metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Q8oD12, D4oDic6, D4.10D6, Q8.16D6, C6.13C24, D6.8C23, C3:22- 1+4, C12.27C23, D12.14C22, Dic3.8C23, Dic6.14C22, C4oD4:6S3, (S3xQ8):5C2, C4oD12:9C2, (C2xC4).25D6, C3:D4.C22, D4:2S3:5C2, (C2xC6).5C23, (C2xDic6):14C2, (C4xS3).6C22, C4.34(C22xS3), C2.14(S3xC23), (C2xC12).49C22, (C3xD4).10C22, C22.4(C22xS3), (C3xQ8).11C22, (C2xDic3).22C22, (C3xC4oD4):5C2, SmallGroup(96,217)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Q8oD12
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, C2xC4, C2xC4, D4, D4, Q8, Q8, Dic3, C12, C12, D6, C2xC6, C2xQ8, C4oD4, C4oD4, Dic6, C4xS3, D12, C2xDic3, C3:D4, C2xC12, C3xD4, C3xQ8, 2- 1+4, C2xDic6, C4oD12, D4:2S3, S3xQ8, C3xC4oD4, Q8oD12
Quotients: C1, C2, C22, S3, C23, D6, C24, C22xS3, 2- 1+4, S3xC23, Q8oD12
Character table of Q8oD12
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 |
(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)]])
Q8oD12 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 S3x2- 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
Q8oD12 is a maximal quotient of
C42.87D6 C42.89D6 C42.90D6 C42.91D6 C42.92D6 C42.94D6 C42.96D6 C42.98D6 C42.99D6 D4xDic6 C42.105D6 C42.106D6 C42.108D6 D12:24D4 Dic6:23D4 D4:6D12 C42.115D6 C42.118D6 Dic6:10Q8 Q8:7Dic6 C42.125D6 Q8xD12 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 Q8oD12 ►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] >;
Q8oD12 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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