metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D8:5D6, SD16:3D6, D12.41D4, D24:2C22, M4(2):9D6, C24.2C23, C12.21C24, Dic6.41D4, D12.14C23, Dic6.14C23, C4oD4:6D6, (S3xD8):2C2, C3:4(D4oD8), D4oD12:7C2, (C2xD4):15D6, C8:D6:2C2, Q8:3D6:2C2, D8:S3:3C2, C8:C22:4S3, C3:D4.4D4, D4:6D6:7C2, (S3xC8):3C22, D12.C4:1C2, D6.32(C2xD4), C4.115(S3xD4), (C3xD8):3C22, (S3xD4):3C22, C3:C8.25C23, C8.2(C22xS3), C8:S3:3C22, C24:C2:3C22, D4:S3:14C22, Q8.7D6:2C2, Q8.13D6:3C2, C12.242(C2xD4), C4oD12:8C22, (C6xD4):23C22, C4.21(S3xC23), C22.14(S3xD4), (C2xD12):36C22, D4:2S3:3C22, (C4xS3).13C23, D4.S3:13C22, Dic3.37(C2xD4), Q8:3S3:3C22, (C3xSD16):3C22, D4.14(C22xS3), C3:Q16:12C22, (C3xD4).14C23, C6.122(C22xD4), Q8.24(C22xS3), (C3xQ8).14C23, (C2xC12).112C23, Q8:2S3:13C22, (C3xM4(2)):3C22, C2.95(C2xS3xD4), (C2xD4:S3):29C2, (C3xC8:C22):3C2, (C2xC3:C8):17C22, (C2xC6).67(C2xD4), (C3xC4oD4):6C22, (C2xC4).96(C22xS3), SmallGroup(192,1333)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D8:5D6
G = < a,b,c,d | a8=b2=c6=d2=1, bab=a-1, cac-1=a5, dad=a3, bc=cb, dbd=a6b, dcd=c-1 >
Subgroups: 832 in 268 conjugacy classes, 99 normal (45 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C8, C2xC4, C2xC4, D4, D4, D4, Q8, Q8, C23, Dic3, Dic3, C12, C12, D6, D6, C2xC6, C2xC6, C2xC8, M4(2), M4(2), D8, D8, SD16, SD16, Q16, C2xD4, C2xD4, C4oD4, C4oD4, C3:C8, C24, Dic6, C4xS3, C4xS3, D12, D12, D12, C2xDic3, C3:D4, C3:D4, C2xC12, C2xC12, C3xD4, C3xD4, C3xD4, C3xQ8, C22xS3, C22xC6, C8oD4, C2xD8, C4oD8, C8:C22, C8:C22, 2+ 1+4, S3xC8, C8:S3, C24:C2, D24, C2xC3:C8, D4:S3, D4:S3, D4.S3, Q8:2S3, C3:Q16, C3xM4(2), C3xD8, C3xSD16, C2xD12, C2xD12, C4oD12, C4oD12, S3xD4, S3xD4, D4:2S3, D4:2S3, Q8:3S3, C2xC3:D4, C6xD4, C3xC4oD4, D4oD8, D12.C4, C8:D6, S3xD8, D8:S3, Q8:3D6, Q8.7D6, C2xD4:S3, Q8.13D6, C3xC8:C22, D4:6D6, D4oD12, D8:5D6
Quotients: C1, C2, C22, S3, D4, C23, D6, C2xD4, C24, C22xS3, C22xD4, S3xD4, S3xC23, D4oD8, C2xS3xD4, D8:5D6
►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)
(2 8)(3 7)(4 6)(9 11)(12 16)(13 15)(18 24)(19 23)(20 22)(25 31)(26 30)(27 29)(33 35)(36 40)(37 39)(41 43)(44 48)(45 47)
(1 46 32 21 38 14)(2 43 25 18 39 11)(3 48 26 23 40 16)(4 45 27 20 33 13)(5 42 28 17 34 10)(6 47 29 22 35 15)(7 44 30 19 36 12)(8 41 31 24 37 9)
(1 11)(2 14)(3 9)(4 12)(5 15)(6 10)(7 13)(8 16)(17 29)(18 32)(19 27)(20 30)(21 25)(22 28)(23 31)(24 26)(33 44)(34 47)(35 42)(36 45)(37 48)(38 43)(39 46)(40 41)
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), (2,8)(3,7)(4,6)(9,11)(12,16)(13,15)(18,24)(19,23)(20,22)(25,31)(26,30)(27,29)(33,35)(36,40)(37,39)(41,43)(44,48)(45,47), (1,46,32,21,38,14)(2,43,25,18,39,11)(3,48,26,23,40,16)(4,45,27,20,33,13)(5,42,28,17,34,10)(6,47,29,22,35,15)(7,44,30,19,36,12)(8,41,31,24,37,9), (1,11)(2,14)(3,9)(4,12)(5,15)(6,10)(7,13)(8,16)(17,29)(18,32)(19,27)(20,30)(21,25)(22,28)(23,31)(24,26)(33,44)(34,47)(35,42)(36,45)(37,48)(38,43)(39,46)(40,41)>;
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), (2,8)(3,7)(4,6)(9,11)(12,16)(13,15)(18,24)(19,23)(20,22)(25,31)(26,30)(27,29)(33,35)(36,40)(37,39)(41,43)(44,48)(45,47), (1,46,32,21,38,14)(2,43,25,18,39,11)(3,48,26,23,40,16)(4,45,27,20,33,13)(5,42,28,17,34,10)(6,47,29,22,35,15)(7,44,30,19,36,12)(8,41,31,24,37,9), (1,11)(2,14)(3,9)(4,12)(5,15)(6,10)(7,13)(8,16)(17,29)(18,32)(19,27)(20,30)(21,25)(22,28)(23,31)(24,26)(33,44)(34,47)(35,42)(36,45)(37,48)(38,43)(39,46)(40,41) );
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)], [(2,8),(3,7),(4,6),(9,11),(12,16),(13,15),(18,24),(19,23),(20,22),(25,31),(26,30),(27,29),(33,35),(36,40),(37,39),(41,43),(44,48),(45,47)], [(1,46,32,21,38,14),(2,43,25,18,39,11),(3,48,26,23,40,16),(4,45,27,20,33,13),(5,42,28,17,34,10),(6,47,29,22,35,15),(7,44,30,19,36,12),(8,41,31,24,37,9)], [(1,11),(2,14),(3,9),(4,12),(5,15),(6,10),(7,13),(8,16),(17,29),(18,32),(19,27),(20,30),(21,25),(22,28),(23,31),(24,26),(33,44),(34,47),(35,42),(36,45),(37,48),(38,43),(39,46),(40,41)]])
33 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 6A | 6B | 6C | 6D | 6E | 8A | 8B | 8C | 8D | 8E | 12A | 12B | 12C | 24A | 24B |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 24 | 24 |
| size | 1 | 1 | 2 | 4 | 4 | 4 | 6 | 6 | 12 | 12 | 12 | 2 | 2 | 2 | 4 | 6 | 6 | 12 | 2 | 4 | 8 | 8 | 8 | 4 | 4 | 6 | 6 | 12 | 4 | 4 | 8 | 8 | 8 |
33 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 | 8 |
| 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 | D4oD8 | D8:5D6 |
| kernel | D8:5D6 | D12.C4 | C8:D6 | S3xD8 | D8:S3 | Q8:3D6 | Q8.7D6 | C2xD4:S3 | Q8.13D6 | C3xC8:C22 | D4:6D6 | D4oD12 | C8:C22 | Dic6 | D12 | C3:D4 | M4(2) | D8 | SD16 | C2xD4 | C4oD4 | C4 | C22 | C3 | C1 |
| # reps | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 1 |
Matrix representation of D8:5D6 ►in GL6(F73)
| 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 57 | 0 | 0 |
| 0 | 0 | 16 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 57 | 16 |
| 0 | 0 | 0 | 0 | 57 | 57 |
| 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 72 |
| 0 | 72 | 0 | 0 | 0 | 0 |
| 1 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 1 | 72 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 16 |
| 0 | 0 | 0 | 0 | 16 | 57 |
| 0 | 0 | 16 | 16 | 0 | 0 |
| 0 | 0 | 16 | 57 | 0 | 0 |
G:=sub<GL(6,GF(73))| [72,0,0,0,0,0,0,72,0,0,0,0,0,0,16,16,0,0,0,0,57,16,0,0,0,0,0,0,57,57,0,0,0,0,16,57],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,72],[0,1,0,0,0,0,72,72,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0],[1,0,0,0,0,0,72,72,0,0,0,0,0,0,0,0,16,16,0,0,0,0,16,57,0,0,16,16,0,0,0,0,16,57,0,0] >;
D8:5D6 in GAP, Magma, Sage, TeX
D_8\rtimes_5D_6
% in TeX
G:=Group("D8:5D6"); // GroupNames label
G:=SmallGroup(192,1333);
// by ID
G=gap.SmallGroup(192,1333);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,477,570,185,438,235,102,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^8=b^2=c^6=d^2=1,b*a*b=a^-1,c*a*c^-1=a^5,d*a*d=a^3,b*c=c*b,d*b*d=a^6*b,d*c*d=c^-1>;
// generators/relations