metabelian, supersoluble, monomial
Aliases: C12.13D12, (C2xD12).5S3, (C6xD12).1C2, (C2xC12).71D6, (C3xC12).28D4, C4.Dic3:2S3, C6.32(D6:C4), C62.27(C2xC4), (C22xS3).Dic3, C4.9(D6:S3), C12.58D6:2C2, C3:1(C12.D4), C12.24(C3:D4), C2.3(D6:Dic3), (C6xC12).20C22, C32:2(C4.D4), C4.19(C3:D12), C3:3(C12.46D4), C22.3(S3xDic3), C6.2(C6.D4), (C2xC4).1S32, (S3xC2xC6).1C4, (C2xC6).66(C4xS3), (C2xC6).4(C2xDic3), (C3xC4.Dic3):13C2, (C3xC6).25(C22:C4), SmallGroup(288,206)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C12.D12
G = < a,b,c | a12=1, b12=a6, c2=a9, bab-1=a-1, cac-1=a5, cbc-1=a3b11 >
Subgroups: 370 in 102 conjugacy classes, 34 normal (30 characteristic)
C1, C2, C2, C3, C3, C4, C22, C22, S3, C6, C6, C8, C2xC4, D4, C23, C32, C12, C12, D6, C2xC6, C2xC6, M4(2), C2xD4, C3xS3, C3xC6, C3xC6, C3:C8, C24, D12, C2xC12, C2xC12, C3xD4, C22xS3, C22xC6, C4.D4, C3xC12, S3xC6, C62, C4.Dic3, C4.Dic3, C3xM4(2), C2xD12, C6xD4, C3xC3:C8, C32:4C8, C3xD12, C6xC12, S3xC2xC6, C12.46D4, C12.D4, C3xC4.Dic3, C12.58D6, C6xD12, C12.D12
Quotients: C1, C2, C4, C22, S3, C2xC4, D4, Dic3, D6, C22:C4, C4xS3, D12, C2xDic3, C3:D4, C4.D4, S32, D6:C4, C6.D4, S3xDic3, D6:S3, C3:D12, C12.46D4, C12.D4, D6:Dic3, C12.D12
►On 48 points
(1 11 21 7 17 3 13 23 9 19 5 15)(2 16 6 20 10 24 14 4 18 8 22 12)(25 47 45 43 41 39 37 35 33 31 29 27)(26 28 30 32 34 36 38 40 42 44 46 48)
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24)(25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48)
(1 39 19 45 13 27 7 33)(2 32 8 26 14 44 20 38)(3 37 21 43 15 25 9 31)(4 30 10 48 16 42 22 36)(5 35 23 41 17 47 11 29)(6 28 12 46 18 40 24 34)
G:=sub<Sym(48)| (1,11,21,7,17,3,13,23,9,19,5,15)(2,16,6,20,10,24,14,4,18,8,22,12)(25,47,45,43,41,39,37,35,33,31,29,27)(26,28,30,32,34,36,38,40,42,44,46,48), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48), (1,39,19,45,13,27,7,33)(2,32,8,26,14,44,20,38)(3,37,21,43,15,25,9,31)(4,30,10,48,16,42,22,36)(5,35,23,41,17,47,11,29)(6,28,12,46,18,40,24,34)>;
G:=Group( (1,11,21,7,17,3,13,23,9,19,5,15)(2,16,6,20,10,24,14,4,18,8,22,12)(25,47,45,43,41,39,37,35,33,31,29,27)(26,28,30,32,34,36,38,40,42,44,46,48), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48), (1,39,19,45,13,27,7,33)(2,32,8,26,14,44,20,38)(3,37,21,43,15,25,9,31)(4,30,10,48,16,42,22,36)(5,35,23,41,17,47,11,29)(6,28,12,46,18,40,24,34) );
G=PermutationGroup([[(1,11,21,7,17,3,13,23,9,19,5,15),(2,16,6,20,10,24,14,4,18,8,22,12),(25,47,45,43,41,39,37,35,33,31,29,27),(26,28,30,32,34,36,38,40,42,44,46,48)], [(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24),(25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48)], [(1,39,19,45,13,27,7,33),(2,32,8,26,14,44,20,38),(3,37,21,43,15,25,9,31),(4,30,10,48,16,42,22,36),(5,35,23,41,17,47,11,29),(6,28,12,46,18,40,24,34)]])
39 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3A | 3B | 3C | 4A | 4B | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 6H | 6I | 6J | 6K | 6L | 8A | 8B | 8C | 8D | 12A | 12B | 12C | ··· | 12I | 24A | 24B | 24C | 24D |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | ··· | 12 | 24 | 24 | 24 | 24 |
| size | 1 | 1 | 2 | 12 | 12 | 2 | 2 | 4 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 12 | 12 | 12 | 12 | 12 | 12 | 36 | 36 | 2 | 2 | 4 | ··· | 4 | 12 | 12 | 12 | 12 |
39 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | - | + | + | + | - | + | - | + | |||||
| image | C1 | C2 | C2 | C2 | C4 | S3 | S3 | D4 | D6 | Dic3 | D12 | C3:D4 | C4xS3 | C4.D4 | S32 | D6:S3 | C3:D12 | S3xDic3 | C12.46D4 | C12.D4 | C12.D12 |
| kernel | C12.D12 | C3xC4.Dic3 | C12.58D6 | C6xD12 | S3xC2xC6 | C4.Dic3 | C2xD12 | C3xC12 | C2xC12 | C22xS3 | C12 | C12 | C2xC6 | C32 | C2xC4 | C4 | C4 | C22 | C3 | C3 | C1 |
| # reps | 1 | 1 | 1 | 1 | 4 | 1 | 1 | 2 | 2 | 2 | 2 | 6 | 2 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 4 |
Matrix representation of C12.D12 ►in GL4(F73) generated by
| 24 | 0 | 0 | 0 |
| 0 | 70 | 0 | 0 |
| 0 | 15 | 49 | 0 |
| 15 | 0 | 0 | 3 |
| 0 | 64 | 0 | 0 |
| 24 | 0 | 0 | 0 |
| 1 | 0 | 0 | 8 |
| 0 | 27 | 70 | 0 |
| 1 | 0 | 0 | 57 |
| 0 | 27 | 6 | 0 |
| 0 | 29 | 46 | 0 |
| 20 | 0 | 0 | 72 |
G:=sub<GL(4,GF(73))| [24,0,0,15,0,70,15,0,0,0,49,0,0,0,0,3],[0,24,1,0,64,0,0,27,0,0,0,70,0,0,8,0],[1,0,0,20,0,27,29,0,0,6,46,0,57,0,0,72] >;
C12.D12 in GAP, Magma, Sage, TeX
C_{12}.D_{12} % in TeX
G:=Group("C12.D12"); // GroupNames label
G:=SmallGroup(288,206);
// by ID
G=gap.SmallGroup(288,206);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,141,36,422,100,346,1356,9414]);
// Polycyclic
G:=Group<a,b,c|a^12=1,b^12=a^6,c^2=a^9,b*a*b^-1=a^-1,c*a*c^-1=a^5,c*b*c^-1=a^3*b^11>;
// generators/relations