Subsections

• Construction of a Standard Permutation Group
• Construction of Extensions

A number of functions are provided which construct various standard groups. The effect of these functions is to construct the group on some standard set of generating permutations.

AbelianGroup(GrpPerm, Q) : Cat, [ RngIntElt ] -> GrpPerm

Construct the abelian group defined by the sequence Q = [n_1, ..., n_r] of positive integers. The function constructs the direct product of cyclic groups Z(n_1) x Z(n_2) x ... x Z(n_r).

AlternatingGroup(n) : RngIntElt -> GrpPerm

Alt(n) : RngIntElt -> GrpPerm

Construct the alternating group of degree n on generators (3, 4, ..., n) and (1, 2, 3), if n is odd, or (1, 2)(3, 4, ..., n) and (1, 2, 3), if n is even.

CyclicGroup(n) : RngIntElt -> GrpPerm

Construct the cyclic group of order n with generator (1, 2, ..., n).

DihedralGroup(n) : RngIntElt -> GrpPerm

Construct the dihedral group of degree n and order 2 * n on generators (1, 2, ..., n) and (1, n)(2, n - 1) ... .

SymmetricGroup(GrpPerm, n) : Cat, RngIntElt -> GrpPerm

Sym(n) : Cat, RngIntElt -> GrpPerm

SymmetricGroup(n) : RngIntElt -> GrpPerm

Construct the symmetric group of degree n on generators (1, 2, ..., n) and (1, 2).

Construct an extra-special group G of order p^(2n + 1) as a permutation group. If p = 2, then G is the central product of n copies of the quaternion group Q_8. If p > 2, then G is the central product of n copies of the extra-special group of order p^3 and exponent p.

• The abelian group Z_2 x Z_2 x Z_4:

> A := AbelianGroup(GrpPerm, [2, 2, 4] );
> A;
Permutation group A acting on a set of cardinality 8
Order = 16 = 2^4
    (1, 2)
    (3, 4)
    (5, 6, 7, 8)

• The alternating group of degree 12:

> A12 := AlternatingGroup(GrpPerm, 12);
> A12;
Permutation group A12 acting on a set of cardinality 12
Order = 239500800 = 2^9 * 3^5 * 5^2 * 7 * 11
    (1, 2)(3, 4, 5, 6, 7, 8, 9, 10, 11, 12)
    (1, 2, 3)

• The cyclic group Z_(24):

> Z24 := CyclicGroup(GrpPerm, 24);
> Z24;
Permutation group Z24 on a set of cardinality 24
Order = 24 = 2^3 * 3
    (1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15,
        16, 17, 18, 19, 20, 21, 22, 23, 24)

• The dihedral group of order 24:

> D12 := DihedralGroup(GrpPerm, 12);
> D12;
Permutation group D12 acting on a set of cardinality 12
Order = 24 = 2^3 * 3
    (1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12)
    (1, 12)(2, 11)(3, 10)(4, 9)(5, 8)(6, 7)

• The symmetric group of degree 8:

> S8 := SymmetricGroup(GrpPerm, 8);
> S8;
Symmetric group S8 acting on a set of cardinality 8
Order = 40320 = 2^7 * 3^2 * 5 * 7

Given two permutation groups G and H, construct the direct product D of G and H as an intransitive group having degree equal to the sum of the degrees of G and H. In addition, the sequences I of inclusions and P of projections are returned, satisfying I[i]: K_i -> D(K_i) and P[i]: D -> K_i (where K_1 = G, K_2 = H and D(K) is the group K represented naturally as a subgroup of D).

DirectProduct(Q) : [ GrpPerm ] -> GrpPerm, [ Hom(Grp) ], [ Hom(Grp) ]

Given a sequence Q of n permutation groups, construct the direct product Q[1] x Q[2] x ... x Q[n] as an intransitive group of degree equal to the sum of the degrees of the groups Q[i], (i = 1, ..., n). In addtion, the sequences I of inclusion and P of projections are returned, satisfying I[i]: Q[i] -> D(Q[i]) and P[i]: D -> Q[i] (where D(K) is the group K represented naturally as a subgroup of D).

Given permutation groups G and H, construct the wreath product G wreath H of G and H, where G wreath H has product action.

PrimitiveWreathProduct(Q) : [ GrpPerm ] -> GrpPerm

Given a sequence Q of n permutation groups, construct the iterated wreath product T = ( ... (Q[1] wreath Q[2]) wreath ... wreath Q[n]), where T has product action.

Given permutation groups G and H, construct the wreath product G wreath H of G and H, where G wreath H has imprimitive action.

WreathProduct(Q) : [ GrpPerm ] -> GrpPerm

Given a sequence Q of n permutation groups, construct the iterated wreath product W = ( ... (Q[1] wreath Q[2]) wreath ... wreath Q[n]), where W has imprimitive action.

• We define G to be the symmetric group of degree 4 and H to be the dihedral group of order 8. We then proceed to form the direct, subdirect, primitive-wreath and wreath products of G and H.

> G := SymmetricGroup(GrpPerm, 4);
> H := DihedralGroup(GrpPerm, 3);
> D := DirectProduct(G, H);
> D;
Permutation group D acting on a set of cardinality 7
    (1, 2, 3, 4)
    (1, 2)
    (5, 6, 7)
    (5, 6)
> Order(D);
144
> T := PrimitiveWreathProduct(G, H);
> T;
Permutation group T acting on a set of cardinality 64
    (2, 5, 17)(3, 9, 33)(4, 13, 49)(6, 21, 18)(7, 25, 34)(8, 29, 50)
       (10, 37, 19) (11, 41, 35)(12, 45, 51) (14, 53, 20)(15, 57, 36)
       (16, 61, 52)(23, 26, 38) (24, 30, 54)(27, 42, 39)(28, 46, 55)
       (31, 58, 40) (32, 62, 56)(44, 47, 59)(48, 63, 60)
    (2, 5)(3, 9)(4, 13)(7, 10)(8, 14)(12, 15)(18, 21)(19 , 25)(20, 29)
       (23, 26)(24, 30)(28, 31)(34, 37)(35 , 41)(36, 45)(39, 42)(40, 46)
       (44, 47)(50, 53)(51 , 57)(52, 61)(55, 58)(56, 62)(60, 63)
    (1, 2, 3, 4)(5, 6, 7, 8)(9, 10, 11, 12)(13, 14, 15, 16)(17, 18, 19, 20)
       (21, 22, 23, 24)(25, 26, 27, 28)(29, 30, 31, 32)(33, 34, 35, 36)
       (37, 38, 39, 40)(41, 42, 43, 44)(45, 46, 47, 48)(49, 50, 51, 52)
       (53, 54, 55, 56)(57, 58, 59, 60)(61, 62, 63, 64)
    (1, 2)(5, 6)(9, 10)(13, 14)(17, 18)(21, 22)(25, 26)( 29, 30)(33, 34)
       (37, 38)(41, 42)(45, 46)(49, 50)( 53, 54)(57, 58)(61, 62)
> Order(T);
82944
> W := WreathProduct(G, H);
> W;
Permutation group W acting on a set of cardinality 12
    (1, 5, 9)(2, 6, 10)(3, 7, 11)(4, 8, 12)
    (1, 5)(2, 6)(3, 7)(4, 8)
    (1, 2, 3, 4)
    (1, 2)
> Order(W);
82944