Order = 1451520 =
Mult = 2.
Out = 1.

Porting notes

Porting incomplete.

Standard generators

Standard generators of S6(2) are a, b where a is in class 2A, b has order 7 and ab has order 9.

Standard generators of 2.S6(2) are preimages A, B where B has order 7 and AB has order 9.

Black box algorithms

Checking generators (semi-presentations)

Group Semi-presentation File
S6(2) 〈〈 a, b | some conditions 〉〉 Download


Group Presentation Link
S6(2) a, b | a2 = b7 = (ab)9 = (ab2)12 = [a, b]3 = [a, b2]2 = 1 〉 Details
2.S6(2) A, B | A2 = [A2, B] = B7 = (AB)9 = (AB2)12 = [A, BABAB]2A−2 = [A, B]3B−2 = [A, B2]2B−2 = 1 〉 Details

(S6(2)) A shorter, and more coset enumeration friendly, presentation may be obtained by replacing (ab2)12 = 1 with [a, babab]2 = 1.


Representations of S6(2)

Representations of 2.S6(2)

Maximal subgroups

Maximal subgroups of S6(2)

Subgroup Order Index Programs/reps
U4(2):2 Program: Generators
A8:2 =S8 Program: Generators
25:A6:2 =25:S6 Program: Generators
U3(3):2 Program: Generators
26:L3(2) Program: Generators
(21+4 × 22):(S3 × S3) Program: Generators
S3 × A6:2 =S3 × S6 Program: Generators
L2(8):3 Program: Generators

Conjugacy classes

Conjugacy classes of S6(2)

Conjugacy class Centraliser order Power up Class rep(s)
1A1 451 520 abababbababbbbabababbababbbbabababbababbbbabababbababbbb
2A23 040 (abbabbabbb)3
2B4 608 ababbbababbbababbbababbb
2C1 536 abababbababbbbabababbababbbb
2D384 (ababbba(b)−1)3
3A2 160 abbabbabbbabbabbabbb
3B648 abbabbabbabb
3C108 abbabbabba(b)−1abbabbabba(b)−1
4A384 ababbbbababbbb
4B192 (ababbbbb)3
4C192 (ababa(b)−1)3
4D128 ababbbababbb
4E32 abababbababbbb
5A30 abababbbbba(b)−1abababbbbba(b)−1
6A144 abbabbabbb
6B144 ababababbbb
6C72 abbabb
6D48 ababa(b)−1ababa(b)−1
6E36 abababba(b)−1
6F36 abbabbabba(b)−1
6G12 ababbba(b)−1
7A7 aab
8A16 ababbbb
8B16 ababbb
9A9 ab
10A10 abababbbbba(b)−1
12A24 ababbbbb
12B24 ababa(b)−1
12C12 abb
15A15 abbb

Download words for class representatives.