metabelian, supersoluble, monomial
Aliases: D12.10D6, Q8⋊5S32, C3⋊C8⋊11D6, (C3×Q8)⋊6D6, C6.62(S3×D4), D6⋊D6⋊5C2, C3⋊4(Q8⋊3D6), Q8⋊2S3⋊3S3, C3⋊D24⋊14C2, C3⋊Dic3.59D4, C12⋊S3⋊6C22, C12.31D6⋊4C2, C12.26D6⋊1C2, C32⋊15(C8⋊C22), C12.18(C22×S3), (C3×C12).18C23, C2.22(Dic3⋊D6), (Q8×C32)⋊4C22, (C3×D12).18C22, C4.18(C2×S32), (C3×C3⋊C8)⋊9C22, (C2×C3⋊S3).24D4, (C3×Q8⋊2S3)⋊2C2, (C3×C6).133(C2×D4), (C4×C3⋊S3).18C22, SmallGroup(288,589)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D12.10D6
G = < a,b,c,d | a12=b2=1, c6=a6, d2=a3, bab=a-1, cac-1=a7, ad=da, cbc-1=dbd-1=a3b, dcd-1=a9c5 >
Subgroups: 826 in 156 conjugacy classes, 38 normal (16 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, C6, C6, C8, C2×C4, D4, Q8, C23, C32, Dic3, C12, C12, D6, C2×C6, M4(2), D8, SD16, C2×D4, C4○D4, C3×S3, C3⋊S3, C3×C6, C3⋊C8, C24, C4×S3, D12, D12, C3⋊D4, C3×D4, C3×Q8, C3×Q8, C22×S3, C8⋊C22, C3⋊Dic3, C3×C12, C3×C12, S32, S3×C6, C2×C3⋊S3, C2×C3⋊S3, C8⋊S3, D24, D4⋊S3, Q8⋊2S3, C3×SD16, S3×D4, Q8⋊3S3, C3×C3⋊C8, D6⋊S3, C3×D12, C4×C3⋊S3, C4×C3⋊S3, C12⋊S3, C12⋊S3, Q8×C32, C2×S32, Q8⋊3D6, C12.31D6, C3⋊D24, C3×Q8⋊2S3, D6⋊D6, C12.26D6, D12.10D6
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C22×S3, C8⋊C22, S32, S3×D4, C2×S32, Q8⋊3D6, Dic3⋊D6, D12.10D6
Character table of D12.10D6
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | 3B | 3C | 4A | 4B | 4C | 6A | 6B | 6C | 6D | 6E | 8A | 8B | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 24A | 24B | 24C | 24D | |
| size | 1 | 1 | 12 | 12 | 18 | 36 | 2 | 2 | 4 | 2 | 4 | 18 | 2 | 2 | 4 | 24 | 24 | 12 | 12 | 4 | 4 | 8 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 1 | 1 | -1 | 1 | -1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | -1 | 1 | -1 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | -1 | -1 | 1 | 1 | -1 | linear of order 2 |
| ρ3 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | linear of order 2 |
| ρ4 | 1 | 1 | -1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | -1 | 1 | 1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | linear of order 2 |
| ρ5 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | linear of order 2 |
| ρ6 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | 1 | -1 | linear of order 2 |
| ρ7 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | linear of order 2 |
| ρ8 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | 1 | 1 | -1 | -1 | 1 | -1 | -1 | 1 | -1 | -1 | 1 | linear of order 2 |
| ρ9 | 2 | 2 | 2 | 0 | 0 | 0 | -1 | 2 | -1 | 2 | 2 | 0 | 2 | -1 | -1 | -1 | 0 | 0 | 2 | -1 | 2 | 2 | -1 | -1 | -1 | -1 | 0 | -1 | -1 | 0 | orthogonal lifted from S3 |
| ρ10 | 2 | 2 | -2 | 0 | 0 | 0 | -1 | 2 | -1 | 2 | -2 | 0 | 2 | -1 | -1 | 1 | 0 | 0 | 2 | -1 | 2 | -2 | 1 | -1 | 1 | 1 | 0 | -1 | -1 | 0 | orthogonal lifted from D6 |
| ρ11 | 2 | 2 | 0 | -2 | 0 | 0 | 2 | -1 | -1 | 2 | -2 | 0 | -1 | 2 | -1 | 0 | 1 | 2 | 0 | 2 | -1 | 1 | 1 | -1 | 1 | -2 | -1 | 0 | 0 | -1 | orthogonal lifted from D6 |
| ρ12 | 2 | 2 | -2 | 0 | 0 | 0 | -1 | 2 | -1 | 2 | 2 | 0 | 2 | -1 | -1 | 1 | 0 | 0 | -2 | -1 | 2 | 2 | -1 | -1 | -1 | -1 | 0 | 1 | 1 | 0 | orthogonal lifted from D6 |
| ρ13 | 2 | 2 | 2 | 0 | 0 | 0 | -1 | 2 | -1 | 2 | -2 | 0 | 2 | -1 | -1 | -1 | 0 | 0 | -2 | -1 | 2 | -2 | 1 | -1 | 1 | 1 | 0 | 1 | 1 | 0 | orthogonal lifted from D6 |
| ρ14 | 2 | 2 | 0 | 2 | 0 | 0 | 2 | -1 | -1 | 2 | 2 | 0 | -1 | 2 | -1 | 0 | -1 | 2 | 0 | 2 | -1 | -1 | -1 | -1 | -1 | 2 | -1 | 0 | 0 | -1 | orthogonal lifted from S3 |
| ρ15 | 2 | 2 | 0 | -2 | 0 | 0 | 2 | -1 | -1 | 2 | 2 | 0 | -1 | 2 | -1 | 0 | 1 | -2 | 0 | 2 | -1 | -1 | -1 | -1 | -1 | 2 | 1 | 0 | 0 | 1 | orthogonal lifted from D6 |
| ρ16 | 2 | 2 | 0 | 2 | 0 | 0 | 2 | -1 | -1 | 2 | -2 | 0 | -1 | 2 | -1 | 0 | -1 | -2 | 0 | 2 | -1 | 1 | 1 | -1 | 1 | -2 | 1 | 0 | 0 | 1 | orthogonal lifted from D6 |
| ρ17 | 2 | 2 | 0 | 0 | 2 | 0 | 2 | 2 | 2 | -2 | 0 | -2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | -2 | -2 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ18 | 2 | 2 | 0 | 0 | -2 | 0 | 2 | 2 | 2 | -2 | 0 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | -2 | -2 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ19 | 4 | 4 | 0 | 0 | 0 | 0 | -2 | -2 | 1 | 4 | -4 | 0 | -2 | -2 | 1 | 0 | 0 | 0 | 0 | -2 | -2 | 2 | -1 | 1 | -1 | 2 | 0 | 0 | 0 | 0 | orthogonal lifted from C2×S32 |
| ρ20 | 4 | 4 | 0 | 0 | 0 | 0 | -2 | 4 | -2 | -4 | 0 | 0 | 4 | -2 | -2 | 0 | 0 | 0 | 0 | 2 | -4 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from S3×D4 |
| ρ21 | 4 | 4 | 0 | 0 | 0 | 0 | -2 | -2 | 1 | -4 | 0 | 0 | -2 | -2 | 1 | 0 | 0 | 0 | 0 | 2 | 2 | 0 | -3 | -1 | 3 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from Dic3⋊D6 |
| ρ22 | 4 | 4 | 0 | 0 | 0 | 0 | 4 | -2 | -2 | -4 | 0 | 0 | -2 | 4 | -2 | 0 | 0 | 0 | 0 | -4 | 2 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from S3×D4 |
| ρ23 | 4 | 4 | 0 | 0 | 0 | 0 | -2 | -2 | 1 | 4 | 4 | 0 | -2 | -2 | 1 | 0 | 0 | 0 | 0 | -2 | -2 | -2 | 1 | 1 | 1 | -2 | 0 | 0 | 0 | 0 | orthogonal lifted from S32 |
| ρ24 | 4 | -4 | 0 | 0 | 0 | 0 | 4 | 4 | 4 | 0 | 0 | 0 | -4 | -4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C8⋊C22 |
| ρ25 | 4 | 4 | 0 | 0 | 0 | 0 | -2 | -2 | 1 | -4 | 0 | 0 | -2 | -2 | 1 | 0 | 0 | 0 | 0 | 2 | 2 | 0 | 3 | -1 | -3 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from Dic3⋊D6 |
| ρ26 | 4 | -4 | 0 | 0 | 0 | 0 | -2 | 4 | -2 | 0 | 0 | 0 | -4 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | √6 | -√6 | 0 | orthogonal lifted from Q8⋊3D6 |
| ρ27 | 4 | -4 | 0 | 0 | 0 | 0 | -2 | 4 | -2 | 0 | 0 | 0 | -4 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -√6 | √6 | 0 | orthogonal lifted from Q8⋊3D6 |
| ρ28 | 4 | -4 | 0 | 0 | 0 | 0 | 4 | -2 | -2 | 0 | 0 | 0 | 2 | -4 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | √6 | 0 | 0 | -√6 | orthogonal lifted from Q8⋊3D6 |
| ρ29 | 4 | -4 | 0 | 0 | 0 | 0 | 4 | -2 | -2 | 0 | 0 | 0 | 2 | -4 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -√6 | 0 | 0 | √6 | orthogonal lifted from Q8⋊3D6 |
| ρ30 | 8 | -8 | 0 | 0 | 0 | 0 | -4 | -4 | 2 | 0 | 0 | 0 | 4 | 4 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal faithful, Schur index 2 |
►On 48 points
Generators in S48
(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 27)(2 26)(3 25)(4 36)(5 35)(6 34)(7 33)(8 32)(9 31)(10 30)(11 29)(12 28)(13 39)(14 38)(15 37)(16 48)(17 47)(18 46)(19 45)(20 44)(21 43)(22 42)(23 41)(24 40)
(1 24 11 22 9 20 7 18 5 16 3 14)(2 19 12 17 10 15 8 13 6 23 4 21)(25 47 27 37 29 39 31 41 33 43 35 45)(26 42 28 44 30 46 32 48 34 38 36 40)
(1 48 4 39 7 42 10 45)(2 37 5 40 8 43 11 46)(3 38 6 41 9 44 12 47)(13 36 16 27 19 30 22 33)(14 25 17 28 20 31 23 34)(15 26 18 29 21 32 24 35)
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), (1,27)(2,26)(3,25)(4,36)(5,35)(6,34)(7,33)(8,32)(9,31)(10,30)(11,29)(12,28)(13,39)(14,38)(15,37)(16,48)(17,47)(18,46)(19,45)(20,44)(21,43)(22,42)(23,41)(24,40), (1,24,11,22,9,20,7,18,5,16,3,14)(2,19,12,17,10,15,8,13,6,23,4,21)(25,47,27,37,29,39,31,41,33,43,35,45)(26,42,28,44,30,46,32,48,34,38,36,40), (1,48,4,39,7,42,10,45)(2,37,5,40,8,43,11,46)(3,38,6,41,9,44,12,47)(13,36,16,27,19,30,22,33)(14,25,17,28,20,31,23,34)(15,26,18,29,21,32,24,35)>;
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), (1,27)(2,26)(3,25)(4,36)(5,35)(6,34)(7,33)(8,32)(9,31)(10,30)(11,29)(12,28)(13,39)(14,38)(15,37)(16,48)(17,47)(18,46)(19,45)(20,44)(21,43)(22,42)(23,41)(24,40), (1,24,11,22,9,20,7,18,5,16,3,14)(2,19,12,17,10,15,8,13,6,23,4,21)(25,47,27,37,29,39,31,41,33,43,35,45)(26,42,28,44,30,46,32,48,34,38,36,40), (1,48,4,39,7,42,10,45)(2,37,5,40,8,43,11,46)(3,38,6,41,9,44,12,47)(13,36,16,27,19,30,22,33)(14,25,17,28,20,31,23,34)(15,26,18,29,21,32,24,35) );
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)], [(1,27),(2,26),(3,25),(4,36),(5,35),(6,34),(7,33),(8,32),(9,31),(10,30),(11,29),(12,28),(13,39),(14,38),(15,37),(16,48),(17,47),(18,46),(19,45),(20,44),(21,43),(22,42),(23,41),(24,40)], [(1,24,11,22,9,20,7,18,5,16,3,14),(2,19,12,17,10,15,8,13,6,23,4,21),(25,47,27,37,29,39,31,41,33,43,35,45),(26,42,28,44,30,46,32,48,34,38,36,40)], [(1,48,4,39,7,42,10,45),(2,37,5,40,8,43,11,46),(3,38,6,41,9,44,12,47),(13,36,16,27,19,30,22,33),(14,25,17,28,20,31,23,34),(15,26,18,29,21,32,24,35)]])
Matrix representation of D12.10D6 ►in GL8(𝔽73)
| 0 | 72 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 72 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 1 | 2 | 71 |
| 0 | 0 | 0 | 0 | 72 | 0 | 2 | 0 |
| 0 | 0 | 0 | 0 | 72 | 1 | 1 | 72 |
| 0 | 0 | 0 | 0 | 72 | 0 | 1 | 0 |
| 0 | 0 | 1 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 | 0 | 0 |
| 1 | 72 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 0 | 24 | 0 |
| 0 | 0 | 0 | 0 | 9 | 64 | 24 | 49 |
| 0 | 0 | 0 | 0 | 21 | 0 | 64 | 0 |
| 0 | 0 | 0 | 0 | 21 | 52 | 64 | 9 |
| 1 | 72 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 41 | 64 | 45 | 56 |
| 0 | 0 | 0 | 0 | 9 | 32 | 17 | 28 |
| 0 | 0 | 0 | 0 | 27 | 19 | 32 | 9 |
| 0 | 0 | 0 | 0 | 54 | 46 | 64 | 41 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 1 | 0 | 0 | 0 | 0 |
| 1 | 72 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 68 | 10 | 5 | 63 |
| 0 | 0 | 0 | 0 | 63 | 5 | 10 | 68 |
| 0 | 0 | 0 | 0 | 34 | 5 | 0 | 0 |
| 0 | 0 | 0 | 0 | 68 | 39 | 0 | 0 |
G:=sub<GL(8,GF(73))| [0,1,0,0,0,0,0,0,72,72,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,72,72,0,0,0,0,0,0,0,0,72,72,72,72,0,0,0,0,1,0,1,0,0,0,0,0,2,2,1,1,0,0,0,0,71,0,72,0],[0,0,1,0,0,0,0,0,0,0,72,72,0,0,0,0,1,0,0,0,0,0,0,0,72,72,0,0,0,0,0,0,0,0,0,0,9,9,21,21,0,0,0,0,0,64,0,52,0,0,0,0,24,24,64,64,0,0,0,0,0,49,0,9],[1,1,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,41,9,27,54,0,0,0,0,64,32,19,46,0,0,0,0,45,17,32,64,0,0,0,0,56,28,9,41],[0,0,1,1,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,0,0,68,63,34,68,0,0,0,0,10,5,5,39,0,0,0,0,5,10,0,0,0,0,0,0,63,68,0,0] >;
D12.10D6 in GAP, Magma, Sage, TeX
D_{12}._{10}D_6 % in TeX
G:=Group("D12.10D6"); // GroupNames label
G:=SmallGroup(288,589);
// by ID
G=gap.SmallGroup(288,589);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,254,303,100,675,346,185,80,1356,9414]);
// Polycyclic
G:=Group<a,b,c,d|a^12=b^2=1,c^6=a^6,d^2=a^3,b*a*b=a^-1,c*a*c^-1=a^7,a*d=d*a,c*b*c^-1=d*b*d^-1=a^3*b,d*c*d^-1=a^9*c^5>;
// generators/relations
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