metabelian, supersoluble, monomial
Aliases: D12:23D6, Dic6:22D6, C62.138C23, (C2xC12):4D6, C3:D4:6D6, C4oD12:7S3, (C4xS3):11D6, Dic3:D6:8C2, (C6xC12):6C22, D6:D6:15C2, (S3xC12):1C22, D6.4D6:8C2, D6.D6:4C2, (S3xC6).5C23, (C3xC6).13C24, C6.13(S3xC23), D12:S3:13C2, D6.6(C22xS3), (C3xD12):30C22, (S3xDic3):6C22, Dic3.D6:15C2, D6:S3:13C22, C3:D12:14C22, C12.151(C22xS3), (C3xC12).118C23, (C3xDic6):29C22, C32:2Q8:12C22, C3:Dic3.39C23, (C3xDic3).8C23, Dic3.5(C22xS3), C6.D6.9C22, (C4xS32):3C2, (C2xC4):10S32, C4.98(C2xS32), C3:2(S3xC4oD4), C32:6(C2xC4oD4), C3:S3:1(C4oD4), C2.15(C22xS32), C22.12(C2xS32), (C3xC4oD12):12C2, (C4xC3:S3):13C22, (C2xS32).10C22, (C3xC3:D4):7C22, (C2xC3:S3).45C23, (C2xC6).14(C22xS3), (C2xC3:Dic3):22C22, (C22xC3:S3).104C22, (C2xC4xC3:S3):6C2, SmallGroup(288,954)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D12:23D6
G = < a,b,c,d | a12=b2=c6=d2=1, bab=a-1, ac=ca, dad=a5, cbc-1=a6b, dbd=a10b, dcd=c-1 >
Subgroups: 1298 in 355 conjugacy classes, 110 normal (26 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, C22, S3, C6, C6, C2xC4, C2xC4, D4, Q8, C23, C32, Dic3, Dic3, C12, C12, D6, D6, C2xC6, C2xC6, C22xC4, C2xD4, C2xQ8, C4oD4, C3xS3, C3:S3, C3:S3, C3xC6, C3xC6, Dic6, Dic6, C4xS3, C4xS3, D12, D12, C2xDic3, C3:D4, C3:D4, C2xC12, C2xC12, C3xD4, C3xQ8, C22xS3, C2xC4oD4, C3xDic3, C3:Dic3, C3xC12, S32, S3xC6, C2xC3:S3, C2xC3:S3, C62, S3xC2xC4, C4oD12, C4oD12, S3xD4, D4:2S3, S3xQ8, Q8:3S3, C3xC4oD4, S3xDic3, C6.D6, D6:S3, C3:D12, C32:2Q8, C3xDic6, S3xC12, C3xD12, C3xC3:D4, C4xC3:S3, C2xC3:Dic3, C6xC12, C2xS32, C22xC3:S3, S3xC4oD4, D12:S3, Dic3.D6, D6.D6, C4xS32, D6:D6, D6.4D6, Dic3:D6, C3xC4oD12, C2xC4xC3:S3, D12:23D6
Quotients: C1, C2, C22, S3, C23, D6, C4oD4, C24, C22xS3, C2xC4oD4, S32, S3xC23, C2xS32, S3xC4oD4, C22xS32, D12:23D6
►On 24 points - transitive group 24T610
(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 24)(2 23)(3 22)(4 21)(5 20)(6 19)(7 18)(8 17)(9 16)(10 15)(11 14)(12 13)
(1 9 5)(2 10 6)(3 11 7)(4 12 8)(13 23 21 19 17 15)(14 24 22 20 18 16)
(1 5)(2 10)(4 8)(7 11)(13 15)(14 20)(16 18)(17 23)(19 21)(22 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), (1,24)(2,23)(3,22)(4,21)(5,20)(6,19)(7,18)(8,17)(9,16)(10,15)(11,14)(12,13), (1,9,5)(2,10,6)(3,11,7)(4,12,8)(13,23,21,19,17,15)(14,24,22,20,18,16), (1,5)(2,10)(4,8)(7,11)(13,15)(14,20)(16,18)(17,23)(19,21)(22,24)>;
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), (1,24)(2,23)(3,22)(4,21)(5,20)(6,19)(7,18)(8,17)(9,16)(10,15)(11,14)(12,13), (1,9,5)(2,10,6)(3,11,7)(4,12,8)(13,23,21,19,17,15)(14,24,22,20,18,16), (1,5)(2,10)(4,8)(7,11)(13,15)(14,20)(16,18)(17,23)(19,21)(22,24) );
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)], [(1,24),(2,23),(3,22),(4,21),(5,20),(6,19),(7,18),(8,17),(9,16),(10,15),(11,14),(12,13)], [(1,9,5),(2,10,6),(3,11,7),(4,12,8),(13,23,21,19,17,15),(14,24,22,20,18,16)], [(1,5),(2,10),(4,8),(7,11),(13,15),(14,20),(16,18),(17,23),(19,21),(22,24)]])
G:=TransitiveGroup(24,610);
48 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 3A | 3B | 3C | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 6A | 6B | 6C | ··· | 6G | 6H | 6I | 6J | 6K | 12A | 12B | 12C | 12D | 12E | ··· | 12J | 12K | 12L | 12M | 12N |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 12 | 12 | 12 | 12 |
| size | 1 | 1 | 2 | 6 | 6 | 6 | 6 | 9 | 9 | 18 | 2 | 2 | 4 | 1 | 1 | 2 | 6 | 6 | 6 | 6 | 9 | 9 | 18 | 2 | 2 | 4 | ··· | 4 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 12 | 12 | 12 | 12 |
48 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D6 | D6 | D6 | D6 | D6 | C4oD4 | S32 | C2xS32 | C2xS32 | S3xC4oD4 | D12:23D6 |
| kernel | D12:23D6 | D12:S3 | Dic3.D6 | D6.D6 | C4xS32 | D6:D6 | D6.4D6 | Dic3:D6 | C3xC4oD12 | C2xC4xC3:S3 | C4oD12 | Dic6 | C4xS3 | D12 | C3:D4 | C2xC12 | C3:S3 | C2xC4 | C4 | C22 | C3 | C1 |
| # reps | 1 | 2 | 1 | 2 | 2 | 1 | 2 | 2 | 2 | 1 | 2 | 2 | 4 | 2 | 4 | 2 | 4 | 1 | 2 | 1 | 4 | 4 |
Matrix representation of D12:23D6 ►in GL4(F5) generated by
| 4 | 4 | 0 | 3 |
| 1 | 0 | 4 | 2 |
| 4 | 4 | 1 | 1 |
| 2 | 2 | 0 | 0 |
| 3 | 4 | 0 | 1 |
| 0 | 2 | 2 | 4 |
| 1 | 1 | 4 | 4 |
| 2 | 0 | 2 | 1 |
| 3 | 4 | 2 | 2 |
| 4 | 0 | 4 | 3 |
| 0 | 3 | 4 | 2 |
| 0 | 1 | 3 | 3 |
| 1 | 3 | 1 | 3 |
| 2 | 1 | 3 | 0 |
| 2 | 3 | 0 | 3 |
| 4 | 2 | 0 | 3 |
G:=sub<GL(4,GF(5))| [4,1,4,2,4,0,4,2,0,4,1,0,3,2,1,0],[3,0,1,2,4,2,1,0,0,2,4,2,1,4,4,1],[3,4,0,0,4,0,3,1,2,4,4,3,2,3,2,3],[1,2,2,4,3,1,3,2,1,3,0,0,3,0,3,3] >;
D12:23D6 in GAP, Magma, Sage, TeX
D_{12}\rtimes_{23}D_6 % in TeX
G:=Group("D12:23D6"); // GroupNames label
G:=SmallGroup(288,954);
// by ID
G=gap.SmallGroup(288,954);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,100,675,346,1356,9414]);
// Polycyclic
G:=Group<a,b,c,d|a^12=b^2=c^6=d^2=1,b*a*b=a^-1,a*c=c*a,d*a*d=a^5,c*b*c^-1=a^6*b,d*b*d=a^10*b,d*c*d=c^-1>;
// generators/relations