metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: (C2xC6):8D8, (C3xD4):13D4, C6.72(C2xD8), D4:5(C3:D4), C3:5(C22:D8), (C22xD4):4S3, C12:7D4:25C2, C6.71C22wrC2, C22:4(D4:S3), (C2xD4).198D6, C12.205(C2xD4), (C2xC12).300D4, D4:Dic3:38C2, (C2xD12):14C22, C4:Dic3:21C22, (C22xC6).196D4, (C22xC4).165D6, C12.55D4:15C2, C6.102(C8:C22), (C2xC12).472C23, C2.4(C24:4S3), (C6xD4).240C22, C23.92(C3:D4), C2.22(D12:6C22), (C22xC12).197C22, (D4xC2xC6):1C2, (C2xD4:S3):23C2, (C2xC3:C8):10C22, C2.26(C2xD4:S3), C4.58(C2xC3:D4), (C2xC6).553(C2xD4), (C2xC4).83(C3:D4), (C2xC4).558(C22xS3), C22.216(C2xC3:D4), SmallGroup(192,776)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for (C2xC6):8D8
G = < a,b,c,d | a2=b6=c8=d2=1, ab=ba, cac-1=dad=ab3, cbc-1=dbd=b-1, dcd=c-1 >
Subgroups: 584 in 198 conjugacy classes, 51 normal (25 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C22, S3, C6, C6, C8, C2xC4, C2xC4, D4, D4, C23, C23, Dic3, C12, C12, D6, C2xC6, C2xC6, C2xC6, C22:C4, C4:C4, C2xC8, D8, C22xC4, C2xD4, C2xD4, C24, C3:C8, D12, C2xDic3, C3:D4, C2xC12, C2xC12, C3xD4, C3xD4, C22xS3, C22xC6, C22xC6, C22:C8, D4:C4, C4:D4, C2xD8, C22xD4, C2xC3:C8, C4:Dic3, D6:C4, D4:S3, C2xD12, C2xC3:D4, C22xC12, C6xD4, C6xD4, C23xC6, C22:D8, C12.55D4, D4:Dic3, C12:7D4, C2xD4:S3, D4xC2xC6, (C2xC6):8D8
Quotients: C1, C2, C22, S3, D4, C23, D6, D8, C2xD4, C3:D4, C22xS3, C22wrC2, C2xD8, C8:C22, D4:S3, C2xC3:D4, C22:D8, C2xD4:S3, D12:6C22, C24:4S3, (C2xC6):8D8
►On 48 points
(2 20)(4 22)(6 24)(8 18)(10 36)(12 38)(14 40)(16 34)(25 44)(27 46)(29 48)(31 42)
(1 13 45 19 39 26)(2 27 40 20 46 14)(3 15 47 21 33 28)(4 29 34 22 48 16)(5 9 41 23 35 30)(6 31 36 24 42 10)(7 11 43 17 37 32)(8 25 38 18 44 12)
(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 24)(4 23)(5 22)(6 21)(7 20)(8 19)(9 34)(10 33)(11 40)(12 39)(13 38)(14 37)(15 36)(16 35)(25 45)(26 44)(27 43)(28 42)(29 41)(30 48)(31 47)(32 46)
G:=sub<Sym(48)| (2,20)(4,22)(6,24)(8,18)(10,36)(12,38)(14,40)(16,34)(25,44)(27,46)(29,48)(31,42), (1,13,45,19,39,26)(2,27,40,20,46,14)(3,15,47,21,33,28)(4,29,34,22,48,16)(5,9,41,23,35,30)(6,31,36,24,42,10)(7,11,43,17,37,32)(8,25,38,18,44,12), (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)(3,24)(4,23)(5,22)(6,21)(7,20)(8,19)(9,34)(10,33)(11,40)(12,39)(13,38)(14,37)(15,36)(16,35)(25,45)(26,44)(27,43)(28,42)(29,41)(30,48)(31,47)(32,46)>;
G:=Group( (2,20)(4,22)(6,24)(8,18)(10,36)(12,38)(14,40)(16,34)(25,44)(27,46)(29,48)(31,42), (1,13,45,19,39,26)(2,27,40,20,46,14)(3,15,47,21,33,28)(4,29,34,22,48,16)(5,9,41,23,35,30)(6,31,36,24,42,10)(7,11,43,17,37,32)(8,25,38,18,44,12), (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)(3,24)(4,23)(5,22)(6,21)(7,20)(8,19)(9,34)(10,33)(11,40)(12,39)(13,38)(14,37)(15,36)(16,35)(25,45)(26,44)(27,43)(28,42)(29,41)(30,48)(31,47)(32,46) );
G=PermutationGroup([[(2,20),(4,22),(6,24),(8,18),(10,36),(12,38),(14,40),(16,34),(25,44),(27,46),(29,48),(31,42)], [(1,13,45,19,39,26),(2,27,40,20,46,14),(3,15,47,21,33,28),(4,29,34,22,48,16),(5,9,41,23,35,30),(6,31,36,24,42,10),(7,11,43,17,37,32),(8,25,38,18,44,12)], [(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),(3,24),(4,23),(5,22),(6,21),(7,20),(8,19),(9,34),(10,33),(11,40),(12,39),(13,38),(14,37),(15,36),(16,35),(25,45),(26,44),(27,43),(28,42),(29,41),(30,48),(31,47),(32,46)]])
39 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 3 | 4A | 4B | 4C | 4D | 6A | ··· | 6G | 6H | ··· | 6O | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 24 | 2 | 2 | 2 | 4 | 24 | 2 | ··· | 2 | 4 | ··· | 4 | 12 | 12 | 12 | 12 | 4 | 4 | 4 | 4 |
39 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D4 | D6 | D6 | D8 | C3:D4 | C3:D4 | C3:D4 | C8:C22 | D4:S3 | D12:6C22 |
| kernel | (C2xC6):8D8 | C12.55D4 | D4:Dic3 | C12:7D4 | C2xD4:S3 | D4xC2xC6 | C22xD4 | C2xC12 | C3xD4 | C22xC6 | C22xC4 | C2xD4 | C2xC6 | C2xC4 | D4 | C23 | C6 | C22 | C2 |
| # reps | 1 | 1 | 2 | 1 | 2 | 1 | 1 | 1 | 4 | 1 | 1 | 2 | 4 | 2 | 8 | 2 | 1 | 2 | 2 |
Matrix representation of (C2xC6):8D8 ►in GL4(F73) generated by
| 1 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 65 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 72 | 0 | 0 |
| 72 | 0 | 0 | 0 |
| 0 | 0 | 57 | 16 |
| 0 | 0 | 57 | 57 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 57 | 16 |
| 0 | 0 | 16 | 16 |
G:=sub<GL(4,GF(73))| [1,0,0,0,0,72,0,0,0,0,1,0,0,0,0,1],[65,0,0,0,0,9,0,0,0,0,1,0,0,0,0,1],[0,72,0,0,72,0,0,0,0,0,57,57,0,0,16,57],[0,1,0,0,1,0,0,0,0,0,57,16,0,0,16,16] >;
(C2xC6):8D8 in GAP, Magma, Sage, TeX
(C_2\times C_6)\rtimes_8D_8
% in TeX
G:=Group("(C2xC6):8D8"); // GroupNames label
G:=SmallGroup(192,776);
// by ID
G=gap.SmallGroup(192,776);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,253,254,1684,851,102,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^6=c^8=d^2=1,a*b=b*a,c*a*c^-1=d*a*d=a*b^3,c*b*c^-1=d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations