metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D8⋊15D6, Q16⋊13D6, SD16⋊11D6, D12.45D4, D24⋊19C22, C12.16C24, C24.38C23, Dic6.45D4, D12.11C23, C4○D4⋊4D6, C4○D8⋊4S3, (S3×D8)⋊7C2, (C2×C8)⋊13D6, C3⋊3(D4○D8), C8○D12⋊7C2, D4○D12⋊5C2, Q8⋊3D6⋊6C2, C3⋊D4.1D4, C3⋊C8.7C23, (C2×D24)⋊23C2, D4⋊D6⋊7C2, (S3×C8)⋊8C22, D4⋊S3⋊3C22, D6.29(C2×D4), C4.143(S3×D4), (S3×D4)⋊2C22, D24⋊C2⋊7C2, C22.8(S3×D4), (C2×C24)⋊12C22, C12.349(C2×D4), (C3×D8)⋊13C22, (C4×S3).9C23, C4.16(S3×C23), C8.16(C22×S3), C8⋊S3⋊11C22, (C2×D12)⋊34C22, Dic3.34(C2×D4), Q8⋊3S3⋊2C22, (C3×Q16)⋊11C22, Q8⋊2S3⋊2C22, (C3×D4).10C23, D4.10(C22×S3), C6.117(C22×D4), Q8.20(C22×S3), (C3×Q8).10C23, (C2×C12).533C23, C4○D12.54C22, (C3×SD16)⋊11C22, C4.Dic3⋊30C22, C2.90(C2×S3×D4), (C3×C4○D8)⋊5C2, (C2×C6).13(C2×D4), (C3×C4○D4)⋊3C22, (C2×C4).232(C22×S3), SmallGroup(192,1328)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 856 in 268 conjugacy classes, 99 normal (31 characteristic)
C1, C2, C2 [×9], C3, C4 [×2], C4 [×4], C22, C22 [×14], S3 [×6], C6, C6 [×3], C8 [×2], C8 [×2], C2×C4, C2×C4 [×8], D4 [×2], D4 [×19], Q8 [×2], Q8, C23 [×6], Dic3 [×2], C12 [×2], C12 [×2], D6 [×2], D6 [×10], C2×C6, C2×C6 [×2], C2×C8, C2×C8 [×2], M4(2) [×3], D8, D8 [×8], SD16 [×2], SD16 [×4], Q16, C2×D4 [×12], C4○D4 [×2], C4○D4 [×7], C3⋊C8 [×2], C24 [×2], Dic6, C4×S3 [×2], C4×S3 [×4], D12, D12 [×4], D12 [×6], C3⋊D4 [×2], C3⋊D4 [×4], C2×C12, C2×C12 [×2], C3×D4 [×2], C3×D4 [×2], C3×Q8 [×2], C22×S3 [×6], C8○D4, C2×D8 [×3], C4○D8, C4○D8 [×2], C8⋊C22 [×6], 2+ (1+4) [×2], S3×C8 [×2], C8⋊S3 [×2], D24 [×4], C4.Dic3, D4⋊S3 [×4], Q8⋊2S3 [×4], C2×C24, C3×D8, C3×SD16 [×2], C3×Q16, C2×D12 [×2], C2×D12 [×2], C4○D12, C4○D12 [×2], S3×D4 [×4], S3×D4 [×4], Q8⋊3S3 [×4], C3×C4○D4 [×2], D4○D8, C8○D12, C2×D24, S3×D8 [×2], Q8⋊3D6 [×4], D24⋊C2 [×2], D4⋊D6 [×2], C3×C4○D8, D4○D12 [×2], D8⋊15D6
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⋊15D6
Generators and relations
G = < a,b,c,d | a8=b2=c6=d2=1, bab=dad=a-1, ac=ca, cbc-1=a4b, dbd=a2b, 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 15)(10 14)(11 13)(17 21)(18 20)(22 24)(25 29)(26 28)(30 32)(34 40)(35 39)(36 38)(41 47)(42 46)(43 45)
(1 46 27 17 37 10)(2 47 28 18 38 11)(3 48 29 19 39 12)(4 41 30 20 40 13)(5 42 31 21 33 14)(6 43 32 22 34 15)(7 44 25 23 35 16)(8 45 26 24 36 9)
(1 11)(2 10)(3 9)(4 16)(5 15)(6 14)(7 13)(8 12)(17 28)(18 27)(19 26)(20 25)(21 32)(22 31)(23 30)(24 29)(33 43)(34 42)(35 41)(36 48)(37 47)(38 46)(39 45)(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)(9,15)(10,14)(11,13)(17,21)(18,20)(22,24)(25,29)(26,28)(30,32)(34,40)(35,39)(36,38)(41,47)(42,46)(43,45), (1,46,27,17,37,10)(2,47,28,18,38,11)(3,48,29,19,39,12)(4,41,30,20,40,13)(5,42,31,21,33,14)(6,43,32,22,34,15)(7,44,25,23,35,16)(8,45,26,24,36,9), (1,11)(2,10)(3,9)(4,16)(5,15)(6,14)(7,13)(8,12)(17,28)(18,27)(19,26)(20,25)(21,32)(22,31)(23,30)(24,29)(33,43)(34,42)(35,41)(36,48)(37,47)(38,46)(39,45)(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)(9,15)(10,14)(11,13)(17,21)(18,20)(22,24)(25,29)(26,28)(30,32)(34,40)(35,39)(36,38)(41,47)(42,46)(43,45), (1,46,27,17,37,10)(2,47,28,18,38,11)(3,48,29,19,39,12)(4,41,30,20,40,13)(5,42,31,21,33,14)(6,43,32,22,34,15)(7,44,25,23,35,16)(8,45,26,24,36,9), (1,11)(2,10)(3,9)(4,16)(5,15)(6,14)(7,13)(8,12)(17,28)(18,27)(19,26)(20,25)(21,32)(22,31)(23,30)(24,29)(33,43)(34,42)(35,41)(36,48)(37,47)(38,46)(39,45)(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),(9,15),(10,14),(11,13),(17,21),(18,20),(22,24),(25,29),(26,28),(30,32),(34,40),(35,39),(36,38),(41,47),(42,46),(43,45)], [(1,46,27,17,37,10),(2,47,28,18,38,11),(3,48,29,19,39,12),(4,41,30,20,40,13),(5,42,31,21,33,14),(6,43,32,22,34,15),(7,44,25,23,35,16),(8,45,26,24,36,9)], [(1,11),(2,10),(3,9),(4,16),(5,15),(6,14),(7,13),(8,12),(17,28),(18,27),(19,26),(20,25),(21,32),(22,31),(23,30),(24,29),(33,43),(34,42),(35,41),(36,48),(37,47),(38,46),(39,45),(40,44)])
Matrix representation ►G ⊆ GL4(𝔽73) generated by
| 0 | 0 | 32 | 0 |
| 0 | 0 | 0 | 32 |
| 57 | 0 | 32 | 0 |
| 0 | 57 | 0 | 32 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 72 | 0 |
| 0 | 1 | 0 | 72 |
| 7 | 66 | 59 | 14 |
| 7 | 14 | 59 | 45 |
| 7 | 66 | 66 | 7 |
| 7 | 14 | 66 | 59 |
| 0 | 0 | 68 | 5 |
| 0 | 0 | 10 | 5 |
| 34 | 39 | 0 | 0 |
| 5 | 39 | 0 | 0 |
G:=sub<GL(4,GF(73))| [0,0,57,0,0,0,0,57,32,0,32,0,0,32,0,32],[1,0,1,0,0,1,0,1,0,0,72,0,0,0,0,72],[7,7,7,7,66,14,66,14,59,59,66,66,14,45,7,59],[0,0,34,5,0,0,39,39,68,10,0,0,5,5,0,0] >;
36 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 | 8A | 8B | 8C | 8D | 8E | 12A | 12B | 12C | 12D | 12E | 24A | 24B | 24C | 24D |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 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 | 12 | 12 | 2 | 2 | 2 | 4 | 4 | 6 | 6 | 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 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D4 | D6 | D6 | D6 | D6 | D6 | S3×D4 | S3×D4 | D4○D8 | D8⋊15D6 |
| kernel | D8⋊15D6 | C8○D12 | C2×D24 | S3×D8 | Q8⋊3D6 | D24⋊C2 | D4⋊D6 | C3×C4○D8 | D4○D12 | C4○D8 | Dic6 | D12 | C3⋊D4 | C2×C8 | D8 | SD16 | Q16 | C4○D4 | C4 | C22 | C3 | C1 |
| # reps | 1 | 1 | 1 | 2 | 4 | 2 | 2 | 1 | 2 | 1 | 1 | 1 | 2 | 1 | 1 | 2 | 1 | 2 | 1 | 1 | 2 | 4 |
In GAP, Magma, Sage, TeX
D_8\rtimes_{15}D_6 % in TeX
G:=Group("D8:15D6"); // GroupNames label
G:=SmallGroup(192,1328);
// by ID
G=gap.SmallGroup(192,1328);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,477,387,570,185,438,235,102,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^8=b^2=c^6=d^2=1,b*a*b=d*a*d=a^-1,a*c=c*a,c*b*c^-1=a^4*b,d*b*d=a^2*b,d*c*d=c^-1>;
// generators/relations