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, D4⋊6D6⋊8C2, D8⋊3S3⋊2C2, D12.C4⋊2C2, (S3×C8)⋊4C22, (S3×SD16)⋊2C2, D6.33(C2×D4), D4.D6⋊2C2, C3⋊4(D4○SD16), 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, D4.15(C22×S3), C3⋊Q16⋊13C22, (C3×D4).15C23, C6.123(C22×D4), (C3×Q8).15C23, Q8.25(C22×S3), (C2×C12).113C23, Q8⋊2S3⋊14C22, C4○D12.29C22, (C2×Dic6)⋊40C22, (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
Subgroups: 704 in 258 conjugacy classes, 99 normal (45 characteristic)
C1, C2, C2 [×7], C3, C4 [×2], C4 [×6], C22, C22 [×9], S3 [×3], C6, C6 [×4], C8 [×2], C8 [×2], C2×C4, C2×C4 [×11], D4, D4 [×2], D4 [×13], Q8, Q8 [×7], C23 [×3], Dic3 [×2], Dic3 [×3], C12 [×2], C12, D6 [×2], D6 [×3], C2×C6, C2×C6 [×4], C2×C8 [×3], M4(2), M4(2) [×2], D8 [×2], D8, SD16 [×2], SD16 [×8], Q16 [×3], C2×D4, C2×D4 [×5], C2×Q8 [×4], C4○D4, C4○D4 [×10], C3⋊C8 [×2], C24 [×2], Dic6 [×2], Dic6 [×2], Dic6 [×3], C4×S3 [×2], C4×S3 [×3], D12 [×2], C2×Dic3 [×5], C3⋊D4 [×2], C3⋊D4 [×7], C2×C12, C2×C12, C3×D4, C3×D4 [×2], C3×D4 [×2], C3×Q8, C22×S3 [×2], C22×C6, C8○D4, C2×SD16 [×3], C4○D8 [×3], C8⋊C22, C8⋊C22 [×2], C8.C22 [×3], 2+ (1+4), 2- (1+4), S3×C8 [×2], C8⋊S3 [×2], C24⋊C2 [×2], Dic12 [×2], C2×C3⋊C8, D4⋊S3, D4.S3, D4.S3 [×4], Q8⋊2S3, C3⋊Q16, C3×M4(2), C3×D8 [×2], C3×SD16 [×2], C2×Dic6, C2×Dic6, C4○D12 [×2], C4○D12, S3×D4 [×2], S3×D4, D4⋊2S3 [×4], D4⋊2S3 [×3], S3×Q8 [×2], C2×C3⋊D4 [×2], C6×D4, C3×C4○D4, D4○SD16, D12.C4, C8.D6, D8⋊S3 [×2], D8⋊3S3 [×2], S3×SD16 [×2], D4.D6 [×2], C2×D4.S3, Q8.13D6, C3×C8⋊C22, D4⋊6D6, Q8○D12, D8⋊6D6
Quotients:
C1, C2 [×15], C22 [×35], S3, D4 [×4], C23 [×15], D6 [×7], C2×D4 [×6], C24, C22×S3 [×7], C22×D4, S3×D4 [×2], S3×C23, D4○SD16, C2×S3×D4, D8⋊6D6
Generators and relations
G = < a,b,c,d | a8=b2=c6=d2=1, bab=a-1, cac-1=dad=a3, bc=cb, dbd=a4b, dcd=c-1 >
►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)(10 16)(11 15)(12 14)(17 19)(20 24)(21 23)(26 32)(27 31)(28 30)(33 35)(36 40)(37 39)(42 48)(43 47)(44 46)
(1 38 45 22 13 29)(2 33 46 17 14 32)(3 36 47 20 15 27)(4 39 48 23 16 30)(5 34 41 18 9 25)(6 37 42 21 10 28)(7 40 43 24 11 31)(8 35 44 19 12 26)
(1 43)(2 46)(3 41)(4 44)(5 47)(6 42)(7 45)(8 48)(9 15)(11 13)(12 16)(17 32)(18 27)(19 30)(20 25)(21 28)(22 31)(23 26)(24 29)(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)(10,16)(11,15)(12,14)(17,19)(20,24)(21,23)(26,32)(27,31)(28,30)(33,35)(36,40)(37,39)(42,48)(43,47)(44,46), (1,38,45,22,13,29)(2,33,46,17,14,32)(3,36,47,20,15,27)(4,39,48,23,16,30)(5,34,41,18,9,25)(6,37,42,21,10,28)(7,40,43,24,11,31)(8,35,44,19,12,26), (1,43)(2,46)(3,41)(4,44)(5,47)(6,42)(7,45)(8,48)(9,15)(11,13)(12,16)(17,32)(18,27)(19,30)(20,25)(21,28)(22,31)(23,26)(24,29)(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)(10,16)(11,15)(12,14)(17,19)(20,24)(21,23)(26,32)(27,31)(28,30)(33,35)(36,40)(37,39)(42,48)(43,47)(44,46), (1,38,45,22,13,29)(2,33,46,17,14,32)(3,36,47,20,15,27)(4,39,48,23,16,30)(5,34,41,18,9,25)(6,37,42,21,10,28)(7,40,43,24,11,31)(8,35,44,19,12,26), (1,43)(2,46)(3,41)(4,44)(5,47)(6,42)(7,45)(8,48)(9,15)(11,13)(12,16)(17,32)(18,27)(19,30)(20,25)(21,28)(22,31)(23,26)(24,29)(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),(10,16),(11,15),(12,14),(17,19),(20,24),(21,23),(26,32),(27,31),(28,30),(33,35),(36,40),(37,39),(42,48),(43,47),(44,46)], [(1,38,45,22,13,29),(2,33,46,17,14,32),(3,36,47,20,15,27),(4,39,48,23,16,30),(5,34,41,18,9,25),(6,37,42,21,10,28),(7,40,43,24,11,31),(8,35,44,19,12,26)], [(1,43),(2,46),(3,41),(4,44),(5,47),(6,42),(7,45),(8,48),(9,15),(11,13),(12,16),(17,32),(18,27),(19,30),(20,25),(21,28),(22,31),(23,26),(24,29),(34,36),(35,39),(38,40)])
Matrix representation ►G ⊆ 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] >;
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 |
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