ATLAS: Suzuki group Sz(32)

Order = 32537600 = 2¹⁰.5².31.41.
Mult = 1.
Out = 5.

The following information is available for Sz(32):

Standard generators.
Representations.
Maximal subgroups.
Conjugacy classes.

Standard generators

Standard generators of Sz(32) are a and b where a has order 2, b has order 4, ab has order 5, ababb has order 25 and abababbababbabb has order 25.
Standard generators of Sz(32):5 are c and d where c has order 2, d has order 5, cd has order 5, cdcdddd has order 5, cddcddd has order 31 and cdcddcddd has order 10.

Representations

The representations of Sz(32) available are:

Dimension 4 over GF(32) - the natural representation: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
Dimension 124 over GF(5): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
Dimension 124 over GF(41): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
Permutations on 1025 points: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
Permutations on 198400 points: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).

The representations of Sz(32):5 available are:

Dimension 20 over GF(2): c and d (Meataxe), c and d (Meataxe binary), c and d (GAP).
Dimension 124 over GF(5): c and d (Meataxe), c and d (Meataxe binary), c and d (GAP).
Dimension 124 over GF(41): c and d (Meataxe), c and d (Meataxe binary), c and d (GAP).
Dimension 248 over GF(5): c and d (Meataxe), c and d (Meataxe binary), c and d (GAP).
- exhibiting the inclusion in E8(5).
The given basis is a Chevalley basis, and the Lie algebra coproduct is here as a 248 x 248² matrix.
Permutations on 1025 points: c and d (Meataxe), c and d (Meataxe binary), c and d (GAP).

Maximal subgroups

The maximal subgroups of Sz(32) are:

2⁵⁺⁵:31, with generators a, (ababb)^-5b(ababb)^5
41:4, with generators a, (ababb)^-5b(ababb)^5
25:4, with generators ababbabab, (abb)^-5b(abb)^5
D₆₂, with generators bb, ababbabab

The maximal subgroups of Sz(32):5 are:

Sz(32).
2⁵⁺⁵:31:5.
41:20.
25:20.
31:10.

Conjugacy classes

A set of generators for the maximal cyclic subgroups of Sz(32) can be obtained by running this program on the standard generators. All conjugacy classes can therefore be obtained as suitable powers of these elements.

A set of generators for the maximal cyclic subgroups of Sz(32):5 can be obtained by running this program on the standard generators. All conjugacy classes can therefore be obtained as suitable powers of these elements.

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Version 2.0 created on 28th April 2000.
Last updated 18.07.00 by JNB.
Information checked to Level 0 on 28.04.00 by RAW.
R.A.Wilson, R.A.Parker and J.N.Bray.