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, C3, C4, C4, C22, C22, S3, C6, C6, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, Dic3, C12, C12, D6, C2×C6, C2×C6, C42, C4⋊C4, M4(2), M4(2), SD16, Q16, C2×Q8, C4○D4, C4○D4, C3⋊C8, C24, Dic6, Dic6, C4×S3, D12, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C3×D4, C3×D4, C3×Q8, C4.10D4, C4≀C2, C4≀C2, C4⋊Q8, C8.C22, 2- 1+4, C24⋊C2, Dic12, C4.Dic3, C4⋊Dic3, D4.S3, C3⋊Q16, C4×C12, C3×M4(2), C2×Dic6, C2×Dic6, C4○D12, C4○D12, D4⋊2S3, 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, C22, S3, D4, C23, D6, C2×D4, D12, C22×S3, C22≀C2, C2×D12, S3×D4, 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 43 37 31)(26 32 38 44)(27 45 39 33)(28 34 40 46)(29 47 41 35)(30 36 42 48)
(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 34 37 46)(26 29 38 41)(27 36 39 48)(28 31 40 43)(30 33 42 45)(32 35 44 47)
(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 44 13 32)(2 31 14 43)(3 42 15 30)(4 29 16 41)(5 40 17 28)(6 27 18 39)(7 38 19 26)(8 25 20 37)(9 36 21 48)(10 47 22 35)(11 34 23 46)(12 45 24 33)
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)(25,43,37,31)(26,32,38,44)(27,45,39,33)(28,34,40,46)(29,47,41,35)(30,36,42,48), (1,4,13,16)(2,11,14,23)(3,6,15,18)(5,8,17,20)(7,10,19,22)(9,12,21,24)(25,34,37,46)(26,29,38,41)(27,36,39,48)(28,31,40,43)(30,33,42,45)(32,35,44,47), (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), (1,44,13,32)(2,31,14,43)(3,42,15,30)(4,29,16,41)(5,40,17,28)(6,27,18,39)(7,38,19,26)(8,25,20,37)(9,36,21,48)(10,47,22,35)(11,34,23,46)(12,45,24,33)>;
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)(25,43,37,31)(26,32,38,44)(27,45,39,33)(28,34,40,46)(29,47,41,35)(30,36,42,48), (1,4,13,16)(2,11,14,23)(3,6,15,18)(5,8,17,20)(7,10,19,22)(9,12,21,24)(25,34,37,46)(26,29,38,41)(27,36,39,48)(28,31,40,43)(30,33,42,45)(32,35,44,47), (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), (1,44,13,32)(2,31,14,43)(3,42,15,30)(4,29,16,41)(5,40,17,28)(6,27,18,39)(7,38,19,26)(8,25,20,37)(9,36,21,48)(10,47,22,35)(11,34,23,46)(12,45,24,33) );
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),(25,43,37,31),(26,32,38,44),(27,45,39,33),(28,34,40,46),(29,47,41,35),(30,36,42,48)], [(1,4,13,16),(2,11,14,23),(3,6,15,18),(5,8,17,20),(7,10,19,22),(9,12,21,24),(25,34,37,46),(26,29,38,41),(27,36,39,48),(28,31,40,43),(30,33,42,45),(32,35,44,47)], [(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)], [(1,44,13,32),(2,31,14,43),(3,42,15,30),(4,29,16,41),(5,40,17,28),(6,27,18,39),(7,38,19,26),(8,25,20,37),(9,36,21,48),(10,47,22,35),(11,34,23,46),(12,45,24,33)]])
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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