direct product, metabelian, supersoluble, monomial
Aliases: C3xD4:S3, C32:6D8, D12:2C6, C12.33D6, C3:C8:1C6, D4:(C3xS3), C3:2(C3xD8), (C3xD4):1C6, (C3xD4):4S3, C4.1(S3xC6), C6.7(C3xD4), (C3xD12):3C2, C12.1(C2xC6), (C3xC6).28D4, (D4xC32):1C2, C6.29(C3:D4), (C3xC12).8C22, (C3xC3:C8):4C2, C2.4(C3xC3:D4), SmallGroup(144,80)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3xD4:S3
G = < a,b,c,d,e | a3=b4=c2=d3=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=ebe=b-1, bd=db, cd=dc, ece=bc, ede=d-1 >
Quotients: C1, C2, C3, C22, S3, C6, D4, D6, C2xC6, D8, C3xS3, C3:D4, C3xD4, S3xC6, D4:S3, C3xD8, C3xC3:D4, C3xD4:S3
Permutation representations of C3xD4:S3
►On 24 points - transitive group 24T246
(1 19 14)(2 20 15)(3 17 16)(4 18 13)(5 10 21)(6 11 22)(7 12 23)(8 9 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 4)(2 3)(6 8)(9 11)(13 14)(15 16)(17 20)(18 19)(22 24)
(1 14 19)(2 15 20)(3 16 17)(4 13 18)(5 10 21)(6 11 22)(7 12 23)(8 9 24)
(1 22)(2 21)(3 24)(4 23)(5 20)(6 19)(7 18)(8 17)(9 16)(10 15)(11 14)(12 13)
G:=sub<Sym(24)| (1,19,14)(2,20,15)(3,17,16)(4,18,13)(5,10,21)(6,11,22)(7,12,23)(8,9,24), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,4)(2,3)(6,8)(9,11)(13,14)(15,16)(17,20)(18,19)(22,24), (1,14,19)(2,15,20)(3,16,17)(4,13,18)(5,10,21)(6,11,22)(7,12,23)(8,9,24), (1,22)(2,21)(3,24)(4,23)(5,20)(6,19)(7,18)(8,17)(9,16)(10,15)(11,14)(12,13)>;
G:=Group( (1,19,14)(2,20,15)(3,17,16)(4,18,13)(5,10,21)(6,11,22)(7,12,23)(8,9,24), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,4)(2,3)(6,8)(9,11)(13,14)(15,16)(17,20)(18,19)(22,24), (1,14,19)(2,15,20)(3,16,17)(4,13,18)(5,10,21)(6,11,22)(7,12,23)(8,9,24), (1,22)(2,21)(3,24)(4,23)(5,20)(6,19)(7,18)(8,17)(9,16)(10,15)(11,14)(12,13) );
G=PermutationGroup([[(1,19,14),(2,20,15),(3,17,16),(4,18,13),(5,10,21),(6,11,22),(7,12,23),(8,9,24)], [(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16),(17,18,19,20),(21,22,23,24)], [(1,4),(2,3),(6,8),(9,11),(13,14),(15,16),(17,20),(18,19),(22,24)], [(1,14,19),(2,15,20),(3,16,17),(4,13,18),(5,10,21),(6,11,22),(7,12,23),(8,9,24)], [(1,22),(2,21),(3,24),(4,23),(5,20),(6,19),(7,18),(8,17),(9,16),(10,15),(11,14),(12,13)]])
G:=TransitiveGroup(24,246);
C3xD4:S3 is a maximal subgroup of
Dic6:3D6 D12:D6 D12.D6 D12:9D6 D12.22D6 D12.8D6 D12:5D6 C3xS3xD8 He3:6D8 D36:C6 He3:7D8
C3xD4:S3 is a maximal quotient of
He3:6D8 D36:C6
36 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 3D | 3E | 4 | 6A | 6B | 6C | 6D | 6E | 6F | ··· | 6M | 6N | 6O | 8A | 8B | 12A | 12B | 12C | 12D | 12E | 24A | 24B | 24C | 24D |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | ··· | 6 | 6 | 6 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | 24 | 24 | 24 | 24 |
| size | 1 | 1 | 4 | 12 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 2 | 2 | 2 | 4 | ··· | 4 | 12 | 12 | 6 | 6 | 2 | 2 | 4 | 4 | 4 | 6 | 6 | 6 | 6 |
36 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | |||||||||||
| image | C1 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | S3 | D4 | D6 | D8 | C3xS3 | C3:D4 | C3xD4 | S3xC6 | C3xD8 | C3xC3:D4 | D4:S3 | C3xD4:S3 |
| kernel | C3xD4:S3 | C3xC3:C8 | C3xD12 | D4xC32 | D4:S3 | C3:C8 | D12 | C3xD4 | C3xD4 | C3xC6 | C12 | C32 | D4 | C6 | C6 | C4 | C3 | C2 | C3 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 1 | 2 |
Matrix representation of C3xD4:S3 ►in GL4(F7) generated by
| 4 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 4 |
| 3 | 6 | 1 | 3 |
| 5 | 1 | 3 | 3 |
| 3 | 3 | 3 | 1 |
| 2 | 5 | 6 | 0 |
| 0 | 1 | 1 | 0 |
| 1 | 0 | 1 | 0 |
| 0 | 0 | 6 | 0 |
| 0 | 0 | 0 | 1 |
| 3 | 6 | 3 | 2 |
| 6 | 3 | 4 | 2 |
| 0 | 0 | 2 | 0 |
| 0 | 0 | 0 | 4 |
| 2 | 0 | 1 | 6 |
| 2 | 2 | 4 | 1 |
| 2 | 5 | 6 | 6 |
| 5 | 5 | 1 | 4 |
G:=sub<GL(4,GF(7))| [4,0,0,0,0,4,0,0,0,0,4,0,0,0,0,4],[3,5,3,2,6,1,3,5,1,3,3,6,3,3,1,0],[0,1,0,0,1,0,0,0,1,1,6,0,0,0,0,1],[3,6,0,0,6,3,0,0,3,4,2,0,2,2,0,4],[2,2,2,5,0,2,5,5,1,4,6,1,6,1,6,4] >;
C3xD4:S3 in GAP, Magma, Sage, TeX
C_3\times D_4\rtimes S_3
% in TeX
G:=Group("C3xD4:S3"); // GroupNames label
G:=SmallGroup(144,80);
// by ID
G=gap.SmallGroup(144,80);
# by ID
G:=PCGroup([6,-2,-2,-3,-2,-2,-3,169,867,441,69,3461]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^4=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=e*b*e=b^-1,b*d=d*b,c*d=d*c,e*c*e=b*c,e*d*e=d^-1>;
// generators/relations
Export