direct product, metabelian, soluble, monomial, A-group
Aliases: C6xC32:C4, C3:S3:3C12, (C3xC6):2C12, C33:2(C2xC4), (C32xC6):1C4, C32:3(C2xC12), (C3xC3:S3):2C4, (C6xC3:S3).4C2, (C2xC3:S3).3C6, C3:S3.3(C2xC6), (C3xC3:S3).7C22, SmallGroup(216,168)
Series: Derived ►Chief ►Lower central ►Upper central
| C32 — C6xC32:C4 |
Generators and relations for C6xC32:C4
G = < a,b,c,d | a6=b3=c3=d4=1, ab=ba, ac=ca, ad=da, dcd-1=bc=cb, dbd-1=b-1c >
Subgroups: 248 in 60 conjugacy classes, 20 normal (16 characteristic)
C1, C2, C2, C3, C3, C4, C22, S3, C6, C6, C2xC4, C32, C32, C12, D6, C2xC6, C3xS3, C3:S3, C3xC6, C3xC6, C2xC12, C33, C32:C4, S3xC6, C2xC3:S3, C3xC3:S3, C32xC6, C2xC32:C4, C3xC32:C4, C6xC3:S3, C6xC32:C4
Quotients: C1, C2, C3, C4, C22, C6, C2xC4, C12, C2xC6, C2xC12, C32:C4, C2xC32:C4, C3xC32:C4, C6xC32:C4
►On 24 points - transitive group 24T555
(1 2 3 4 5 6)(7 8 9 10 11 12)(13 14 15 16 17 18)(19 20 21 22 23 24)
(7 9 11)(8 10 12)(19 23 21)(20 24 22)
(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 8 16 24)(2 9 17 19)(3 10 18 20)(4 11 13 21)(5 12 14 22)(6 7 15 23)
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), (7,9,11)(8,10,12)(19,23,21)(20,24,22), (1,3,5)(2,4,6)(7,9,11)(8,10,12)(13,17,15)(14,18,16)(19,23,21)(20,24,22), (1,8,16,24)(2,9,17,19)(3,10,18,20)(4,11,13,21)(5,12,14,22)(6,7,15,23)>;
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), (7,9,11)(8,10,12)(19,23,21)(20,24,22), (1,3,5)(2,4,6)(7,9,11)(8,10,12)(13,17,15)(14,18,16)(19,23,21)(20,24,22), (1,8,16,24)(2,9,17,19)(3,10,18,20)(4,11,13,21)(5,12,14,22)(6,7,15,23) );
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)], [(7,9,11),(8,10,12),(19,23,21),(20,24,22)], [(1,3,5),(2,4,6),(7,9,11),(8,10,12),(13,17,15),(14,18,16),(19,23,21),(20,24,22)], [(1,8,16,24),(2,9,17,19),(3,10,18,20),(4,11,13,21),(5,12,14,22),(6,7,15,23)]])
G:=TransitiveGroup(24,555);
C6xC32:C4 is a maximal subgroup of
D6:(C32:C4) C33:(C4:C4) (C3xC6).8D12 (C3xC6).9D12 C6.PSU3(F2) C6.2PSU3(F2)
36 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | ··· | 3H | 4A | 4B | 4C | 4D | 6A | 6B | 6C | ··· | 6H | 6I | 6J | 6K | 6L | 12A | ··· | 12H |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | ··· | 3 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | 12 | ··· | 12 |
| size | 1 | 1 | 9 | 9 | 1 | 1 | 4 | ··· | 4 | 9 | 9 | 9 | 9 | 1 | 1 | 4 | ··· | 4 | 9 | 9 | 9 | 9 | 9 | ··· | 9 |
36 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | |||||||||
| image | C1 | C2 | C2 | C3 | C4 | C4 | C6 | C6 | C12 | C12 | C32:C4 | C2xC32:C4 | C3xC32:C4 | C6xC32:C4 |
| kernel | C6xC32:C4 | C3xC32:C4 | C6xC3:S3 | C2xC32:C4 | C3xC3:S3 | C32xC6 | C32:C4 | C2xC3:S3 | C3:S3 | C3xC6 | C6 | C3 | C2 | C1 |
| # reps | 1 | 2 | 1 | 2 | 2 | 2 | 4 | 2 | 4 | 4 | 2 | 2 | 4 | 4 |
Matrix representation of C6xC32:C4 ►in GL4(F7) generated by
| 3 | 0 | 0 | 0 |
| 0 | 3 | 0 | 0 |
| 0 | 0 | 3 | 0 |
| 0 | 0 | 0 | 3 |
| 0 | 0 | 4 | 6 |
| 2 | 3 | 1 | 0 |
| 6 | 1 | 2 | 2 |
| 4 | 4 | 5 | 0 |
| 4 | 2 | 5 | 0 |
| 2 | 3 | 5 | 2 |
| 2 | 2 | 6 | 2 |
| 2 | 4 | 3 | 2 |
| 5 | 0 | 5 | 3 |
| 6 | 4 | 6 | 4 |
| 5 | 4 | 6 | 6 |
| 1 | 6 | 3 | 6 |
G:=sub<GL(4,GF(7))| [3,0,0,0,0,3,0,0,0,0,3,0,0,0,0,3],[0,2,6,4,0,3,1,4,4,1,2,5,6,0,2,0],[4,2,2,2,2,3,2,4,5,5,6,3,0,2,2,2],[5,6,5,1,0,4,4,6,5,6,6,3,3,4,6,6] >;
C6xC32:C4 in GAP, Magma, Sage, TeX
C_6\times C_3^2\rtimes C_4
% in TeX
G:=Group("C6xC3^2:C4"); // GroupNames label
G:=SmallGroup(216,168);
// by ID
G=gap.SmallGroup(216,168);
# by ID
G:=PCGroup([6,-2,-2,-3,-2,-3,3,72,5044,142,6917,455]);
// Polycyclic
G:=Group<a,b,c,d|a^6=b^3=c^3=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,d*c*d^-1=b*c=c*b,d*b*d^-1=b^-1*c>;
// generators/relations