metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D8⋊11D6, Q16⋊10D6, SD16⋊15D6, D12.46D4, C24.43C23, C12.17C24, Dic6.46D4, D12.12C23, Dic6.11C23, C4○D8⋊5S3, C4○D4⋊5D6, (C2×C8)⋊14D6, D4○D12⋊6C2, D8⋊S3⋊6C2, Q8○D12⋊5C2, C3⋊D4.2D4, C3⋊C8.8C23, C8○D12⋊10C2, D4⋊D6⋊8C2, D4⋊S3⋊4C22, (S3×SD16)⋊6C2, D6.30(C2×D4), Q16⋊S3⋊6C2, C4.144(S3×D4), C3⋊3(D4○SD16), (S3×Q8)⋊2C22, C22.9(S3×D4), (S3×C8)⋊10C22, (C2×C24)⋊17C22, Q8.14D6⋊7C2, Q8.7D6⋊6C2, C12.350(C2×D4), (C3×D8)⋊16C22, C8.17(C22×S3), C4.17(S3×C23), D4.S3⋊3C22, C3⋊Q16⋊2C22, (S3×D4).2C22, C24⋊C2⋊21C22, C8⋊S3⋊16C22, D4⋊2S3⋊2C22, (C4×S3).10C23, Dic3.35(C2×D4), Q8⋊2S3⋊3C22, (C3×Q16)⋊14C22, D4.11(C22×S3), (C3×D4).11C23, C6.118(C22×D4), Q8.21(C22×S3), (C3×Q8).11C23, (C2×C12).534C23, (C3×SD16)⋊16C22, C4○D12.55C22, (C2×Dic6)⋊38C22, C4.Dic3⋊31C22, Q8⋊3S3.2C22, (C2×D12).181C22, C2.91(C2×S3×D4), (C3×C4○D8)⋊7C2, (C2×C6).14(C2×D4), (C2×C24⋊C2)⋊27C2, (C3×C4○D4)⋊4C22, (C2×C4).233(C22×S3), SmallGroup(192,1329)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 728 in 258 conjugacy classes, 99 normal (53 characteristic)
C1, C2, C2 [×7], C3, C4 [×2], C4 [×6], C22, C22 [×9], S3 [×4], C6, C6 [×3], C8 [×2], C8 [×2], C2×C4, C2×C4 [×11], D4 [×2], D4 [×14], Q8 [×2], Q8 [×6], C23 [×3], Dic3 [×2], Dic3 [×2], C12 [×2], C12 [×2], D6 [×2], D6 [×5], C2×C6, C2×C6 [×2], C2×C8, C2×C8 [×2], M4(2) [×3], D8, D8 [×2], SD16 [×2], SD16 [×8], Q16, Q16 [×2], C2×D4 [×6], C2×Q8 [×4], C4○D4 [×2], C4○D4 [×9], C3⋊C8 [×2], C24 [×2], Dic6, Dic6 [×2], Dic6 [×3], C4×S3 [×2], C4×S3 [×4], D12, D12 [×2], D12 [×3], C2×Dic3 [×3], C3⋊D4 [×2], C3⋊D4 [×4], C2×C12, C2×C12 [×2], C3×D4 [×2], C3×D4 [×2], C3×Q8 [×2], C22×S3 [×3], C8○D4, C2×SD16 [×3], C4○D8, C4○D8 [×2], C8⋊C22 [×3], C8.C22 [×3], 2+ (1+4), 2- (1+4), S3×C8 [×2], C8⋊S3 [×2], C24⋊C2 [×4], C4.Dic3, D4⋊S3 [×2], D4.S3 [×2], Q8⋊2S3 [×2], C3⋊Q16 [×2], C2×C24, C3×D8, C3×SD16 [×2], C3×Q16, C2×Dic6, C2×Dic6, C2×D12, C2×D12, C4○D12, C4○D12 [×2], S3×D4 [×2], S3×D4 [×2], D4⋊2S3 [×2], D4⋊2S3 [×2], S3×Q8 [×2], Q8⋊3S3 [×2], C3×C4○D4 [×2], D4○SD16, C8○D12, C2×C24⋊C2, D8⋊S3 [×2], S3×SD16 [×2], Q8.7D6 [×2], Q16⋊S3 [×2], D4⋊D6, Q8.14D6, C3×C4○D8, D4○D12, Q8○D12, D8⋊11D6
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⋊11D6
Generators and relations
G = < a,b,c,d | a8=b2=c6=d2=1, bab=a-1, ac=ca, dad=a3, cbc-1=a4b, 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)
(1 8)(2 7)(3 6)(4 5)(9 10)(11 16)(12 15)(13 14)(17 18)(19 24)(20 23)(21 22)(25 30)(26 29)(27 28)(31 32)(33 38)(34 37)(35 36)(39 40)(41 48)(42 47)(43 46)(44 45)
(1 16 45 26 18 34)(2 9 46 27 19 35)(3 10 47 28 20 36)(4 11 48 29 21 37)(5 12 41 30 22 38)(6 13 42 31 23 39)(7 14 43 32 24 40)(8 15 44 25 17 33)
(1 47)(2 42)(3 45)(4 48)(5 43)(6 46)(7 41)(8 44)(9 13)(10 16)(12 14)(18 20)(19 23)(22 24)(25 33)(26 36)(27 39)(28 34)(29 37)(30 40)(31 35)(32 38)
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), (1,8)(2,7)(3,6)(4,5)(9,10)(11,16)(12,15)(13,14)(17,18)(19,24)(20,23)(21,22)(25,30)(26,29)(27,28)(31,32)(33,38)(34,37)(35,36)(39,40)(41,48)(42,47)(43,46)(44,45), (1,16,45,26,18,34)(2,9,46,27,19,35)(3,10,47,28,20,36)(4,11,48,29,21,37)(5,12,41,30,22,38)(6,13,42,31,23,39)(7,14,43,32,24,40)(8,15,44,25,17,33), (1,47)(2,42)(3,45)(4,48)(5,43)(6,46)(7,41)(8,44)(9,13)(10,16)(12,14)(18,20)(19,23)(22,24)(25,33)(26,36)(27,39)(28,34)(29,37)(30,40)(31,35)(32,38)>;
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), (1,8)(2,7)(3,6)(4,5)(9,10)(11,16)(12,15)(13,14)(17,18)(19,24)(20,23)(21,22)(25,30)(26,29)(27,28)(31,32)(33,38)(34,37)(35,36)(39,40)(41,48)(42,47)(43,46)(44,45), (1,16,45,26,18,34)(2,9,46,27,19,35)(3,10,47,28,20,36)(4,11,48,29,21,37)(5,12,41,30,22,38)(6,13,42,31,23,39)(7,14,43,32,24,40)(8,15,44,25,17,33), (1,47)(2,42)(3,45)(4,48)(5,43)(6,46)(7,41)(8,44)(9,13)(10,16)(12,14)(18,20)(19,23)(22,24)(25,33)(26,36)(27,39)(28,34)(29,37)(30,40)(31,35)(32,38) );
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)], [(1,8),(2,7),(3,6),(4,5),(9,10),(11,16),(12,15),(13,14),(17,18),(19,24),(20,23),(21,22),(25,30),(26,29),(27,28),(31,32),(33,38),(34,37),(35,36),(39,40),(41,48),(42,47),(43,46),(44,45)], [(1,16,45,26,18,34),(2,9,46,27,19,35),(3,10,47,28,20,36),(4,11,48,29,21,37),(5,12,41,30,22,38),(6,13,42,31,23,39),(7,14,43,32,24,40),(8,15,44,25,17,33)], [(1,47),(2,42),(3,45),(4,48),(5,43),(6,46),(7,41),(8,44),(9,13),(10,16),(12,14),(18,20),(19,23),(22,24),(25,33),(26,36),(27,39),(28,34),(29,37),(30,40),(31,35),(32,38)])
Matrix representation ►G ⊆ GL4(𝔽73) generated by
| 42 | 62 | 31 | 11 |
| 11 | 31 | 62 | 42 |
| 42 | 62 | 42 | 62 |
| 11 | 31 | 11 | 31 |
| 42 | 62 | 31 | 11 |
| 11 | 31 | 62 | 42 |
| 31 | 11 | 31 | 11 |
| 62 | 42 | 62 | 42 |
| 0 | 0 | 14 | 66 |
| 0 | 0 | 7 | 7 |
| 59 | 7 | 0 | 0 |
| 66 | 66 | 0 | 0 |
| 0 | 0 | 1 | 72 |
| 0 | 0 | 0 | 72 |
| 1 | 72 | 0 | 0 |
| 0 | 72 | 0 | 0 |
G:=sub<GL(4,GF(73))| [42,11,42,11,62,31,62,31,31,62,42,11,11,42,62,31],[42,11,31,62,62,31,11,42,31,62,31,62,11,42,11,42],[0,0,59,66,0,0,7,66,14,7,0,0,66,7,0,0],[0,0,1,0,0,0,72,72,1,0,0,0,72,72,0,0] >;
36 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 | 8A | 8B | 8C | 8D | 8E | 12A | 12B | 12C | 12D | 12E | 24A | 24B | 24C | 24D |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 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 | 2 | 2 | 2 | 4 | 4 | 6 | 6 | 12 | 12 | 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 | 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 | C2 | C2 | C2 | S3 | D4 | D4 | D4 | D6 | D6 | D6 | D6 | D6 | S3×D4 | S3×D4 | D4○SD16 | D8⋊11D6 |
| kernel | D8⋊11D6 | C8○D12 | C2×C24⋊C2 | D8⋊S3 | S3×SD16 | Q8.7D6 | Q16⋊S3 | D4⋊D6 | Q8.14D6 | C3×C4○D8 | D4○D12 | Q8○D12 | C4○D8 | Dic6 | D12 | C3⋊D4 | C2×C8 | D8 | SD16 | Q16 | C4○D4 | C4 | C22 | C3 | C1 |
| # reps | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 2 | 1 | 2 | 1 | 1 | 2 | 4 |
In GAP, Magma, Sage, TeX
D_8\rtimes_{11}D_6 % in TeX
G:=Group("D8:11D6"); // GroupNames label
G:=SmallGroup(192,1329);
// by ID
G=gap.SmallGroup(192,1329);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,477,387,570,185,136,438,235,102,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^8=b^2=c^6=d^2=1,b*a*b=a^-1,a*c=c*a,d*a*d=a^3,c*b*c^-1=a^4*b,d*b*d=a^6*b,d*c*d=c^-1>;
// generators/relations