metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42:10D6, C4:C4:42D6, (C4xD12):7C2, (C2xC12):10D4, (C2xC4):11D12, D6:1(C4oD4), (C4xC12):5C22, C4.70(C2xD12), C12:D4:42C2, D6:D4:29C2, C42:C2:8S3, D6:C4:51C22, C4.D12:47C2, C12.223(C2xD4), (C2xC6).68C24, C22:C4.92D6, C6.12(C22xD4), (C2xD12):52C22, C4:Dic3:55C22, (C22xC4).381D6, C2.14(C22xD12), C22.19(C2xD12), (C2xC12).143C23, C3:1(C22.19C24), (C2xDic6):61C22, C22.97(S3xC23), C23.21D6:32C2, (C22xS3).18C23, C23.166(C22xS3), (C22xC6).138C23, (S3xC23).104C22, (C22xC12).228C22, (C2xDic3).197C23, (C22xDic3).217C22, (S3xC22xC4):2C2, C2.9(S3xC4oD4), (S3xC2xC4):44C22, (C2xC6).49(C2xD4), (C2xC4oD12):17C2, (C3xC4:C4):52C22, C6.133(C2xC4oD4), (C3xC42:C2):10C2, (C2xC4).575(C22xS3), (C2xC3:D4).99C22, (C3xC22:C4).100C22, SmallGroup(192,1083)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42:10D6
G = < a,b,c,d | a4=b4=c6=d2=1, ab=ba, cac-1=ab2, dad=a-1, bc=cb, bd=db, dcd=c-1 >
Subgroups: 904 in 330 conjugacy classes, 115 normal (23 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C22, S3, C6, 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, C22xC4, C22xC4, C2xD4, C2xQ8, C4oD4, C24, Dic6, C4xS3, D12, C2xDic3, C2xDic3, C3:D4, C2xC12, C2xC12, C22xS3, C22xS3, C22xC6, C42:C2, C4xD4, C22wrC2, C4:D4, C22:Q8, C22.D4, C23xC4, C2xC4oD4, C4:Dic3, D6:C4, C4xC12, C3xC22:C4, C3xC4:C4, C2xDic6, S3xC2xC4, S3xC2xC4, C2xD12, C2xD12, C4oD12, C22xDic3, C2xC3:D4, C22xC12, S3xC23, C22.19C24, C4xD12, D6:D4, C23.21D6, C12:D4, C4.D12, C3xC42:C2, S3xC22xC4, C2xC4oD12, C42:10D6
Quotients: C1, C2, C22, S3, D4, C23, D6, C2xD4, C4oD4, C24, D12, C22xS3, C22xD4, C2xC4oD4, C2xD12, S3xC23, C22.19C24, C22xD12, S3xC4oD4, C42:10D6
►On 48 points
(1 26 4 19)(2 23 5 30)(3 28 6 21)(7 32 44 35)(8 40 45 37)(9 34 46 31)(10 42 47 39)(11 36 48 33)(12 38 43 41)(13 22 16 29)(14 27 17 20)(15 24 18 25)
(1 34 13 41)(2 35 14 42)(3 36 15 37)(4 31 16 38)(5 32 17 39)(6 33 18 40)(7 27 47 23)(8 28 48 24)(9 29 43 19)(10 30 44 20)(11 25 45 21)(12 26 46 22)
(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 18)(2 17)(3 16)(4 15)(5 14)(6 13)(7 47)(8 46)(9 45)(10 44)(11 43)(12 48)(19 25)(20 30)(21 29)(22 28)(23 27)(24 26)(31 37)(32 42)(33 41)(34 40)(35 39)(36 38)
G:=sub<Sym(48)| (1,26,4,19)(2,23,5,30)(3,28,6,21)(7,32,44,35)(8,40,45,37)(9,34,46,31)(10,42,47,39)(11,36,48,33)(12,38,43,41)(13,22,16,29)(14,27,17,20)(15,24,18,25), (1,34,13,41)(2,35,14,42)(3,36,15,37)(4,31,16,38)(5,32,17,39)(6,33,18,40)(7,27,47,23)(8,28,48,24)(9,29,43,19)(10,30,44,20)(11,25,45,21)(12,26,46,22), (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), (1,18)(2,17)(3,16)(4,15)(5,14)(6,13)(7,47)(8,46)(9,45)(10,44)(11,43)(12,48)(19,25)(20,30)(21,29)(22,28)(23,27)(24,26)(31,37)(32,42)(33,41)(34,40)(35,39)(36,38)>;
G:=Group( (1,26,4,19)(2,23,5,30)(3,28,6,21)(7,32,44,35)(8,40,45,37)(9,34,46,31)(10,42,47,39)(11,36,48,33)(12,38,43,41)(13,22,16,29)(14,27,17,20)(15,24,18,25), (1,34,13,41)(2,35,14,42)(3,36,15,37)(4,31,16,38)(5,32,17,39)(6,33,18,40)(7,27,47,23)(8,28,48,24)(9,29,43,19)(10,30,44,20)(11,25,45,21)(12,26,46,22), (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), (1,18)(2,17)(3,16)(4,15)(5,14)(6,13)(7,47)(8,46)(9,45)(10,44)(11,43)(12,48)(19,25)(20,30)(21,29)(22,28)(23,27)(24,26)(31,37)(32,42)(33,41)(34,40)(35,39)(36,38) );
G=PermutationGroup([[(1,26,4,19),(2,23,5,30),(3,28,6,21),(7,32,44,35),(8,40,45,37),(9,34,46,31),(10,42,47,39),(11,36,48,33),(12,38,43,41),(13,22,16,29),(14,27,17,20),(15,24,18,25)], [(1,34,13,41),(2,35,14,42),(3,36,15,37),(4,31,16,38),(5,32,17,39),(6,33,18,40),(7,27,47,23),(8,28,48,24),(9,29,43,19),(10,30,44,20),(11,25,45,21),(12,26,46,22)], [(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)], [(1,18),(2,17),(3,16),(4,15),(5,14),(6,13),(7,47),(8,46),(9,45),(10,44),(11,43),(12,48),(19,25),(20,30),(21,29),(22,28),(23,27),(24,26),(31,37),(32,42),(33,41),(34,40),(35,39),(36,38)]])
48 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 2K | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 4P | 6A | 6B | 6C | 6D | 6E | 12A | 12B | 12C | 12D | 12E | ··· | 12N |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 12 | 12 | 12 | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 6 | 6 | 6 | 6 | 12 | 12 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 12 | 12 | 2 | 2 | 2 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | ··· | 4 |
48 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D6 | D6 | D6 | D6 | C4oD4 | D12 | S3xC4oD4 |
| kernel | C42:10D6 | C4xD12 | D6:D4 | C23.21D6 | C12:D4 | C4.D12 | C3xC42:C2 | S3xC22xC4 | C2xC4oD12 | C42:C2 | C2xC12 | C42 | C22:C4 | C4:C4 | C22xC4 | D6 | C2xC4 | C2 |
| # reps | 1 | 4 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 4 | 2 | 2 | 2 | 1 | 8 | 8 | 4 |
Matrix representation of C42:10D6 ►in GL6(F13)
| 0 | 12 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 8 | 0 | 0 | 0 |
| 0 | 0 | 0 | 8 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 0 | 0 | 0 | 0 | 1 | 12 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 12 |
| 0 | 0 | 0 | 0 | 0 | 12 |
G:=sub<GL(6,GF(13))| [0,1,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,8,0,0,0,0,0,0,8,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,12,12],[12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,12,12] >;
C42:10D6 in GAP, Magma, Sage, TeX
C_4^2\rtimes_{10}D_6 % in TeX
G:=Group("C4^2:10D6"); // GroupNames label
G:=SmallGroup(192,1083);
// by ID
G=gap.SmallGroup(192,1083);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,184,675,570,80,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^6=d^2=1,a*b=b*a,c*a*c^-1=a*b^2,d*a*d=a^-1,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations