direct product, metabelian, supersoluble, monomial, A-group
Aliases: S32×C6, C33⋊2C23, C6⋊1(S3×C6), (C3×C6)⋊6D6, (S3×C6)⋊5C6, (C32×C6)⋊1C22, C32⋊7(C22×S3), C32⋊2(C22×C6), (S3×C32)⋊2C22, C3⋊1(S3×C2×C6), (C3×S3)⋊(C2×C6), (S3×C3×C6)⋊7C2, (C6×C3⋊S3)⋊7C2, (C2×C3⋊S3)⋊7C6, C3⋊S3⋊2(C2×C6), (C3×C6)⋊2(C2×C6), (C3×C3⋊S3)⋊2C22, SmallGroup(216,170)
Series: Derived ►Chief ►Lower central ►Upper central
| C32 — S32×C6 |
Generators and relations for S32×C6
G = < a,b,c,d,e | a6=b3=c2=d3=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >
Subgroups: 492 in 162 conjugacy classes, 56 normal (12 characteristic)
C1, C2, C2, C3, C3, C3, C22, S3, S3, C6, C6, C6, C23, C32, C32, C32, D6, D6, C2×C6, C3×S3, C3×S3, C3⋊S3, C3×C6, C3×C6, C3×C6, C22×S3, C22×C6, C33, S32, S3×C6, S3×C6, C2×C3⋊S3, C62, S3×C32, C3×C3⋊S3, C32×C6, C2×S32, S3×C2×C6, C3×S32, S3×C3×C6, C6×C3⋊S3, S32×C6
Quotients: C1, C2, C3, C22, S3, C6, C23, D6, C2×C6, C3×S3, C22×S3, C22×C6, S32, S3×C6, C2×S32, S3×C2×C6, C3×S32, S32×C6
►On 24 points - transitive group 24T547
(1 2 3 4 5 6)(7 8 9 10 11 12)(13 14 15 16 17 18)(19 20 21 22 23 24)
(1 5 3)(2 6 4)(7 9 11)(8 10 12)(13 17 15)(14 18 16)(19 21 23)(20 22 24)
(1 11)(2 12)(3 7)(4 8)(5 9)(6 10)(13 19)(14 20)(15 21)(16 22)(17 23)(18 24)
(1 3 5)(2 4 6)(7 9 11)(8 10 12)(13 17 15)(14 18 16)(19 23 21)(20 24 22)
(1 17)(2 18)(3 13)(4 14)(5 15)(6 16)(7 19)(8 20)(9 21)(10 22)(11 23)(12 24)
G:=sub<Sym(24)| (1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18)(19,20,21,22,23,24), (1,5,3)(2,6,4)(7,9,11)(8,10,12)(13,17,15)(14,18,16)(19,21,23)(20,22,24), (1,11)(2,12)(3,7)(4,8)(5,9)(6,10)(13,19)(14,20)(15,21)(16,22)(17,23)(18,24), (1,3,5)(2,4,6)(7,9,11)(8,10,12)(13,17,15)(14,18,16)(19,23,21)(20,24,22), (1,17)(2,18)(3,13)(4,14)(5,15)(6,16)(7,19)(8,20)(9,21)(10,22)(11,23)(12,24)>;
G:=Group( (1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18)(19,20,21,22,23,24), (1,5,3)(2,6,4)(7,9,11)(8,10,12)(13,17,15)(14,18,16)(19,21,23)(20,22,24), (1,11)(2,12)(3,7)(4,8)(5,9)(6,10)(13,19)(14,20)(15,21)(16,22)(17,23)(18,24), (1,3,5)(2,4,6)(7,9,11)(8,10,12)(13,17,15)(14,18,16)(19,23,21)(20,24,22), (1,17)(2,18)(3,13)(4,14)(5,15)(6,16)(7,19)(8,20)(9,21)(10,22)(11,23)(12,24) );
G=PermutationGroup([[(1,2,3,4,5,6),(7,8,9,10,11,12),(13,14,15,16,17,18),(19,20,21,22,23,24)], [(1,5,3),(2,6,4),(7,9,11),(8,10,12),(13,17,15),(14,18,16),(19,21,23),(20,22,24)], [(1,11),(2,12),(3,7),(4,8),(5,9),(6,10),(13,19),(14,20),(15,21),(16,22),(17,23),(18,24)], [(1,3,5),(2,4,6),(7,9,11),(8,10,12),(13,17,15),(14,18,16),(19,23,21),(20,24,22)], [(1,17),(2,18),(3,13),(4,14),(5,15),(6,16),(7,19),(8,20),(9,21),(10,22),(11,23),(12,24)]])
G:=TransitiveGroup(24,547);
S32×C6 is a maximal subgroup of
S32⋊Dic3 D6⋊4S32 D6⋊S32
54 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3A | 3B | 3C | ··· | 3H | 3I | 3J | 3K | 6A | 6B | 6C | ··· | 6H | 6I | ··· | 6P | 6Q | 6R | 6S | 6T | ··· | 6AE | 6AF | 6AG | 6AH | 6AI |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | ··· | 3 | 3 | 3 | 3 | 6 | 6 | 6 | ··· | 6 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | ··· | 6 | 6 | 6 | 6 | 6 |
| size | 1 | 1 | 3 | 3 | 3 | 3 | 9 | 9 | 1 | 1 | 2 | ··· | 2 | 4 | 4 | 4 | 1 | 1 | 2 | ··· | 2 | 3 | ··· | 3 | 4 | 4 | 4 | 6 | ··· | 6 | 9 | 9 | 9 | 9 |
54 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | |||||||||
| image | C1 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | S3 | D6 | D6 | C3×S3 | S3×C6 | S3×C6 | S32 | C2×S32 | C3×S32 | S32×C6 |
| kernel | S32×C6 | C3×S32 | S3×C3×C6 | C6×C3⋊S3 | C2×S32 | S32 | S3×C6 | C2×C3⋊S3 | S3×C6 | C3×S3 | C3×C6 | D6 | S3 | C6 | C6 | C3 | C2 | C1 |
| # reps | 1 | 4 | 2 | 1 | 2 | 8 | 4 | 2 | 2 | 4 | 2 | 4 | 8 | 4 | 1 | 1 | 2 | 2 |
Matrix representation of S32×C6 ►in GL4(𝔽7) generated by
| 4 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 5 | 0 |
| 0 | 0 | 0 | 5 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 6 | 6 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
| 0 | 1 | 0 | 0 |
| 6 | 6 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 6 | 6 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(7))| [4,0,0,0,0,4,0,0,0,0,5,0,0,0,0,5],[1,0,0,0,0,1,0,0,0,0,0,6,0,0,1,6],[1,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0],[0,6,0,0,1,6,0,0,0,0,1,0,0,0,0,1],[1,6,0,0,0,6,0,0,0,0,1,0,0,0,0,1] >;
S32×C6 in GAP, Magma, Sage, TeX
S_3^2\times C_6
% in TeX
G:=Group("S3^2xC6"); // GroupNames label
G:=SmallGroup(216,170);
// by ID
G=gap.SmallGroup(216,170);
# by ID
G:=PCGroup([6,-2,-2,-2,-3,-3,-3,730,5189]);
// Polycyclic
G:=Group<a,b,c,d,e|a^6=b^3=c^2=d^3=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
// generators/relations