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, C4○D4⋊6D6, (S3×D8)⋊2C2, C3⋊4(D4○D8), D4○D12⋊7C2, (C2×D4)⋊15D6, C8⋊D6⋊2C2, D8⋊S3⋊3C2, Q8⋊3D6⋊2C2, C8⋊C22⋊4S3, C3⋊D4.4D4, D4⋊6D6⋊7C2, (S3×C8)⋊3C22, D12.C4⋊1C2, D6.32(C2×D4), C4.115(S3×D4), (C3×D8)⋊3C22, (S3×D4)⋊3C22, C3⋊C8.25C23, C8.2(C22×S3), C24⋊C2⋊3C22, C8⋊S3⋊3C22, D4⋊S3⋊14C22, Q8.13D6⋊3C2, Q8.7D6⋊2C2, C12.242(C2×D4), C4○D12⋊8C22, (C6×D4)⋊23C22, C4.21(S3×C23), C22.14(S3×D4), D4⋊2S3⋊3C22, (C2×D12)⋊36C22, (C4×S3).13C23, D4.S3⋊13C22, Dic3.37(C2×D4), Q8⋊3S3⋊3C22, (C3×SD16)⋊3C22, (C3×D4).14C23, C3⋊Q16⋊12C22, D4.14(C22×S3), C6.122(C22×D4), Q8.24(C22×S3), (C3×Q8).14C23, (C2×C12).112C23, Q8⋊2S3⋊13C22, (C3×M4(2))⋊3C22, C2.95(C2×S3×D4), (C2×D4⋊S3)⋊29C2, (C3×C8⋊C22)⋊3C2, (C2×C3⋊C8)⋊17C22, (C2×C6).67(C2×D4), (C3×C4○D4)⋊6C22, (C2×C4).96(C22×S3), SmallGroup(192,1333)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 832 in 268 conjugacy classes, 99 normal (45 characteristic)
C1, C2, C2 [×9], C3, C4 [×2], C4 [×4], C22, C22 [×14], S3 [×5], C6, C6 [×4], C8 [×2], C8 [×2], C2×C4, C2×C4 [×8], D4, D4 [×2], D4 [×18], Q8, Q8 [×2], C23 [×6], Dic3 [×2], Dic3, C12 [×2], C12, D6 [×2], D6 [×8], C2×C6, C2×C6 [×4], C2×C8 [×3], M4(2), M4(2) [×2], D8 [×2], D8 [×7], SD16 [×2], SD16 [×4], Q16, C2×D4, C2×D4 [×11], C4○D4, C4○D4 [×8], C3⋊C8 [×2], C24 [×2], Dic6 [×2], C4×S3 [×2], C4×S3 [×3], D12 [×2], D12 [×2], D12 [×3], C2×Dic3 [×2], C3⋊D4 [×2], C3⋊D4 [×7], C2×C12, C2×C12, C3×D4, C3×D4 [×2], C3×D4 [×2], C3×Q8, C22×S3 [×5], C22×C6, C8○D4, C2×D8 [×3], C4○D8 [×3], C8⋊C22, C8⋊C22 [×5], 2+ (1+4) [×2], S3×C8 [×2], C8⋊S3 [×2], C24⋊C2 [×2], D24 [×2], C2×C3⋊C8, D4⋊S3, D4⋊S3 [×4], D4.S3, Q8⋊2S3, C3⋊Q16, C3×M4(2), C3×D8 [×2], C3×SD16 [×2], C2×D12, C2×D12, C4○D12 [×2], C4○D12, S3×D4 [×4], S3×D4 [×3], D4⋊2S3 [×2], D4⋊2S3, Q8⋊3S3 [×2], C2×C3⋊D4 [×2], C6×D4, C3×C4○D4, D4○D8, D12.C4, C8⋊D6, S3×D8 [×2], D8⋊S3 [×2], Q8⋊3D6 [×2], Q8.7D6 [×2], C2×D4⋊S3, Q8.13D6, C3×C8⋊C22, D4⋊6D6, D4○D12, D8⋊5D6
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○D8, C2×S3×D4, D8⋊5D6
Generators and relations
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 >
►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)(17 19)(20 24)(21 23)(25 31)(26 30)(27 29)(34 40)(35 39)(36 38)(41 43)(44 48)(45 47)
(1 46 32 18 37 14)(2 43 25 23 38 11)(3 48 26 20 39 16)(4 45 27 17 40 13)(5 42 28 22 33 10)(6 47 29 19 34 15)(7 44 30 24 35 12)(8 41 31 21 36 9)
(1 11)(2 14)(3 9)(4 12)(5 15)(6 10)(7 13)(8 16)(17 30)(18 25)(19 28)(20 31)(21 26)(22 29)(23 32)(24 27)(33 47)(34 42)(35 45)(36 48)(37 43)(38 46)(39 41)(40 44)
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)(17,19)(20,24)(21,23)(25,31)(26,30)(27,29)(34,40)(35,39)(36,38)(41,43)(44,48)(45,47), (1,46,32,18,37,14)(2,43,25,23,38,11)(3,48,26,20,39,16)(4,45,27,17,40,13)(5,42,28,22,33,10)(6,47,29,19,34,15)(7,44,30,24,35,12)(8,41,31,21,36,9), (1,11)(2,14)(3,9)(4,12)(5,15)(6,10)(7,13)(8,16)(17,30)(18,25)(19,28)(20,31)(21,26)(22,29)(23,32)(24,27)(33,47)(34,42)(35,45)(36,48)(37,43)(38,46)(39,41)(40,44)>;
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)(17,19)(20,24)(21,23)(25,31)(26,30)(27,29)(34,40)(35,39)(36,38)(41,43)(44,48)(45,47), (1,46,32,18,37,14)(2,43,25,23,38,11)(3,48,26,20,39,16)(4,45,27,17,40,13)(5,42,28,22,33,10)(6,47,29,19,34,15)(7,44,30,24,35,12)(8,41,31,21,36,9), (1,11)(2,14)(3,9)(4,12)(5,15)(6,10)(7,13)(8,16)(17,30)(18,25)(19,28)(20,31)(21,26)(22,29)(23,32)(24,27)(33,47)(34,42)(35,45)(36,48)(37,43)(38,46)(39,41)(40,44) );
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),(17,19),(20,24),(21,23),(25,31),(26,30),(27,29),(34,40),(35,39),(36,38),(41,43),(44,48),(45,47)], [(1,46,32,18,37,14),(2,43,25,23,38,11),(3,48,26,20,39,16),(4,45,27,17,40,13),(5,42,28,22,33,10),(6,47,29,19,34,15),(7,44,30,24,35,12),(8,41,31,21,36,9)], [(1,11),(2,14),(3,9),(4,12),(5,15),(6,10),(7,13),(8,16),(17,30),(18,25),(19,28),(20,31),(21,26),(22,29),(23,32),(24,27),(33,47),(34,42),(35,45),(36,48),(37,43),(38,46),(39,41),(40,44)])
Matrix representation ►G ⊆ 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] >;
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 | S3×D4 | S3×D4 | D4○D8 | D8⋊5D6 |
| kernel | D8⋊5D6 | D12.C4 | C8⋊D6 | S3×D8 | D8⋊S3 | Q8⋊3D6 | Q8.7D6 | C2×D4⋊S3 | Q8.13D6 | C3×C8⋊C22 | D4⋊6D6 | D4○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_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