metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D4.9D12, D12.34D4, Q8.14D12, C42.25D6, Dic6.34D4, M4(2).7D6, C4≀C2⋊3S3, (C3×D4).4D4, C12.5(C2×D4), (C3×Q8).4D4, C4○D4.19D6, C4.11(C2×D12), C8.D6⋊9C2, C4.127(S3×D4), Q8○D12.1C2, C42⋊4S3⋊9C2, C6.29C22≀C2, C12⋊2Q8⋊10C2, Q8.14D6⋊2C2, (C2×Dic3).2D4, C22.31(S3×D4), C12.47D4⋊1C2, C3⋊2(D4.10D4), (C4×C12).52C22, C2.32(D6⋊D4), (C2×C12).266C23, C4○D12.15C22, (C2×Dic6).76C22, (C3×M4(2)).4C22, C4.Dic3.10C22, (C3×C4≀C2)⋊3C2, (C2×C6).28(C2×D4), (C3×C4○D4).7C22, (C2×C4).111(C22×S3), SmallGroup(192,385)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Q8.14D12
G = < a,b,c,d | a4=1, b2=c12=d2=a2, bab-1=cac-1=dad-1=a-1, cbc-1=dbd-1=ab, dcd-1=c11 >
Subgroups: 416 in 142 conjugacy classes, 39 normal (37 characteristic)
C1, C2, C2 [×3], C3, C4 [×2], C4 [×7], C22, C22 [×2], S3, C6, C6 [×2], C8 [×2], C2×C4, C2×C4 [×10], D4, D4 [×5], Q8, Q8 [×7], Dic3 [×4], C12 [×2], C12 [×3], D6, C2×C6, C2×C6, C42, C4⋊C4 [×2], M4(2), M4(2), SD16 [×2], Q16 [×2], C2×Q8 [×4], C4○D4, C4○D4 [×5], C3⋊C8, C24, Dic6, Dic6 [×6], C4×S3 [×3], D12, C2×Dic3 [×2], C2×Dic3 [×3], C3⋊D4 [×3], C2×C12, C2×C12 [×2], C3×D4, C3×D4, C3×Q8, C4.10D4, C4≀C2, C4≀C2, C4⋊Q8, C8.C22 [×2], 2- 1+4, C24⋊C2, Dic12, C4.Dic3, C4⋊Dic3 [×2], D4.S3, C3⋊Q16, C4×C12, C3×M4(2), C2×Dic6 [×2], C2×Dic6, C4○D12, C4○D12, D4⋊2S3 [×3], S3×Q8, C3×C4○D4, D4.10D4, C42⋊4S3, C12.47D4, C3×C4≀C2, C12⋊2Q8, C8.D6, Q8.14D6, Q8○D12, Q8.14D12
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×6], C23, D6 [×3], C2×D4 [×3], D12 [×2], C22×S3, C22≀C2, C2×D12, S3×D4 [×2], D4.10D4, D6⋊D4, Q8.14D12
Character table of Q8.14D12
| class | 1 | 2A | 2B | 2C | 2D | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 6A | 6B | 6C | 8A | 8B | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 24A | 24B | |
| size | 1 | 1 | 2 | 4 | 12 | 2 | 2 | 2 | 4 | 4 | 4 | 12 | 12 | 12 | 24 | 2 | 4 | 8 | 8 | 24 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | |
| 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 | -2 | 0 | -1 | 2 | 2 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | -1 | -1 | 1 | -2 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | orthogonal lifted from D6 |
| ρ10 | 2 | 2 | -2 | 0 | 2 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 2 | 2 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ11 | 2 | 2 | 2 | 0 | 0 | 2 | -2 | -2 | 0 | 0 | 0 | -2 | 0 | 2 | 0 | 2 | 2 | 0 | 0 | 0 | -2 | -2 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ12 | 2 | 2 | 2 | -2 | 0 | -1 | 2 | 2 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | -1 | -1 | 1 | 2 | 0 | -1 | -1 | 1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | orthogonal lifted from D6 |
| ρ13 | 2 | 2 | 2 | 2 | 0 | -1 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | 2 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ14 | 2 | 2 | 2 | 0 | 0 | 2 | -2 | -2 | 0 | 0 | 0 | 2 | 0 | -2 | 0 | 2 | 2 | 0 | 0 | 0 | -2 | -2 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ15 | 2 | 2 | -2 | -2 | 0 | 2 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | -2 | 0 | 0 | -2 | -2 | 0 | 2 | 0 | 0 | 0 | 2 | 0 | 0 | orthogonal lifted from D4 |
| ρ16 | 2 | 2 | 2 | 2 | 0 | -1 | 2 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -2 | 0 | -1 | -1 | 1 | -1 | 1 | 1 | 1 | -1 | 1 | 1 | orthogonal lifted from D6 |
| ρ17 | 2 | 2 | -2 | 2 | 0 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | 2 | 0 | 0 | -2 | -2 | 0 | 2 | 0 | 0 | 0 | -2 | 0 | 0 | orthogonal lifted from D4 |
| ρ18 | 2 | 2 | -2 | 0 | -2 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 2 | 2 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ19 | 2 | 2 | -2 | 2 | 0 | -1 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 1 | -1 | 0 | 0 | 1 | 1 | -√3 | -1 | √3 | -√3 | √3 | 1 | -√3 | √3 | orthogonal lifted from D12 |
| ρ20 | 2 | 2 | -2 | -2 | 0 | -1 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 1 | 1 | 0 | 0 | 1 | 1 | √3 | -1 | -√3 | √3 | -√3 | -1 | -√3 | √3 | orthogonal lifted from D12 |
| ρ21 | 2 | 2 | -2 | 2 | 0 | -1 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 1 | -1 | 0 | 0 | 1 | 1 | √3 | -1 | -√3 | √3 | -√3 | 1 | √3 | -√3 | orthogonal lifted from D12 |
| ρ22 | 2 | 2 | -2 | -2 | 0 | -1 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 1 | 1 | 0 | 0 | 1 | 1 | -√3 | -1 | √3 | -√3 | √3 | -1 | √3 | -√3 | orthogonal lifted from D12 |
| ρ23 | 4 | 4 | 4 | 0 | 0 | -2 | -4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | 0 | 0 | 0 | 2 | 2 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from S3×D4 |
| ρ24 | 4 | 4 | -4 | 0 | 0 | -2 | -4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | -2 | -2 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from S3×D4 |
| ρ25 | 4 | -4 | 0 | 0 | 0 | 4 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 0 | 2 | 2 | -2 | 0 | 0 | 0 | symplectic lifted from D4.10D4, Schur index 2 |
| ρ26 | 4 | -4 | 0 | 0 | 0 | 4 | 0 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | -2 | -2 | 2 | 0 | 0 | 0 | symplectic lifted from D4.10D4, Schur index 2 |
| ρ27 | 4 | -4 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | -2√3 | 2√3 | -1+√3 | 0 | 1+√3 | 1-√3 | -1-√3 | 0 | 0 | 0 | symplectic faithful, Schur index 2 |
| ρ28 | 4 | -4 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | -2√3 | 2√3 | 1-√3 | 0 | -1-√3 | -1+√3 | 1+√3 | 0 | 0 | 0 | symplectic faithful, Schur index 2 |
| ρ29 | 4 | -4 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 2√3 | -2√3 | 1+√3 | 0 | -1+√3 | -1-√3 | 1-√3 | 0 | 0 | 0 | symplectic faithful, Schur index 2 |
| ρ30 | 4 | -4 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 2√3 | -2√3 | -1-√3 | 0 | 1-√3 | 1+√3 | -1+√3 | 0 | 0 | 0 | symplectic faithful, Schur index 2 |
►On 48 points
(1 7 13 19)(2 20 14 8)(3 9 15 21)(4 22 16 10)(5 11 17 23)(6 24 18 12)(25 31 37 43)(26 44 38 32)(27 33 39 45)(28 46 40 34)(29 35 41 47)(30 48 42 36)
(1 4 13 16)(2 11 14 23)(3 6 15 18)(5 8 17 20)(7 10 19 22)(9 12 21 24)(25 28 37 40)(26 35 38 47)(27 30 39 42)(29 32 41 44)(31 34 43 46)(33 36 45 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 29 13 41)(2 40 14 28)(3 27 15 39)(4 38 16 26)(5 25 17 37)(6 36 18 48)(7 47 19 35)(8 34 20 46)(9 45 21 33)(10 32 22 44)(11 43 23 31)(12 30 24 42)
G:=sub<Sym(48)| (1,7,13,19)(2,20,14,8)(3,9,15,21)(4,22,16,10)(5,11,17,23)(6,24,18,12)(25,31,37,43)(26,44,38,32)(27,33,39,45)(28,46,40,34)(29,35,41,47)(30,48,42,36), (1,4,13,16)(2,11,14,23)(3,6,15,18)(5,8,17,20)(7,10,19,22)(9,12,21,24)(25,28,37,40)(26,35,38,47)(27,30,39,42)(29,32,41,44)(31,34,43,46)(33,36,45,48), (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), (1,29,13,41)(2,40,14,28)(3,27,15,39)(4,38,16,26)(5,25,17,37)(6,36,18,48)(7,47,19,35)(8,34,20,46)(9,45,21,33)(10,32,22,44)(11,43,23,31)(12,30,24,42)>;
G:=Group( (1,7,13,19)(2,20,14,8)(3,9,15,21)(4,22,16,10)(5,11,17,23)(6,24,18,12)(25,31,37,43)(26,44,38,32)(27,33,39,45)(28,46,40,34)(29,35,41,47)(30,48,42,36), (1,4,13,16)(2,11,14,23)(3,6,15,18)(5,8,17,20)(7,10,19,22)(9,12,21,24)(25,28,37,40)(26,35,38,47)(27,30,39,42)(29,32,41,44)(31,34,43,46)(33,36,45,48), (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), (1,29,13,41)(2,40,14,28)(3,27,15,39)(4,38,16,26)(5,25,17,37)(6,36,18,48)(7,47,19,35)(8,34,20,46)(9,45,21,33)(10,32,22,44)(11,43,23,31)(12,30,24,42) );
G=PermutationGroup([(1,7,13,19),(2,20,14,8),(3,9,15,21),(4,22,16,10),(5,11,17,23),(6,24,18,12),(25,31,37,43),(26,44,38,32),(27,33,39,45),(28,46,40,34),(29,35,41,47),(30,48,42,36)], [(1,4,13,16),(2,11,14,23),(3,6,15,18),(5,8,17,20),(7,10,19,22),(9,12,21,24),(25,28,37,40),(26,35,38,47),(27,30,39,42),(29,32,41,44),(31,34,43,46),(33,36,45,48)], [(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)], [(1,29,13,41),(2,40,14,28),(3,27,15,39),(4,38,16,26),(5,25,17,37),(6,36,18,48),(7,47,19,35),(8,34,20,46),(9,45,21,33),(10,32,22,44),(11,43,23,31),(12,30,24,42)])
Matrix representation of Q8.14D12 ►in GL4(𝔽73) generated by
| 66 | 59 | 0 | 0 |
| 14 | 7 | 0 | 0 |
| 66 | 59 | 7 | 14 |
| 14 | 7 | 59 | 66 |
| 1 | 0 | 71 | 0 |
| 0 | 1 | 0 | 71 |
| 1 | 0 | 72 | 0 |
| 0 | 1 | 0 | 72 |
| 72 | 72 | 2 | 2 |
| 1 | 0 | 71 | 0 |
| 69 | 3 | 1 | 1 |
| 70 | 66 | 72 | 0 |
| 8 | 34 | 0 | 0 |
| 26 | 65 | 0 | 0 |
| 68 | 7 | 18 | 20 |
| 12 | 5 | 2 | 55 |
G:=sub<GL(4,GF(73))| [66,14,66,14,59,7,59,7,0,0,7,59,0,0,14,66],[1,0,1,0,0,1,0,1,71,0,72,0,0,71,0,72],[72,1,69,70,72,0,3,66,2,71,1,72,2,0,1,0],[8,26,68,12,34,65,7,5,0,0,18,2,0,0,20,55] >;
Q8.14D12 in GAP, Magma, Sage, TeX
Q_8._{14}D_{12} % in TeX
G:=Group("Q8.14D12"); // GroupNames label
G:=SmallGroup(192,385);
// by ID
G=gap.SmallGroup(192,385);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,232,254,219,58,1123,136,851,438,102,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=1,b^2=c^12=d^2=a^2,b*a*b^-1=c*a*c^-1=d*a*d^-1=a^-1,c*b*c^-1=d*b*d^-1=a*b,d*c*d^-1=c^11>;
// generators/relations
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