metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D8⋊6D6, SD16⋊4D6, D12.42D4, C24.3C23, M4(2)⋊10D6, C12.22C24, Dic6.42D4, Dic12⋊2C22, D12.15C23, Dic6.15C23, C4○D4⋊7D6, D8⋊S3⋊4C2, C8⋊C22⋊5S3, Q8○D12⋊7C2, C3⋊D4.5D4, D8⋊3S3⋊2C2, D4⋊6D6⋊8C2, D12.C4⋊2C2, (S3×C8)⋊4C22, (S3×SD16)⋊2C2, D6.33(C2×D4), C3⋊4(D4○SD16), D4.D6⋊2C2, C4.116(S3×D4), C8.D6⋊2C2, (C3×D8)⋊4C22, C3⋊C8.26C23, C8.3(C22×S3), (S3×Q8)⋊3C22, C8⋊S3⋊4C22, C24⋊C2⋊4C22, D4⋊S3⋊15C22, Q8.13D6⋊4C2, (C2×D4).117D6, C12.243(C2×D4), C4.22(S3×C23), (S3×D4).3C22, C22.15(S3×D4), D4⋊2S3⋊4C22, (C4×S3).14C23, D4.S3⋊14C22, Dic3.38(C2×D4), (C3×SD16)⋊4C22, C3⋊Q16⋊13C22, D4.15(C22×S3), (C3×D4).15C23, C6.123(C22×D4), Q8.25(C22×S3), (C3×Q8).15C23, (C2×C12).113C23, Q8⋊2S3⋊14C22, (C2×Dic6)⋊40C22, C4○D12.29C22, (C6×D4).168C22, (C3×M4(2))⋊4C22, C2.96(C2×S3×D4), (C3×C8⋊C22)⋊4C2, (C2×C3⋊C8)⋊18C22, (C2×C6).68(C2×D4), (C2×D4.S3)⋊29C2, (C3×C4○D4)⋊7C22, (C2×C4).97(C22×S3), SmallGroup(192,1334)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D8⋊6D6
G = < a,b,c,d | a8=b2=c6=d2=1, bab=a-1, cac-1=dad=a3, bc=cb, dbd=a4b, dcd=c-1 >
Subgroups: 704 in 258 conjugacy classes, 99 normal (45 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C8, C2×C4, C2×C4, D4, D4, D4, Q8, Q8, C23, Dic3, Dic3, C12, C12, D6, D6, C2×C6, C2×C6, C2×C8, M4(2), M4(2), D8, D8, SD16, SD16, Q16, C2×D4, C2×D4, C2×Q8, C4○D4, C4○D4, C3⋊C8, C24, Dic6, Dic6, Dic6, C4×S3, C4×S3, D12, C2×Dic3, C3⋊D4, C3⋊D4, C2×C12, C2×C12, C3×D4, C3×D4, C3×D4, C3×Q8, C22×S3, C22×C6, C8○D4, C2×SD16, C4○D8, C8⋊C22, C8⋊C22, C8.C22, 2+ 1+4, 2- 1+4, S3×C8, C8⋊S3, C24⋊C2, Dic12, C2×C3⋊C8, D4⋊S3, D4.S3, D4.S3, Q8⋊2S3, C3⋊Q16, C3×M4(2), C3×D8, C3×SD16, C2×Dic6, C2×Dic6, C4○D12, C4○D12, S3×D4, S3×D4, D4⋊2S3, D4⋊2S3, S3×Q8, C2×C3⋊D4, C6×D4, C3×C4○D4, D4○SD16, D12.C4, C8.D6, D8⋊S3, D8⋊3S3, S3×SD16, D4.D6, C2×D4.S3, Q8.13D6, C3×C8⋊C22, D4⋊6D6, Q8○D12, D8⋊6D6
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C24, C22×S3, C22×D4, S3×D4, S3×C23, D4○SD16, C2×S3×D4, D8⋊6D6
►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 15)(10 14)(11 13)(17 19)(20 24)(21 23)(25 27)(28 32)(29 31)(33 35)(36 40)(37 39)(42 48)(43 47)(44 46)
(1 38 45 18 26 12)(2 33 46 21 27 15)(3 36 47 24 28 10)(4 39 48 19 29 13)(5 34 41 22 30 16)(6 37 42 17 31 11)(7 40 43 20 32 14)(8 35 44 23 25 9)
(1 43)(2 46)(3 41)(4 44)(5 47)(6 42)(7 45)(8 48)(9 19)(10 22)(11 17)(12 20)(13 23)(14 18)(15 21)(16 24)(25 29)(26 32)(28 30)(34 36)(35 39)(38 40)
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)(9,15)(10,14)(11,13)(17,19)(20,24)(21,23)(25,27)(28,32)(29,31)(33,35)(36,40)(37,39)(42,48)(43,47)(44,46), (1,38,45,18,26,12)(2,33,46,21,27,15)(3,36,47,24,28,10)(4,39,48,19,29,13)(5,34,41,22,30,16)(6,37,42,17,31,11)(7,40,43,20,32,14)(8,35,44,23,25,9), (1,43)(2,46)(3,41)(4,44)(5,47)(6,42)(7,45)(8,48)(9,19)(10,22)(11,17)(12,20)(13,23)(14,18)(15,21)(16,24)(25,29)(26,32)(28,30)(34,36)(35,39)(38,40)>;
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)(9,15)(10,14)(11,13)(17,19)(20,24)(21,23)(25,27)(28,32)(29,31)(33,35)(36,40)(37,39)(42,48)(43,47)(44,46), (1,38,45,18,26,12)(2,33,46,21,27,15)(3,36,47,24,28,10)(4,39,48,19,29,13)(5,34,41,22,30,16)(6,37,42,17,31,11)(7,40,43,20,32,14)(8,35,44,23,25,9), (1,43)(2,46)(3,41)(4,44)(5,47)(6,42)(7,45)(8,48)(9,19)(10,22)(11,17)(12,20)(13,23)(14,18)(15,21)(16,24)(25,29)(26,32)(28,30)(34,36)(35,39)(38,40) );
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),(9,15),(10,14),(11,13),(17,19),(20,24),(21,23),(25,27),(28,32),(29,31),(33,35),(36,40),(37,39),(42,48),(43,47),(44,46)], [(1,38,45,18,26,12),(2,33,46,21,27,15),(3,36,47,24,28,10),(4,39,48,19,29,13),(5,34,41,22,30,16),(6,37,42,17,31,11),(7,40,43,20,32,14),(8,35,44,23,25,9)], [(1,43),(2,46),(3,41),(4,44),(5,47),(6,42),(7,45),(8,48),(9,19),(10,22),(11,17),(12,20),(13,23),(14,18),(15,21),(16,24),(25,29),(26,32),(28,30),(34,36),(35,39),(38,40)]])
33 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 | 6E | 8A | 8B | 8C | 8D | 8E | 12A | 12B | 12C | 24A | 24B |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 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 | 2 | 2 | 2 | 4 | 6 | 6 | 12 | 12 | 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 | S3×D4 | S3×D4 | D4○SD16 | D8⋊6D6 |
| kernel | D8⋊6D6 | D12.C4 | C8.D6 | D8⋊S3 | D8⋊3S3 | S3×SD16 | D4.D6 | C2×D4.S3 | Q8.13D6 | C3×C8⋊C22 | D4⋊6D6 | Q8○D12 | C8⋊C22 | Dic6 | D12 | C3⋊D4 | M4(2) | D8 | SD16 | C2×D4 | C4○D4 | 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⋊6D6 ►in GL6(𝔽73)
| 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 61 |
| 0 | 0 | 67 | 0 | 12 | 0 |
| 0 | 0 | 67 | 6 | 6 | 67 |
| 0 | 0 | 0 | 67 | 6 | 67 |
| 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 1 | 0 | 0 | 72 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 72 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 71 | 0 |
| 0 | 0 | 0 | 0 | 72 | 1 |
| 0 | 0 | 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 1 | 72 | 0 |
| 72 | 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 71 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 1 |
| 0 | 0 | 0 | 72 | 1 | 0 |
G:=sub<GL(6,GF(73))| [72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,67,67,0,0,0,0,0,6,67,0,0,12,12,6,6,0,0,61,0,67,67],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,1,0,1,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,72],[0,72,0,0,0,0,1,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,71,72,72,72,0,0,0,1,0,0],[72,0,0,0,0,0,1,1,0,0,0,0,0,0,1,0,0,0,0,0,71,72,72,72,0,0,0,0,0,1,0,0,0,0,1,0] >;
D8⋊6D6 in GAP, Magma, Sage, TeX
D_8\rtimes_6D_6
% in TeX
G:=Group("D8:6D6"); // GroupNames label
G:=SmallGroup(192,1334);
// by ID
G=gap.SmallGroup(192,1334);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,477,570,185,136,438,235,102,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^8=b^2=c^6=d^2=1,b*a*b=a^-1,c*a*c^-1=d*a*d=a^3,b*c=c*b,d*b*d=a^4*b,d*c*d=c^-1>;
// generators/relations