ATLAS: Suzuki group Suz
Order = 448345497600.
Mult = 6.
Out = 2.
The following information is available for Suz:
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 pre-images A
and B where
B has order 3 and
AB has order 13.
Standard generators of 3.Suz are pre-images A
and B where
A has order 2 and
AB has order 13.
Standard generators of 6.Suz are pre-images 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 pre-images C
and D where
D has order 3.
Standard generators of 3.Suz:2 are pre-images C
and D where
D is in class +3B (equivalently,
CDCDD has order 7).
Automorphisms
The outer automorphism of Suz may be realised by mapping
a, b to
(ab)^-2bab, (abb)^-2b(abb)^2.
If c' is the 15th power of this automorphism, and d' =b, then
(c',d') is conjugate to (c,d).
Black box algorithms
To find standard generators for Suz:
- Find any element of order 14. Its 7th power is a 2B-element.
- Find any element of order 9 or 18. This powers up to a 3B-element, y, say.
- Find a conjugate a of x and a conjugate b of y, whose product has order 13,
and whose commutator has order 15.
To find standard generators for Suz.2:
- Find any element of order 30. It powers up to a 2C-element.
- Find any element of order 9 or 18. This powers up to a 3B-element, y, say.
- Find a conjugate a of x and a conjugate b of y, whose product has order 28.
Representations
Representations are available for the following decorations of Suz:
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
b^-1ab,
(abb)^-4(abababababbababbabb)^2(abb)^4.
- U5(2), with standard generators
(abababbababb)^4,
(abababbab)^9(abababbabababababbababb)^6(abababbab)^-9.
- 2^1+6.U4(2), with generators
b^-1ab,
(abb)^-6(ababb)^3(abb)^6.
- 3^5: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.
- 2^4+6:3.A6, with generators
(ab)^-5b(ab)^4, (abb)^-6(abababbababb)^2(abb)^6.
- (A4 x L3(4)):2, with generators
here.
- 2^2+8:(A5 x S3), with generators
here.
- M12:2, with standard generators
here.
- (A6 x A5).2, with generators
here.
- (A6 x 3^2:4).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
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 on
for.mat.bham.ac.uk.
Version 2.0 created on 20th December 2000.
Last updated 19.12.02 by RAW.
Information checked to
Level 0 on 20.12.00 by RAW.
R.A.Wilson, R.A.Parker and J.N.Bray.