2     | 7^2:3xD(8)
    |  26 |     3528 |      |   |     | e.a. |      | 1,24^2     | 7^2:3x(Q_8:3)
    |  27 |     3528 |      | - |     | e.a. |      | 1,12,36    | footnote *6
    |  28 |     4704 |   2  | - |     | e.a. |      | 1^7,2^21   | AYL(1,49)
    |  29 |     7056 |   2  |   |     | e.a. |      | 1^7,3^14   | footnote *7
    |  30 |    16464 |   2  |   |     | e.a. |      | 1^7,7^6    | ASL(2,7)
    |  31 |    32928 |   2  | - |     | e.a. |      | 1^7,14^3   | ASL(2,7):2
    |  32 |    49392 |   2  |   |     | e.a. |      | 1^7,21^2   | ASL(2,7):3
    |  33 |    56448 |      | - |     |      |      | 1,12,36    | footnote *8
    |  34 |    98784 |   2  | - |     | e.a. |      | 1^7,42     | AGL(2,7)
    |  35 | 12700800 |      | - |     |      |      | 1,12,36    | (A(7)xA(7)):2
    |  36 | 25401600 |      |   |     |      |      | 1,12,36    | (A_7xA_7):2^2
    |  37 | 25401600 |      | - |     |      |      | 1,12,36    | (A(7)xA(7)):4
    |  38 | 50803200 |      | - |     |      |      | 1,12,36    | (S(7)xS(7)):2
    |  39 |    49!/2 | s47p |   |     |  -   | 48G3 |            | A(49)
    |  40 |      49! | s49  | - |     | 49G39| 48G4 |            | S(49)
 50 |   1 |    58800 |   2p |   |     |  -   | 49G13| 1^2,24^2   | PSL(2,49)
    |   2 |   117600 |  s3  |   |     | 50G1 | 49G22|            | PGL(2,49)
    |   3 |   117600 |  s3  | - |     | 50G1 | 49G24|            | PSL(2,49):2
    |   4 |   117600 |   2p | - |     | 50G1 | 49G25| 1^2,24^2   | PSL(2,49):2
    |   5 |   126000 |      |   |     |  -   |      | 1,7,42     | PSU(3,5^2)
    |   6 |   235200 |   3  | - |     | 50G1 | 49G28| 1^8,2^21   | PYL(2,49)
    |   7 |   252000 |      |   |     | 50G5 |      | 1,7,42     | PSU(3,5^2)
    |   8 |    50!/2 | s48p |   |     |  -   | 49G39|            | A(50)
    |   9 |      50! | s50  | - |     | 50G8 | 49G40|            | S(50)
    --------------------------------------------------------------------------

Footnotes:
*1 3^2:8=AGL(1,9)
*2 5^2:((Q(8):3)'2)
*3 5^2:((Q(8):3)'4)
*4 7^2:(Q(8)'D(2*3))
*5 AGL(1,49)=7^2:48
*6 (AGL(1,7)xAGL(1,7)):2
*7 7^2:((Q(8)'D(2*3)) x 3)
*8 (PSL(3,2)xPSL(3,2)):2

Example

    load prmgps;
    P := PrmProcessOfDegree(5);
    while not PrmProcessIsEmpty(P) do
	G := PrmProcessGroup(P);
	... do stuff with G ...
	PrmProcessNext(~P);
    end while;

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