metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D8:11D6, Q16:10D6, SD16:15D6, D12.46D4, C12.17C24, C24.43C23, Dic6.46D4, D12.12C23, Dic6.11C23, C4oD4:5D6, C4oD8:5S3, (C2xC8):14D6, D4oD12:6C2, D8:S3:6C2, Q8oD12:5C2, C3:D4.2D4, C3:C8.8C23, C8oD12:10C2, D4:D6:8C2, D4:S3:4C22, (S3xSD16):6C2, D6.30(C2xD4), C3:3(D4oSD16), Q16:S3:6C2, C4.144(S3xD4), (S3xQ8):2C22, C22.9(S3xD4), (S3xC8):10C22, (C2xC24):17C22, Q8.7D6:6C2, Q8.14D6:7C2, C12.350(C2xD4), (C3xD8):16C22, C4.17(S3xC23), C8.17(C22xS3), D4.S3:3C22, (S3xD4).2C22, C3:Q16:2C22, C24:C2:21C22, C8:S3:16C22, D4:2S3:2C22, (C4xS3).10C23, Dic3.35(C2xD4), (C3xQ16):14C22, Q8:2S3:3C22, (C3xD4).11C23, D4.11(C22xS3), C6.118(C22xD4), (C3xQ8).11C23, Q8.21(C22xS3), (C2xC12).534C23, C4oD12.55C22, (C3xSD16):16C22, (C2xDic6):38C22, C4.Dic3:31C22, Q8:3S3.2C22, (C2xD12).181C22, C2.91(C2xS3xD4), (C3xC4oD8):7C2, (C2xC6).14(C2xD4), (C2xC24:C2):27C2, (C3xC4oD4):4C22, (C2xC4).233(C22xS3), SmallGroup(192,1329)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D8:11D6
G = < a,b,c,d | a8=b2=c6=d2=1, bab=a-1, ac=ca, dad=a3, cbc-1=a4b, dbd=a6b, dcd=c-1 >
Subgroups: 728 in 258 conjugacy classes, 99 normal (53 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C8, C2xC4, C2xC4, D4, D4, Q8, Q8, C23, Dic3, Dic3, C12, C12, D6, D6, C2xC6, C2xC6, C2xC8, C2xC8, M4(2), D8, D8, SD16, SD16, Q16, Q16, C2xD4, C2xQ8, C4oD4, C4oD4, C3:C8, C24, Dic6, Dic6, Dic6, C4xS3, C4xS3, D12, D12, D12, C2xDic3, C3:D4, C3:D4, C2xC12, C2xC12, C3xD4, C3xD4, C3xQ8, C22xS3, C8oD4, C2xSD16, C4oD8, C4oD8, C8:C22, C8.C22, 2+ 1+4, 2- 1+4, S3xC8, C8:S3, C24:C2, C4.Dic3, D4:S3, D4.S3, Q8:2S3, C3:Q16, C2xC24, C3xD8, C3xSD16, C3xQ16, C2xDic6, C2xDic6, C2xD12, C2xD12, C4oD12, C4oD12, S3xD4, S3xD4, D4:2S3, D4:2S3, S3xQ8, Q8:3S3, C3xC4oD4, D4oSD16, C8oD12, C2xC24:C2, D8:S3, S3xSD16, Q8.7D6, Q16:S3, D4:D6, Q8.14D6, C3xC4oD8, D4oD12, Q8oD12, D8:11D6
Quotients: C1, C2, C22, S3, D4, C23, D6, C2xD4, C24, C22xS3, C22xD4, S3xD4, S3xC23, D4oSD16, C2xS3xD4, D8:11D6
►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 8)(2 7)(3 6)(4 5)(9 10)(11 16)(12 15)(13 14)(17 18)(19 24)(20 23)(21 22)(25 30)(26 29)(27 28)(31 32)(33 38)(34 37)(35 36)(39 40)(41 48)(42 47)(43 46)(44 45)
(1 16 45 26 18 34)(2 9 46 27 19 35)(3 10 47 28 20 36)(4 11 48 29 21 37)(5 12 41 30 22 38)(6 13 42 31 23 39)(7 14 43 32 24 40)(8 15 44 25 17 33)
(1 47)(2 42)(3 45)(4 48)(5 43)(6 46)(7 41)(8 44)(9 13)(10 16)(12 14)(18 20)(19 23)(22 24)(25 33)(26 36)(27 39)(28 34)(29 37)(30 40)(31 35)(32 38)
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), (1,8)(2,7)(3,6)(4,5)(9,10)(11,16)(12,15)(13,14)(17,18)(19,24)(20,23)(21,22)(25,30)(26,29)(27,28)(31,32)(33,38)(34,37)(35,36)(39,40)(41,48)(42,47)(43,46)(44,45), (1,16,45,26,18,34)(2,9,46,27,19,35)(3,10,47,28,20,36)(4,11,48,29,21,37)(5,12,41,30,22,38)(6,13,42,31,23,39)(7,14,43,32,24,40)(8,15,44,25,17,33), (1,47)(2,42)(3,45)(4,48)(5,43)(6,46)(7,41)(8,44)(9,13)(10,16)(12,14)(18,20)(19,23)(22,24)(25,33)(26,36)(27,39)(28,34)(29,37)(30,40)(31,35)(32,38)>;
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), (1,8)(2,7)(3,6)(4,5)(9,10)(11,16)(12,15)(13,14)(17,18)(19,24)(20,23)(21,22)(25,30)(26,29)(27,28)(31,32)(33,38)(34,37)(35,36)(39,40)(41,48)(42,47)(43,46)(44,45), (1,16,45,26,18,34)(2,9,46,27,19,35)(3,10,47,28,20,36)(4,11,48,29,21,37)(5,12,41,30,22,38)(6,13,42,31,23,39)(7,14,43,32,24,40)(8,15,44,25,17,33), (1,47)(2,42)(3,45)(4,48)(5,43)(6,46)(7,41)(8,44)(9,13)(10,16)(12,14)(18,20)(19,23)(22,24)(25,33)(26,36)(27,39)(28,34)(29,37)(30,40)(31,35)(32,38) );
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)], [(1,8),(2,7),(3,6),(4,5),(9,10),(11,16),(12,15),(13,14),(17,18),(19,24),(20,23),(21,22),(25,30),(26,29),(27,28),(31,32),(33,38),(34,37),(35,36),(39,40),(41,48),(42,47),(43,46),(44,45)], [(1,16,45,26,18,34),(2,9,46,27,19,35),(3,10,47,28,20,36),(4,11,48,29,21,37),(5,12,41,30,22,38),(6,13,42,31,23,39),(7,14,43,32,24,40),(8,15,44,25,17,33)], [(1,47),(2,42),(3,45),(4,48),(5,43),(6,46),(7,41),(8,44),(9,13),(10,16),(12,14),(18,20),(19,23),(22,24),(25,33),(26,36),(27,39),(28,34),(29,37),(30,40),(31,35),(32,38)]])
36 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 6A | 6B | 6C | 6D | 8A | 8B | 8C | 8D | 8E | 12A | 12B | 12C | 12D | 12E | 24A | 24B | 24C | 24D |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | 24 | 24 | 24 | 24 |
| size | 1 | 1 | 2 | 4 | 4 | 6 | 6 | 12 | 12 | 2 | 2 | 2 | 4 | 4 | 6 | 6 | 12 | 12 | 2 | 4 | 8 | 8 | 2 | 2 | 4 | 12 | 12 | 2 | 2 | 4 | 8 | 8 | 4 | 4 | 4 | 4 |
36 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D4 | D6 | D6 | D6 | D6 | D6 | S3xD4 | S3xD4 | D4oSD16 | D8:11D6 |
| kernel | D8:11D6 | C8oD12 | C2xC24:C2 | D8:S3 | S3xSD16 | Q8.7D6 | Q16:S3 | D4:D6 | Q8.14D6 | C3xC4oD8 | D4oD12 | Q8oD12 | C4oD8 | Dic6 | D12 | C3:D4 | C2xC8 | D8 | SD16 | Q16 | C4oD4 | C4 | C22 | C3 | C1 |
| # reps | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 2 | 1 | 2 | 1 | 1 | 2 | 4 |
Matrix representation of D8:11D6 ►in GL4(F73) generated by
| 42 | 62 | 31 | 11 |
| 11 | 31 | 62 | 42 |
| 42 | 62 | 42 | 62 |
| 11 | 31 | 11 | 31 |
| 42 | 62 | 31 | 11 |
| 11 | 31 | 62 | 42 |
| 31 | 11 | 31 | 11 |
| 62 | 42 | 62 | 42 |
| 0 | 0 | 14 | 66 |
| 0 | 0 | 7 | 7 |
| 59 | 7 | 0 | 0 |
| 66 | 66 | 0 | 0 |
| 0 | 0 | 1 | 72 |
| 0 | 0 | 0 | 72 |
| 1 | 72 | 0 | 0 |
| 0 | 72 | 0 | 0 |
G:=sub<GL(4,GF(73))| [42,11,42,11,62,31,62,31,31,62,42,11,11,42,62,31],[42,11,31,62,62,31,11,42,31,62,31,62,11,42,11,42],[0,0,59,66,0,0,7,66,14,7,0,0,66,7,0,0],[0,0,1,0,0,0,72,72,1,0,0,0,72,72,0,0] >;
D8:11D6 in GAP, Magma, Sage, TeX
D_8\rtimes_{11}D_6 % in TeX
G:=Group("D8:11D6"); // GroupNames label
G:=SmallGroup(192,1329);
// by ID
G=gap.SmallGroup(192,1329);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,477,387,570,185,136,438,235,102,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^8=b^2=c^6=d^2=1,b*a*b=a^-1,a*c=c*a,d*a*d=a^3,c*b*c^-1=a^4*b,d*b*d=a^6*b,d*c*d=c^-1>;
// generators/relations