Database of Maximal Finite Subgroups of GL(n, Z)

This database contains the maximal finite subgroups of GL(n, Z) for n = 2, 3, 4, 5. The groups are taken from the tables which appear in R. Buelow, "Ueber Dadegruppen in GL(5, Z)", Diss, RWTH Aachen 1973.

Each group is named according to the dimension of the matrices. For example, the label of the 3rd group of 4-dimensional matrices is 3, 4.

The following is a list of the groups:

Label		Name		Order
-----		----		-----

2, 1		CU2             8 = 2^3
2, 2		SX2             12 = 2^2 * 3

3, 1		CU3             48 = 2^4 * 3
3, 2		SX3             48 = 2^4 * 3
3, 3		PY3             48 = 2^4 * 3
3, 4		SX2 + CU1       24 = 2^3 * 3

4, 1		Q4              1152 = 2^7 * 3^2
4, 2		CU4             384 = 2^7 * 3
4, 3		SX2 + CU2       96 = 2^5 * 3
4, 4		SX3 + CU1       96 = 2^5 * 3
4, 5		SX4             240 = 2^4 * 3 * 5
4, 6		SX2(2)          288 = 2^5 * 3^2
4, 7		PY3 + CU1       96 = 2^5 * 3
4, 8		PY4             240 = 2^4 * 3 * 5
4, 9		SX2^2           144 = 2^4 * 3^2

5, 1		SX2(2) + CU1    576 = 2^6 * 3^2
5, 2		SX4 + CU1       480 = 2^5 * 3 * 5
5, 3		PY4 + CU1       480 = 2^5 * 3 * 5
5, 4		SX2^2 + CU1     288 = 2^5 * 3^2
5, 5		Q4 + CU1        2304 = 2^8 * 3^2
5, 6		SX3 + CU2       384 = 2^7 * 3
5, 7		PY3 + CU2       384 = 2^7 * 3
5, 8		SX2 + CU3       576 = 2^6 * 3^2
5, 9		SX3 + SX2       576 = 2^6 * 3^2
5, 10		PY3 + SX2       576 = 2^6 * 3^2
5, 11		CU5             3840 = 2^8 * 3 * 5
5, 12		SX5             1440 = 2^5 * 3^2 * 5
5, 13		PY5             1440 = 2^5 * 3^2 * 5
5, 14				1440 = 2^5 * 3^2 * 5
5, 15				3840 = 2^8 * 3 * 5
5, 16				1440 = 2^5 * 3^2 * 5

The groups are accessed through the following functions only.

GLZGroup(d, n)

This function returns the n-th group of degree d.

GLZInfo(d, n)

This function returns a string describing the structure of the n-th group of degree d.

GLZGroupOfDegreeCount(d)

This function returns the number of groups of degree d which are stored in the library.

GLZGroupSatisfying(f)

Given a boolean valued function f: GrpMat --> 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.

GLZGroupOfDegreeSatisfying(d, f)

As GLZGroupSatisfying(f), except it only runs through the groups of degree d.

GLZGroupsSatisfying(f)

As GLZGroupSatisfying(f), except a sequence of all such groups is returned.

GLZGroupsOfDegreeSatisfying(d, f)

As GLZGroupOfDegreeSatisfying(d, f), except a sequence of all such groups is returned.

GLZProcess()

Return a "process" for looping over all the stored groups. Initially it points to the first group (of degree 2).

GLZProcessOfDegree(d)

Return a "process" for looping over all the stored groups of degree d. Initially it points to the first group of degree d.

GLZProcessOfDegree()

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 greater than or equal to lo.

GLZProcessIsEmpty(P)

Return whether the process P currently points to a group.

GLZProcessGroup(P)

Given a process P which currently points to a group, return that group.

GLZProcessInfo(P)

Given a process P which currently points to a group, return the string describing the structure of the group.

GLZProcessLabel(P)

Given a process P which currently points to a group, return the degree d of the group and the number n such that the group is GLZGroup(d, n).

GLZProcessNext(~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.

Example

load glnzgps;
P := GLZProcessOfDegree(5);
while not GLZProcessIsEmpty(P) do
    G := GLZProcessGroup(P);
    ... do computations with G ...
    GLZProcessNext(~P);
end while;