ATLAS: Suzuki group Suz

Order = 448345497600 = 2¹³.3⁷.5².7.11.13.
Mult = 6.
Out = 2.

The following information is available for Suz:

Standard generators.
Automorphisms.
Black box algorithms.
Representations.
Maximal subgroups.
Conjugacy classes.

Standard generators

Standard generators of the Suzuki group Suz are a and b where a is in class 2B, b is in class 3B, ab has order 13 and ababb has order 15.
Standard generators of 2.Suz are preimages A and B where B has order 3 and AB has order 13.
Standard generators of 3.Suz are preimages A and B where A has order 2 and AB has order 13.
Standard generators of 6.Suz are preimages A and B where A has order 4, B has order 3 and AB has order 13.

Standard generators of the automorphism group Suz:2 are c and d where c is in class 2C, d is in class 3B and cd has order 28.
Standard generators of 2.Suz:2 are preimages C and D where D has order 3.
Standard generators of 3.Suz:2 are preimages C and D where D is in class +3B (equivalently, CDCDD has order 7).
Standard generators of 6.Suz:2 are preimages C and D where D has order 3 and CDCDD has order 7 or 14. (The order of CDCDD is determined by the sixfold cover.)

Automorphisms

The outer automorphism of Suz may be realised by mapping a, b to a^abab, b^abbabb.
If c' is the 15th power of this automorphism, and d' = b, then (c', d') is conjugate to (c, d).
Another outer automorphism [with shorter words] maps a, b to a, b^ababbab.
If u is this second automorphism, then u is class 8H and ((babu)⁷, b^aba) is conjugate to (c, d).

Black box algorithms

Finding generators

To find standard generators for Suz:

Find any element of order 14. Its 7th power is a 2B-element, x say.
[The probability of success at each attempt is 1 in 28.]
Find any element of order 9 or 18. This powers up to a 3B-element, y say.
[The probability of success at each attempt is 4 in 27 (about 1 in 7).]
Find a conjugate a of x and a conjugate b of y such that ab has order 13 and ababb has order 15.
[The probability of success at each attempt is 18 in 715 (about 1 in 40).]

This algorithm is available in computer readable format: finder for Suz.

To find standard generators for Suz.2:

Find any element of order 30. It powers up to a 2C-element, x say.
[The probability of success at each attempt is 1 in 30.]
Find any element of order 9 or 18. This powers up to a 3B-element, y say.
[The probability of success at each attempt is 2 in 27 (about 1 in 14).]
Find a conjugate c of x and a conjugate d of y whose product has order 28.
[The probability of success at each attempt is 27 in 143 (about 1 in 5).]

This algorithm is available in computer readable format: finder for Suz.2.

Checking generators

To check that elements x and y of Suz are standard generators:

Check o(x) = 2
Check o(y) = 3
Check o(xy) = 13
Check o(xyxyy) = 15
Check o(xyxyxyy) = 12

This algorithm is available in computer readable format: checker for Suz.

To check that elements x and y of Suz.2 are standard generators:

Check o(x) = 2
Check o(y) = 3
Check o(xy) = 28
Check o(xyxyxyyxyy) = 7

This algorithm is available in computer readable format: checker for Suz.2

Representations

Representations are available for the following decorations of Suz:

Suz.
Suz:2.
2.Suz.
2.Suz:2.
3.Suz.
3.Suz:2.
6.Suz.
6.Suz:2.

The representations of Suz available are:

Permutations on 1782 points: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
Permutations on 22880 points: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
Permutations on 32760 points: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
Permutations on 135135 points: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
Permutations on 232960 points: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP). - courtesy of Beth Holmes.
Permutations on 370656 points: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP). - courtesy of Beth Holmes.
Permutations on 405405 points: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
Dimension 110 over GF(4): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
Dimension 142 over GF(2): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
Dimension 572 over GF(4): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
Dimension 572 over GF(4): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
Dimension 638 over GF(2): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
Dimension 64 over GF(3): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
Dimension 78 over GF(3): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
Dimension 286 over GF(3): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
Dimension 429 over GF(3): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
Dimension 649 over GF(3): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
Dimension 143 over GF(5): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
Dimension 363 over GF(5): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
Dimension 780 over GF(5): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
Dimension 143 over GF(7): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
Dimension 364 over GF(7): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
Dimension 780 over GF(7): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
Dimension 143 over GF(11): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
Dimension 364 over GF(11): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
Dimension 779 over GF(11): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
Dimension 143 over GF(13): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
Dimension 364 over GF(13): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
Dimension 780 over GF(13): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).

The representations of 2.Suz available are:

Permutations on 65520 points: A and B (Meataxe), A and B (Meataxe binary), A and B (GAP).
Dimension 12 over GF(3): A and B (Meataxe), A and B (Meataxe binary), A and B (GAP).
Dimension 208 over GF(3): A and B (Meataxe), A and B (Meataxe binary), A and B (GAP).
Dimension 352 over GF(3): A and B (Meataxe), A and B (Meataxe binary), A and B (GAP).

The representations of 3.Suz available are:

Permutations on 5346 points: A and B (Meataxe), A and B (Meataxe binary), A and B (GAP).
Permutations on 98280 points: A and B (Meataxe), A and B (Meataxe binary), A and B (GAP).
Permutations on 405405 points: A and B (Meataxe), A and B (Meataxe binary), A and B (GAP).
Dimension 12 over GF(4): A and B (Meataxe), A and B (Meataxe binary), A and B (GAP).
Dimension 66 over GF(4): A and B (Meataxe), A and B (Meataxe binary), A and B (GAP).
Dimension 429 over GF(4): A and B (Meataxe), A and B (Meataxe binary), A and B (GAP).
Dimension 825 over GF(4): A and B (Meataxe), A and B (Meataxe binary), A and B (GAP).
Dimension 825 over GF(4): A and B (Meataxe), A and B (Meataxe binary), A and B (GAP).
Dimension 66 over GF(25): A and B (Meataxe), A and B (Meataxe binary), A and B (GAP).

The representations of 6.Suz available are:

Dimension 12 over GF(25): A and B (Meataxe), A and B (Meataxe binary), A and B (GAP).
Dimension 12 over GF(7): A and B (Meataxe), A and B (Meataxe binary), A and B (GAP).
Dimension 12 over GF(13): A and B (Meataxe), A and B (Meataxe binary), A and B (GAP).
Permutations on 196560 points: A and B (Meataxe), A and B (Meataxe binary), A and B (GAP).

The representations of Suz:2 available are:

Permutations on 1782 points: c and d (Meataxe), c and d (Meataxe binary), c and d (GAP).
Dimension 64 over GF(3): c and d (Meataxe), c and d (Meataxe binary), c and d (GAP).
Dimension 143 over GF(5): c and d (Meataxe), c and d (Meataxe binary), c and d (GAP).
Dimension 143 over GF(7): c and d (Meataxe), c and d (Meataxe binary), c and d (GAP).
Dimension 143 over GF(11): c and d (Meataxe), c and d (Meataxe binary), c and d (GAP).

The representations of 2.Suz:2 available are:

Dimension 12 over GF(3): C and D (Meataxe), C and D (Meataxe binary), C and D (GAP).

The representations of 3.Suz:2 available are:

Permutations on 5346 points: C and D (Meataxe), C and D (Meataxe binary), C and D (GAP).
Dimension 24 over GF(2): C and D (Meataxe), C and D (Meataxe binary), C and D (GAP).
Dimension 132 over GF(5): C and D (Meataxe), C and D (Meataxe binary), C and D (GAP).
Dimension 132 over GF(7): C and D (Meataxe), C and D (Meataxe binary), C and D (GAP).
Dimension 132 over GF(11): C and D (Meataxe), C and D (Meataxe binary), C and D (GAP).

The representations of 6.Suz:2 available are:

Dimension 24 over GF(3): C and D (Meataxe), C and D (Meataxe binary), C and D (GAP). - a reducible representation
Dimension 24 over GF(5): C and D (Meataxe), C and D (Meataxe binary), C and D (GAP).
Dimension 24 over GF(11): C and D (Meataxe), C and D (Meataxe binary), C and D (GAP).

Maximal subgroups

The maximal subgroups of Suz are:

G₂(4), with standard generators (ab)^-5(abababb)⁶(ab)⁵, (abb)^-4(ababb)³(abb)⁴.
3.U₄(3):2, with generators a^b, (abb)^-4(abababababbababbabb)²(abb)⁴.
U₅(2), with standard generators (abababbababb)⁴, (abababbab)⁹(abababbabababababbababb)⁶(abababbab)^-9.
2¹⁺⁶.U₄(2), with generators a^b, (abb)^-6(ababb)³(abb)⁶.
3⁵:M₁₁, with generators (mapping to standard generators of M₁₁) (ab)^-6(abababbababb)⁴(ab)⁶, (abb)^-5(abababbababb)⁶(abb)⁵.
J₂:2, with generators (ab)^-4a(ab)⁴, (abb)^-6(ababb)³(abb)⁶.
2⁴⁺⁶:3.A₆, with generators (ab)^-5b(ab)⁴, (abb)^-6(abababbababb)²(abb)⁶.
(A₄ × L₃(4)):2, with generators here.
2²⁺⁸:(A₅ × S₃), with generators here.
M₁₂:2, with standard generators (ab)^-2bab, (abb)^-6b(abb)⁶.
3²⁺⁴:2.(A₄ × 2 × 2).2, with generators here.
(A₆ × A₅).2, with generators here.
(A₆ × 3²:4).2, with generators here.
L₃(3):2, with standard generators here.
L₃(3):2, with standard generators here.
[NB: Old (non-standard) generators for one of these L₃(3):2 groups are (ab)^-6b(ab)⁵, (abb)^-3(abababbabababbababb)²(abb)³.]
L₂(25), with generators a, (ababababb)^-1(ababb)⁵(ababababb).
A₇, with (non-standard) generators (abababbab)^-2a(abababbab)², (ababababb)^-5(abababababbababbabb)²(ababababb)⁵.

The maximal subgroups of Suz:2 are:

Suz, with standard generators (cdd)^-2(cd)¹⁴(cdd)², d.
G₂(4):2 with generators here.
3.U₄(3).2.2 with generators here.
U₅(2):2 with generators here.
2¹⁺⁶.U₄(2).2 with generators here.
3⁵:(M₁₁ × 2) with generators here.
J₂:2 × 2 with generators here.
2⁴⁺⁶:3.S₆ with generators here.
(A₄ × L₃(4):2):2 with generators here.
2²⁺⁸:(S₅ × S₃) with generators here.
M₁₂:2 × 2 with generators here.
3²⁺⁴:2.(S₄ × D₈) with generators here.
(PGL₂(9) × A₅).2 with generators here.
(A₆ × 3²:8).2 with generators here.
L₂(25):2 with generators here.
S₇ with generators here.

Conjugacy classes

A set of generators for the maximal cyclic subgroups can be obtained by running this program on the standard generators. All conjugacy classes can therefore be obtained as suitable powers of these elements. Problems of algebraic conjugacy are not yet dealt with.

Go to main ATLAS (version 2.0) page.

Go to sporadic groups page.

Go to old Suz page - ATLAS version 1.

Anonymous ftp access is also available. See here for details.

Version 2.0 created on 20th December 2000.
Last updated 7.1.05 by SJN.
Information checked to Level 0 on 20.12.00 by RAW.
R.A.Wilson, R.A.Parker and J.N.Bray.