metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D12:21D4, C6.1172+ 1+4, C4:C4:10D6, (C2xQ8):19D6, C22:Q8:7S3, C3:7(D4:5D4), D6.20(C2xD4), C4.111(S3xD4), C12:D4:25C2, D6:D4:16C2, (C6xQ8):7C22, D6:C4:20C22, C12.234(C2xD4), C22:C4.57D6, Dic3:5D4:25C2, C6.76(C22xD4), D6.D4:17C2, C2.34(D4oD12), (C2xD12):25C22, (C22xD12):16C2, (C2xC6).174C24, (C2xC12).54C23, (C22xC4).252D6, C12.23D4:12C2, Dic3:C4:53C22, C22:3(Q8:3S3), (C4xDic3):28C22, (S3xC23).52C22, (C22xC6).202C23, C23.199(C22xS3), C22.195(S3xC23), (C22xS3).196C23, (C22xC12).254C22, (C2xDic3).233C23, C6.D4.115C22, C2.49(C2xS3xD4), (C2xC6):7(C4oD4), (C4xC3:D4):22C2, (S3xC22:C4):8C2, (S3xC2xC4):18C22, (C3xC4:C4):19C22, (C2xQ8:3S3):7C2, C6.114(C2xC4oD4), (C3xC22:Q8):10C2, (C2xC4).47(C22xS3), C2.17(C2xQ8:3S3), (C2xC3:D4).122C22, (C3xC22:C4).29C22, SmallGroup(192,1189)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D12:21D4
G = < a,b,c,d | a12=b2=c4=d2=1, bab=a-1, cac-1=dad=a5, cbc-1=dbd=a10b, dcd=c-1 >
Subgroups: 1040 in 334 conjugacy classes, 107 normal (43 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C22, S3, C6, C6, C2xC4, C2xC4, C2xC4, D4, Q8, C23, C23, Dic3, C12, C12, D6, D6, C2xC6, C2xC6, C2xC6, C42, C22:C4, C22:C4, C4:C4, C4:C4, C4:C4, C22xC4, C22xC4, C2xD4, C2xQ8, C4oD4, C24, C4xS3, D12, D12, C2xDic3, C3:D4, C2xC12, C2xC12, C2xC12, C3xQ8, C22xS3, C22xS3, C22xS3, C22xC6, C2xC22:C4, C4xD4, C22wrC2, C4:D4, C22:Q8, C22.D4, C4.4D4, C22xD4, C2xC4oD4, C4xDic3, Dic3:C4, D6:C4, D6:C4, C6.D4, C3xC22:C4, C3xC4:C4, C3xC4:C4, S3xC2xC4, S3xC2xC4, C2xD12, C2xD12, C2xD12, Q8:3S3, C2xC3:D4, C22xC12, C6xQ8, S3xC23, D4:5D4, S3xC22:C4, D6:D4, Dic3:5D4, D6.D4, C12:D4, C12:D4, C4xC3:D4, C12.23D4, C3xC22:Q8, C22xD12, C2xQ8:3S3, D12:21D4
Quotients: C1, C2, C22, S3, D4, C23, D6, C2xD4, C4oD4, C24, C22xS3, C22xD4, C2xC4oD4, 2+ 1+4, S3xD4, Q8:3S3, S3xC23, D4:5D4, C2xS3xD4, C2xQ8:3S3, D4oD12, D12:21D4
►On 48 points
(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 3)(4 12)(5 11)(6 10)(7 9)(13 23)(14 22)(15 21)(16 20)(17 19)(25 31)(26 30)(27 29)(32 36)(33 35)(37 45)(38 44)(39 43)(40 42)(46 48)
(1 36 40 14)(2 29 41 19)(3 34 42 24)(4 27 43 17)(5 32 44 22)(6 25 45 15)(7 30 46 20)(8 35 47 13)(9 28 48 18)(10 33 37 23)(11 26 38 16)(12 31 39 21)
(1 20)(2 13)(3 18)(4 23)(5 16)(6 21)(7 14)(8 19)(9 24)(10 17)(11 22)(12 15)(25 39)(26 44)(27 37)(28 42)(29 47)(30 40)(31 45)(32 38)(33 43)(34 48)(35 41)(36 46)
G:=sub<Sym(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), (1,3)(4,12)(5,11)(6,10)(7,9)(13,23)(14,22)(15,21)(16,20)(17,19)(25,31)(26,30)(27,29)(32,36)(33,35)(37,45)(38,44)(39,43)(40,42)(46,48), (1,36,40,14)(2,29,41,19)(3,34,42,24)(4,27,43,17)(5,32,44,22)(6,25,45,15)(7,30,46,20)(8,35,47,13)(9,28,48,18)(10,33,37,23)(11,26,38,16)(12,31,39,21), (1,20)(2,13)(3,18)(4,23)(5,16)(6,21)(7,14)(8,19)(9,24)(10,17)(11,22)(12,15)(25,39)(26,44)(27,37)(28,42)(29,47)(30,40)(31,45)(32,38)(33,43)(34,48)(35,41)(36,46)>;
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)(25,26,27,28,29,30,31,32,33,34,35,36)(37,38,39,40,41,42,43,44,45,46,47,48), (1,3)(4,12)(5,11)(6,10)(7,9)(13,23)(14,22)(15,21)(16,20)(17,19)(25,31)(26,30)(27,29)(32,36)(33,35)(37,45)(38,44)(39,43)(40,42)(46,48), (1,36,40,14)(2,29,41,19)(3,34,42,24)(4,27,43,17)(5,32,44,22)(6,25,45,15)(7,30,46,20)(8,35,47,13)(9,28,48,18)(10,33,37,23)(11,26,38,16)(12,31,39,21), (1,20)(2,13)(3,18)(4,23)(5,16)(6,21)(7,14)(8,19)(9,24)(10,17)(11,22)(12,15)(25,39)(26,44)(27,37)(28,42)(29,47)(30,40)(31,45)(32,38)(33,43)(34,48)(35,41)(36,46) );
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),(25,26,27,28,29,30,31,32,33,34,35,36),(37,38,39,40,41,42,43,44,45,46,47,48)], [(1,3),(4,12),(5,11),(6,10),(7,9),(13,23),(14,22),(15,21),(16,20),(17,19),(25,31),(26,30),(27,29),(32,36),(33,35),(37,45),(38,44),(39,43),(40,42),(46,48)], [(1,36,40,14),(2,29,41,19),(3,34,42,24),(4,27,43,17),(5,32,44,22),(6,25,45,15),(7,30,46,20),(8,35,47,13),(9,28,48,18),(10,33,37,23),(11,26,38,16),(12,31,39,21)], [(1,20),(2,13),(3,18),(4,23),(5,16),(6,21),(7,14),(8,19),(9,24),(10,17),(11,22),(12,15),(25,39),(26,44),(27,37),(28,42),(29,47),(30,40),(31,45),(32,38),(33,43),(34,48),(35,41),(36,46)]])
39 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 2K | 2L | 3 | 4A | 4B | 4C | ··· | 4G | 4H | 4I | 4J | 4K | 4L | 6A | 6B | 6C | 6D | 6E | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 6 | 6 | 6 | 6 | 12 | 12 | 12 | 2 | 2 | 2 | 4 | ··· | 4 | 6 | 6 | 6 | 6 | 12 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 |
39 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D6 | D6 | D6 | D6 | C4oD4 | 2+ 1+4 | S3xD4 | Q8:3S3 | D4oD12 |
| kernel | D12:21D4 | S3xC22:C4 | D6:D4 | Dic3:5D4 | D6.D4 | C12:D4 | C4xC3:D4 | C12.23D4 | C3xC22:Q8 | C22xD12 | C2xQ8:3S3 | C22:Q8 | D12 | C22:C4 | C4:C4 | C22xC4 | C2xQ8 | C2xC6 | C6 | C4 | C22 | C2 |
| # reps | 1 | 2 | 2 | 1 | 2 | 3 | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 2 | 3 | 1 | 1 | 4 | 1 | 2 | 2 | 2 |
Matrix representation of D12:21D4 ►in GL6(F13)
| 0 | 1 | 0 | 0 | 0 | 0 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 12 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 8 | 0 | 0 | 0 | 0 |
| 5 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 5 | 0 | 0 | 0 | 0 |
| 8 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
G:=sub<GL(6,GF(13))| [0,12,0,0,0,0,1,0,0,0,0,0,0,0,1,12,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[12,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,12,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,5,0,0,0,0,8,0,0,0,0,0,0,0,12,1,0,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,1,0],[0,8,0,0,0,0,5,0,0,0,0,0,0,0,12,1,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;
D12:21D4 in GAP, Magma, Sage, TeX
D_{12}\rtimes_{21}D_4 % in TeX
G:=Group("D12:21D4"); // GroupNames label
G:=SmallGroup(192,1189);
// by ID
G=gap.SmallGroup(192,1189);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,219,184,1571,297,192,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^12=b^2=c^4=d^2=1,b*a*b=a^-1,c*a*c^-1=d*a*d=a^5,c*b*c^-1=d*b*d=a^10*b,d*c*d=c^-1>;
// generators/relations