This database contains various useful matrix groups, almost all over finite fields (the only exception is the Weyl group E6).
Other standard matrix groups can be constructed directly in Magma. These are:
GL(n, q) GL(n, GF(q))
SL(n, q) SL(n, GF(q))
Sp(n, q) Sp(n, GF(q))
GU(n, q) GU(n, GF(q))
SU(n, q) SU(n, GF(q))
Sz(q) Sz(GF(q))
The following is a list of the groups in this library:
co1f2 The first Conway group.
Matrices of degree 24 over GF(2).
f42f2 The Chevalley group F(4, 2).
Matrices of degree 26 over GF(2).
fi22f4 The Fischer(22) (also called M(22)).
Matrices of degree 27 over GF(4).
fib29m Homomorph of the Fibonacci group F(2, 9)
Matrices of degree 19 over GF(5).
hu3o2n1 Huppert's doubly transitive soluble groups. See
hu3o2n2 B. Huppert, "Zweifach transitive, aufloesbare
hu3o4n1 Permutationsgruppen", Math. Zeit. 68(1957) 126-50.
hu3o4n2
hu3o4n3
hu5o2n1
hu5o2n2
hu5o2n3
hu7o2n1
hu7o2n2
hu11o2n1
hu11o2n2
hu23o2n1
j1f11b The first simple group of Janko (J1) given as a 7-dimensional
j1f11c matrix representation over GF(11).
j2f4 Two representations of the second Janko group
j2m1 (Hall-Janko-Wales group).
Matrices of degree 6 over GF(4).
j2a2f5 Two representations of the second Janko group extended by
j2a2f9 an automorphism of degree 2. j2a2f5 has matrices of degree 6
over GF(5), while j2a2f9 has matrices of degree 6 over GF(9).
j3f4 The Schur cover of the third simple group of Janko (J3) given
as a 9-dimensional matrix representation over GF(4).
j4f2 The fourth Janko group.
Matrices of degree 112 over GF(2).
lyf5 The Lyons group.
Matrices of degree 111 over GF(5).
m11z3 The Mathieu group M11.
Matrices of degree 5 over GF(3).
m22c3 3-fold cover of the Mathieu group M22.
Matrices of degree 6 over GF(4).
mat3f7 Matrix groups of degree given by the first integer over the
mat3f9 field GF(q), where q is the second integer. E.g., mat3f9
mat4f9 contains a matrix group of degree 3 over GF(9).
mat5f2
mat5f3
mat6f4
mclf5 The Schur cover of the McLaughlin simple group.
Matrices of degree 111 over GF(5).
onf7 The Schur cover of the O'Nan simple group.
Matrices of degree 45 over GF(7).
rudvalis The double cover of the Rudvalis simple group.
Matrices of degree 28 over GF(17).
rudc2 2-fold cover of the Rudvalis group.
Matrices of degree 28 over GF(17).
ruf2 The Rudvalis group.
Matrices of degree 28 over GF(2).
szf4 The triple cover of the Suzuki simple group.
Matrices of degree 12 over GF(4).
titsf25 The simple group TITS.
Matrices of degree 26 over GF(25).
weyle6 The Weyl group E6.
Matrices of degree 6 over the integers.
> load szf4;
Loading "matgps/szf4"
The triple cover of the sporadic simple group of Suzuki represented
as a matrix group of degree 12 over the field of 4 elements.
Field: K; Primitive element: w; Group: G.
Order: 1345036492800 = 2^13 * 3^8 * 5^2 * 7 * 11 * 13.
Representation is due to Richard Parker.
> print G;
MatrixGroup(12, GF(2^2))
Generators:
[ w 1 0 0 0 0 0 0 0 0 0 0]
[w^2 1 w w^2 0 0 0 0 0 0 0 0]
[w^2 1 0 0 w^2 w^2 0 0 0 0 0 0]
[w^2 w w 0 1 0 w^2 0 0 0 0 0]
[ 0 w 0 0 w w^2 w^2 0 0 0 0 0]
[ 1 w^2 0 0 w^2 w^2 0 1 w^2 0 0 0]
[w^2 1 1 0 w^2 1 w 1 w 1 0 0]
[w^2 w w^2 1 0 w 1 w^2 w^2 w^2 0 0]
[w^2 w^2 1 w^2 0 1 1 0 1 w w 0]
[ 0 0 1 w^2 1 0 w w^2 w^2 w w 0]
[w^2 0 w^2 1 w 0 0 w 0 1 0 0]
[ w 0 w^2 0 0 1 w 0 0 0 w^2 w^2]
[ 1 0 1 0 0 0 0 0 0 0 0 0]
[w^2 1 0 w^2 w^2 0 0 0 0 0 0 0]
[w^2 1 1 0 0 0 0 0 0 0 0 0]
[w^2 1 w w w 0 w^2 w^2 0 0 0 0]
[w^2 0 w w w 0 w^2 w^2 0 0 0 0]
[ w w w^2 1 1 w^2 1 1 1 w 0 0]
[ 1 0 w 1 w^2 0 w^2 w^2 1 w^2 w 0]
[w^2 w 0 w 0 w^2 1 0 0 1 0 1]
[w^2 1 w^2 w w 1 w^2 w w 0 1 0]
[w^2 w^2 w 1 1 w^2 w^2 w 0 1 w 1]
[w^2 1 w w^2 0 w^2 w w 1 w 0 w^2]
[ 1 w w w^2 0 w w^2 0 0 w^2 w^2 w^2]
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