metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Q8○D12, D4○Dic6, D4.10D6, Q8.16D6, C6.13C24, D6.8C23, C3⋊22- 1+4, C12.27C23, D12.14C22, Dic3.8C23, Dic6.14C22, C4○D4⋊6S3, (S3×Q8)⋊5C2, C4○D12⋊9C2, (C2×C4).25D6, C3⋊D4.C22, D4⋊2S3⋊5C2, (C2×C6).5C23, (C2×Dic6)⋊14C2, (C4×S3).6C22, C4.34(C22×S3), C2.14(S3×C23), (C2×C12).49C22, (C3×D4).10C22, C22.4(C22×S3), (C3×Q8).11C22, (C2×Dic3).22C22, (C3×C4○D4)⋊5C2, SmallGroup(96,217)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Q8○D12
G = < a,b,c,d | a4=d2=1, b2=c6=a2, bab-1=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=a2c5 >
Subgroups: 266 in 146 conjugacy classes, 85 normal (12 characteristic)
C1, C2, C2, C3, C4, C4, C4, C22, C22, S3, C6, C6, C2×C4, C2×C4, D4, D4, Q8, Q8, Dic3, C12, C12, D6, C2×C6, C2×Q8, C4○D4, C4○D4, Dic6, C4×S3, D12, C2×Dic3, C3⋊D4, C2×C12, C3×D4, C3×Q8, 2- 1+4, C2×Dic6, C4○D12, D4⋊2S3, S3×Q8, C3×C4○D4, Q8○D12
Quotients: C1, C2, C22, S3, C23, D6, C24, C22×S3, 2- 1+4, S3×C23, Q8○D12
Character table of Q8○D12
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 6A | 6B | 6C | 6D | 12A | 12B | 12C | 12D | 12E | |
| size | 1 | 1 | 2 | 2 | 2 | 6 | 6 | 2 | 2 | 2 | 2 | 2 | 6 | 6 | 6 | 6 | 6 | 6 | 2 | 4 | 4 | 4 | 2 | 2 | 4 | 4 | 4 | |
| 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 | 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 | 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 | 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 | 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 | 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 | 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 | 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 | linear of order 2 |
| ρ9 | 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 |
| ρ10 | 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 |
| ρ11 | 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 |
| ρ12 | 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 |
| ρ13 | 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 |
| ρ14 | 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 |
| ρ15 | 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 |
| ρ16 | 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 |
| ρ17 | 2 | 2 | -2 | 2 | -2 | 0 | 0 | -1 | -2 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 1 | -1 | 1 | -1 | -1 | -1 | 1 | 1 | orthogonal lifted from D6 |
| ρ18 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | -1 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ19 | 2 | 2 | -2 | -2 | 2 | 0 | 0 | -1 | -2 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 1 | 1 | -1 | -1 | -1 | 1 | -1 | 1 | orthogonal lifted from D6 |
| ρ20 | 2 | 2 | 2 | -2 | -2 | 0 | 0 | -1 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | orthogonal lifted from D6 |
| ρ21 | 2 | 2 | -2 | -2 | -2 | 0 | 0 | -1 | 2 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | orthogonal lifted from D6 |
| ρ22 | 2 | 2 | 2 | -2 | 2 | 0 | 0 | -1 | -2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | 1 | -1 | 1 | 1 | -1 | 1 | 1 | orthogonal lifted from D6 |
| ρ23 | 2 | 2 | -2 | 2 | 2 | 0 | 0 | -1 | 2 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | orthogonal lifted from D6 |
| ρ24 | 2 | 2 | 2 | 2 | -2 | 0 | 0 | -1 | -2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | 1 | orthogonal lifted from D6 |
| ρ25 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from 2- 1+4, Schur index 2 |
| ρ26 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | -2√3 | 2√3 | 0 | 0 | 0 | symplectic faithful, Schur index 2 |
| ρ27 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 2√3 | -2√3 | 0 | 0 | 0 | symplectic faithful, Schur index 2 |
►On 48 points
(1 25 7 31)(2 26 8 32)(3 27 9 33)(4 28 10 34)(5 29 11 35)(6 30 12 36)(13 40 19 46)(14 41 20 47)(15 42 21 48)(16 43 22 37)(17 44 23 38)(18 45 24 39)
(1 46 7 40)(2 47 8 41)(3 48 9 42)(4 37 10 43)(5 38 11 44)(6 39 12 45)(13 25 19 31)(14 26 20 32)(15 27 21 33)(16 28 22 34)(17 29 23 35)(18 30 24 36)
(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 3)(4 12)(5 11)(6 10)(7 9)(13 15)(16 24)(17 23)(18 22)(19 21)(25 27)(28 36)(29 35)(30 34)(31 33)(37 45)(38 44)(39 43)(40 42)(46 48)
G:=sub<Sym(48)| (1,25,7,31)(2,26,8,32)(3,27,9,33)(4,28,10,34)(5,29,11,35)(6,30,12,36)(13,40,19,46)(14,41,20,47)(15,42,21,48)(16,43,22,37)(17,44,23,38)(18,45,24,39), (1,46,7,40)(2,47,8,41)(3,48,9,42)(4,37,10,43)(5,38,11,44)(6,39,12,45)(13,25,19,31)(14,26,20,32)(15,27,21,33)(16,28,22,34)(17,29,23,35)(18,30,24,36), (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), (1,3)(4,12)(5,11)(6,10)(7,9)(13,15)(16,24)(17,23)(18,22)(19,21)(25,27)(28,36)(29,35)(30,34)(31,33)(37,45)(38,44)(39,43)(40,42)(46,48)>;
G:=Group( (1,25,7,31)(2,26,8,32)(3,27,9,33)(4,28,10,34)(5,29,11,35)(6,30,12,36)(13,40,19,46)(14,41,20,47)(15,42,21,48)(16,43,22,37)(17,44,23,38)(18,45,24,39), (1,46,7,40)(2,47,8,41)(3,48,9,42)(4,37,10,43)(5,38,11,44)(6,39,12,45)(13,25,19,31)(14,26,20,32)(15,27,21,33)(16,28,22,34)(17,29,23,35)(18,30,24,36), (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), (1,3)(4,12)(5,11)(6,10)(7,9)(13,15)(16,24)(17,23)(18,22)(19,21)(25,27)(28,36)(29,35)(30,34)(31,33)(37,45)(38,44)(39,43)(40,42)(46,48) );
G=PermutationGroup([[(1,25,7,31),(2,26,8,32),(3,27,9,33),(4,28,10,34),(5,29,11,35),(6,30,12,36),(13,40,19,46),(14,41,20,47),(15,42,21,48),(16,43,22,37),(17,44,23,38),(18,45,24,39)], [(1,46,7,40),(2,47,8,41),(3,48,9,42),(4,37,10,43),(5,38,11,44),(6,39,12,45),(13,25,19,31),(14,26,20,32),(15,27,21,33),(16,28,22,34),(17,29,23,35),(18,30,24,36)], [(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)], [(1,3),(4,12),(5,11),(6,10),(7,9),(13,15),(16,24),(17,23),(18,22),(19,21),(25,27),(28,36),(29,35),(30,34),(31,33),(37,45),(38,44),(39,43),(40,42),(46,48)]])
Q8○D12 is a maximal subgroup of
Q8.14D12 D4.10D12 D12.38D4 D12.40D4 D4.11D12 D4.13D12 D8⋊11D6 D8.10D6 D8⋊6D6 SD16.D6 D12.33C23 D12.35C23 C6.C25 D6.C24 S3×2- 1+4 D4.10D18 D12.A4 D12.33D6 D12.34D6 Dic6.24D6 D12.25D6 C32⋊92- 1+4 D20.38D6 D20.39D6 C15⋊2- 1+4 D20.29D6 D4.10D30
Q8○D12 is a maximal quotient of
C42.87D6 C42.89D6 C42.90D6 C42.91D6 C42.92D6 C42.94D6 C42.96D6 C42.98D6 C42.99D6 D4×Dic6 C42.105D6 C42.106D6 C42.108D6 D12⋊24D4 Dic6⋊23D4 D4⋊6D12 C42.115D6 C42.118D6 Dic6⋊10Q8 Q8⋊7Dic6 C42.125D6 Q8×D12 C42.134D6 C42.135D6 C6.322+ 1+4 Dic6⋊19D4 C6.702- 1+4 C6.712- 1+4 C6.722- 1+4 C6.732- 1+4 C6.492+ 1+4 C6.752- 1+4 C6.152- 1+4 C6.162- 1+4 Dic6⋊21D4 C6.772- 1+4 C6.782- 1+4 C6.252- 1+4 C6.792- 1+4 C6.802- 1+4 C6.812- 1+4 C6.822- 1+4 C6.632+ 1+4 C6.652+ 1+4 C6.852- 1+4 C6.692+ 1+4 C42.137D6 C42.139D6 C42.140D6 C42.141D6 Dic6⋊10D4 C42.144D6 C42.145D6 Dic6⋊7Q8 C42.147D6 C42.148D6 C42.152D6 C42.154D6 C42.156D6 C42.157D6 C42.159D6 C42.160D6 C42.161D6 C42.162D6 C42.164D6 C42.165D6 C6.1042- 1+4 C6.1052- 1+4 C6.1442+ 1+4 C6.1072- 1+4 C6.1082- 1+4 D4.10D18 D12.33D6 D12.34D6 Dic6.24D6 D12.25D6 C32⋊92- 1+4 D20.38D6 D20.39D6 C15⋊2- 1+4 D20.29D6 D4.10D30
Matrix representation of Q8○D12 ►in GL4(𝔽13) generated by
| 8 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 |
| 5 | 8 | 5 | 0 |
| 10 | 5 | 0 | 5 |
| 8 | 0 | 1 | 1 |
| 0 | 8 | 11 | 1 |
| 0 | 0 | 5 | 0 |
| 0 | 0 | 0 | 5 |
| 10 | 3 | 0 | 0 |
| 10 | 7 | 0 | 0 |
| 0 | 0 | 7 | 3 |
| 0 | 0 | 10 | 10 |
| 1 | 1 | 0 | 0 |
| 0 | 12 | 0 | 0 |
| 0 | 0 | 12 | 1 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(13))| [8,0,5,10,0,8,8,5,0,0,5,0,0,0,0,5],[8,0,0,0,0,8,0,0,1,11,5,0,1,1,0,5],[10,10,0,0,3,7,0,0,0,0,7,10,0,0,3,10],[1,0,0,0,1,12,0,0,0,0,12,0,0,0,1,1] >;
Q8○D12 in GAP, Magma, Sage, TeX
Q_8\circ D_{12} % in TeX
G:=Group("Q8oD12"); // GroupNames label
G:=SmallGroup(96,217);
// by ID
G=gap.SmallGroup(96,217);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-3,103,188,86,579,69,2309]);
// Polycyclic
G:=Group<a,b,c,d|a^4=d^2=1,b^2=c^6=a^2,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=a^2*c^5>;
// generators/relations
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