metabelian, supersoluble, monomial
Aliases: D12.15D6, Dic6.13D6, C3⋊C8.11D6, Q8.17S32, C3⋊Q16⋊6S3, C6.66(S3×D4), Q8⋊2S3⋊6S3, (C3×Q8).35D6, C3⋊4(D4.D6), C3⋊3(Q16⋊S3), C3⋊Dic3.24D4, D12⋊S3.2C2, C12.31D6⋊2C2, (C3×C12).28C23, C12.28(C22×S3), C32⋊3Q16⋊14C2, D12.S3⋊10C2, C2.26(Dic3⋊D6), (C3×D12).24C22, C32⋊15(C8.C22), (C3×Dic6).23C22, (Q8×C32).10C22, C32⋊4Q8.15C22, C4.28(C2×S32), (Q8×C3⋊S3)⋊2C2, (C2×C3⋊S3).61D4, (C3×C3⋊Q16)⋊8C2, (C3×Q8⋊2S3)⋊4C2, (C3×C6).143(C2×D4), (C3×C3⋊C8).13C22, (C4×C3⋊S3).22C22, SmallGroup(288,599)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D12.15D6
G = < a,b,c,d | a12=b2=1, c6=a6, d2=a9, bab=a-1, cac-1=a7, ad=da, cbc-1=dbd-1=a3b, dcd-1=a9c5 >
Subgroups: 578 in 139 conjugacy classes, 38 normal (all characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, C6, C6, C8, C2×C4, D4, Q8, Q8, C32, Dic3, C12, C12, D6, C2×C6, M4(2), SD16, Q16, C2×Q8, C4○D4, C3×S3, C3⋊S3, C3×C6, C3⋊C8, C24, Dic6, Dic6, C4×S3, D12, D12, C2×Dic3, C3⋊D4, C3×D4, C3×Q8, C3×Q8, C8.C22, C3×Dic3, C3⋊Dic3, C3⋊Dic3, C3×C12, C3×C12, S3×C6, C2×C3⋊S3, C8⋊S3, C24⋊C2, Dic12, D4.S3, Q8⋊2S3, C3⋊Q16, C3⋊Q16, C3×SD16, C3×Q16, D4⋊2S3, S3×Q8, Q8⋊3S3, C3×C3⋊C8, S3×Dic3, C3⋊D12, C3×Dic6, C3×D12, C32⋊4Q8, C32⋊4Q8, C4×C3⋊S3, C4×C3⋊S3, Q8×C32, D4.D6, Q16⋊S3, C12.31D6, D12.S3, C32⋊3Q16, C3×Q8⋊2S3, C3×C3⋊Q16, D12⋊S3, Q8×C3⋊S3, D12.15D6
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C22×S3, C8.C22, S32, S3×D4, C2×S32, D4.D6, Q16⋊S3, Dic3⋊D6, D12.15D6
Character table of D12.15D6
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 4A | 4B | 4C | 4D | 4E | 6A | 6B | 6C | 6D | 8A | 8B | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 24A | 24B | 24C | 24D | |
| size | 1 | 1 | 12 | 18 | 2 | 2 | 4 | 2 | 4 | 12 | 18 | 36 | 2 | 2 | 4 | 24 | 12 | 12 | 4 | 4 | 8 | 8 | 8 | 8 | 8 | 24 | 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 | 2 | -1 | -1 | 2 | 2 | 0 | 0 | 0 | -1 | 2 | -1 | 1 | 0 | -2 | 2 | -1 | -1 | 2 | -1 | -1 | -1 | 0 | 1 | 1 | 0 | 0 | orthogonal lifted from D6 |
| ρ10 | 2 | 2 | 0 | 0 | -1 | 2 | -1 | 2 | 2 | 2 | 0 | 0 | 2 | -1 | -1 | 0 | 2 | 0 | -1 | 2 | -1 | -1 | 2 | -1 | -1 | -1 | 0 | 0 | -1 | -1 | orthogonal lifted from S3 |
| ρ11 | 2 | 2 | 0 | 0 | -1 | 2 | -1 | 2 | 2 | -2 | 0 | 0 | 2 | -1 | -1 | 0 | -2 | 0 | -1 | 2 | -1 | -1 | 2 | -1 | -1 | 1 | 0 | 0 | 1 | 1 | orthogonal lifted from D6 |
| ρ12 | 2 | 2 | 0 | 2 | 2 | 2 | 2 | -2 | 0 | 0 | -2 | 0 | 2 | 2 | 2 | 0 | 0 | 0 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ13 | 2 | 2 | -2 | 0 | 2 | -1 | -1 | 2 | -2 | 0 | 0 | 0 | -1 | 2 | -1 | 1 | 0 | 2 | 2 | -1 | -1 | -2 | 1 | 1 | 1 | 0 | -1 | -1 | 0 | 0 | orthogonal lifted from D6 |
| ρ14 | 2 | 2 | 0 | -2 | 2 | 2 | 2 | -2 | 0 | 0 | 2 | 0 | 2 | 2 | 2 | 0 | 0 | 0 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ15 | 2 | 2 | 2 | 0 | 2 | -1 | -1 | 2 | -2 | 0 | 0 | 0 | -1 | 2 | -1 | -1 | 0 | -2 | 2 | -1 | -1 | -2 | 1 | 1 | 1 | 0 | 1 | 1 | 0 | 0 | orthogonal lifted from D6 |
| ρ16 | 2 | 2 | 0 | 0 | -1 | 2 | -1 | 2 | -2 | 2 | 0 | 0 | 2 | -1 | -1 | 0 | -2 | 0 | -1 | 2 | -1 | 1 | -2 | 1 | 1 | -1 | 0 | 0 | 1 | 1 | orthogonal lifted from D6 |
| ρ17 | 2 | 2 | 0 | 0 | -1 | 2 | -1 | 2 | -2 | -2 | 0 | 0 | 2 | -1 | -1 | 0 | 2 | 0 | -1 | 2 | -1 | 1 | -2 | 1 | 1 | 1 | 0 | 0 | -1 | -1 | orthogonal lifted from D6 |
| ρ18 | 2 | 2 | 2 | 0 | 2 | -1 | -1 | 2 | 2 | 0 | 0 | 0 | -1 | 2 | -1 | -1 | 0 | 2 | 2 | -1 | -1 | 2 | -1 | -1 | -1 | 0 | -1 | -1 | 0 | 0 | orthogonal lifted from S3 |
| ρ19 | 4 | 4 | 0 | 0 | -2 | -2 | 1 | -4 | 0 | 0 | 0 | 0 | -2 | -2 | 1 | 0 | 0 | 0 | 2 | 2 | -1 | 0 | 0 | -3 | 3 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from Dic3⋊D6 |
| ρ20 | 4 | 4 | 0 | 0 | -2 | -2 | 1 | 4 | -4 | 0 | 0 | 0 | -2 | -2 | 1 | 0 | 0 | 0 | -2 | -2 | 1 | 2 | 2 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C2×S32 |
| ρ21 | 4 | 4 | 0 | 0 | -2 | -2 | 1 | -4 | 0 | 0 | 0 | 0 | -2 | -2 | 1 | 0 | 0 | 0 | 2 | 2 | -1 | 0 | 0 | 3 | -3 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from Dic3⋊D6 |
| ρ22 | 4 | 4 | 0 | 0 | -2 | 4 | -2 | -4 | 0 | 0 | 0 | 0 | 4 | -2 | -2 | 0 | 0 | 0 | 2 | -4 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from S3×D4 |
| ρ23 | 4 | 4 | 0 | 0 | -2 | -2 | 1 | 4 | 4 | 0 | 0 | 0 | -2 | -2 | 1 | 0 | 0 | 0 | -2 | -2 | 1 | -2 | -2 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from S32 |
| ρ24 | 4 | 4 | 0 | 0 | 4 | -2 | -2 | -4 | 0 | 0 | 0 | 0 | -2 | 4 | -2 | 0 | 0 | 0 | -4 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from S3×D4 |
| ρ25 | 4 | -4 | 0 | 0 | 4 | 4 | 4 | 0 | 0 | 0 | 0 | 0 | -4 | -4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from C8.C22, Schur index 2 |
| ρ26 | 4 | -4 | 0 | 0 | 4 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 2 | -4 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | √6 | -√6 | 0 | 0 | symplectic lifted from D4.D6, Schur index 2 |
| ρ27 | 4 | -4 | 0 | 0 | 4 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 2 | -4 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -√6 | √6 | 0 | 0 | symplectic lifted from D4.D6, Schur index 2 |
| ρ28 | 4 | -4 | 0 | 0 | -2 | 4 | -2 | 0 | 0 | 0 | 0 | 0 | -4 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -√-6 | √-6 | complex lifted from Q16⋊S3 |
| ρ29 | 4 | -4 | 0 | 0 | -2 | 4 | -2 | 0 | 0 | 0 | 0 | 0 | -4 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | √-6 | -√-6 | complex lifted from Q16⋊S3 |
| ρ30 | 8 | -8 | 0 | 0 | -4 | -4 | 2 | 0 | 0 | 0 | 0 | 0 | 4 | 4 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic faithful, Schur index 2 |
►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 36)(2 35)(3 34)(4 33)(5 32)(6 31)(7 30)(8 29)(9 28)(10 27)(11 26)(12 25)(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 17 3 19 5 21 7 23 9 13 11 15)(2 24 4 14 6 16 8 18 10 20 12 22)(25 39 35 37 33 47 31 45 29 43 27 41)(26 46 36 44 34 42 32 40 30 38 28 48)
(1 38 10 47 7 44 4 41)(2 39 11 48 8 45 5 42)(3 40 12 37 9 46 6 43)(13 29 22 26 19 35 16 32)(14 30 23 27 20 36 17 33)(15 31 24 28 21 25 18 34)
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), (1,36)(2,35)(3,34)(4,33)(5,32)(6,31)(7,30)(8,29)(9,28)(10,27)(11,26)(12,25)(13,39)(14,38)(15,37)(16,48)(17,47)(18,46)(19,45)(20,44)(21,43)(22,42)(23,41)(24,40), (1,17,3,19,5,21,7,23,9,13,11,15)(2,24,4,14,6,16,8,18,10,20,12,22)(25,39,35,37,33,47,31,45,29,43,27,41)(26,46,36,44,34,42,32,40,30,38,28,48), (1,38,10,47,7,44,4,41)(2,39,11,48,8,45,5,42)(3,40,12,37,9,46,6,43)(13,29,22,26,19,35,16,32)(14,30,23,27,20,36,17,33)(15,31,24,28,21,25,18,34)>;
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), (1,36)(2,35)(3,34)(4,33)(5,32)(6,31)(7,30)(8,29)(9,28)(10,27)(11,26)(12,25)(13,39)(14,38)(15,37)(16,48)(17,47)(18,46)(19,45)(20,44)(21,43)(22,42)(23,41)(24,40), (1,17,3,19,5,21,7,23,9,13,11,15)(2,24,4,14,6,16,8,18,10,20,12,22)(25,39,35,37,33,47,31,45,29,43,27,41)(26,46,36,44,34,42,32,40,30,38,28,48), (1,38,10,47,7,44,4,41)(2,39,11,48,8,45,5,42)(3,40,12,37,9,46,6,43)(13,29,22,26,19,35,16,32)(14,30,23,27,20,36,17,33)(15,31,24,28,21,25,18,34) );
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)], [(1,36),(2,35),(3,34),(4,33),(5,32),(6,31),(7,30),(8,29),(9,28),(10,27),(11,26),(12,25),(13,39),(14,38),(15,37),(16,48),(17,47),(18,46),(19,45),(20,44),(21,43),(22,42),(23,41),(24,40)], [(1,17,3,19,5,21,7,23,9,13,11,15),(2,24,4,14,6,16,8,18,10,20,12,22),(25,39,35,37,33,47,31,45,29,43,27,41),(26,46,36,44,34,42,32,40,30,38,28,48)], [(1,38,10,47,7,44,4,41),(2,39,11,48,8,45,5,42),(3,40,12,37,9,46,6,43),(13,29,22,26,19,35,16,32),(14,30,23,27,20,36,17,33),(15,31,24,28,21,25,18,34)]])
Matrix representation of D12.15D6 ►in GL8(𝔽73)
| 0 | 72 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 72 | 0 | 0 | 0 | 0 | 0 | 0 |
| 50 | 0 | 72 | 72 | 0 | 0 | 0 | 0 |
| 0 | 23 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 71 |
| 0 | 0 | 0 | 0 | 72 | 72 | 2 | 2 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 72 |
| 0 | 0 | 0 | 0 | 72 | 72 | 1 | 1 |
| 23 | 23 | 1 | 2 | 0 | 0 | 0 | 0 |
| 23 | 23 | 2 | 1 | 0 | 0 | 0 | 0 |
| 12 | 13 | 50 | 50 | 0 | 0 | 0 | 0 |
| 13 | 12 | 50 | 50 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 63 | 68 |
| 0 | 0 | 0 | 0 | 0 | 0 | 5 | 10 |
| 0 | 0 | 0 | 0 | 68 | 34 | 0 | 0 |
| 0 | 0 | 0 | 0 | 39 | 5 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 72 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 23 | 0 | 1 | 0 | 0 | 0 | 0 |
| 50 | 0 | 72 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 46 | 0 | 54 | 0 |
| 0 | 0 | 0 | 0 | 0 | 46 | 0 | 54 |
| 0 | 0 | 0 | 0 | 0 | 0 | 27 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 27 |
| 50 | 50 | 72 | 71 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 72 | 0 | 0 | 0 | 0 |
| 30 | 31 | 0 | 23 | 0 | 0 | 0 | 0 |
| 30 | 30 | 0 | 23 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 62 | 51 | 11 | 22 |
| 0 | 0 | 0 | 0 | 22 | 11 | 51 | 62 |
| 0 | 0 | 0 | 0 | 31 | 62 | 0 | 0 |
| 0 | 0 | 0 | 0 | 11 | 42 | 0 | 0 |
G:=sub<GL(8,GF(73))| [0,1,50,0,0,0,0,0,72,72,0,23,0,0,0,0,0,0,72,1,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,0,72,0,72,0,0,0,0,1,72,1,72,0,0,0,0,0,2,0,1,0,0,0,0,71,2,72,1],[23,23,12,13,0,0,0,0,23,23,13,12,0,0,0,0,1,2,50,50,0,0,0,0,2,1,50,50,0,0,0,0,0,0,0,0,0,0,68,39,0,0,0,0,0,0,34,5,0,0,0,0,63,5,0,0,0,0,0,0,68,10,0,0],[0,1,0,50,0,0,0,0,72,72,23,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,1,72,0,0,0,0,0,0,0,0,46,0,0,0,0,0,0,0,0,46,0,0,0,0,0,0,54,0,27,0,0,0,0,0,0,54,0,27],[50,0,30,30,0,0,0,0,50,0,31,30,0,0,0,0,72,1,0,0,0,0,0,0,71,72,23,23,0,0,0,0,0,0,0,0,62,22,31,11,0,0,0,0,51,11,62,42,0,0,0,0,11,51,0,0,0,0,0,0,22,62,0,0] >;
D12.15D6 in GAP, Magma, Sage, TeX
D_{12}._{15}D_6 % in TeX
G:=Group("D12.15D6"); // GroupNames label
G:=SmallGroup(288,599);
// by ID
G=gap.SmallGroup(288,599);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,120,254,303,100,675,346,185,80,1356,9414]);
// Polycyclic
G:=Group<a,b,c,d|a^12=b^2=1,c^6=a^6,d^2=a^9,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
Export