metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D4oD12, D4:8D6, Q8:8D6, Q8oDic6, C6.12C24, D6.7C23, C3:22+ 1+4, D12:11C22, C12.26C23, Dic6:12C22, Dic3.7C23, (C2xC4):4D6, C4oD4:5S3, (S3xD4):5C2, C4oD12:8C2, (C2xD12):13C2, (C4xS3):2C22, (C2xC12):5C22, Q8:3S3:5C2, (C3xD4):9C22, C3:D4:5C22, (C2xC6).4C23, (C3xQ8):8C22, C2.13(S3xC23), C4.33(C22xS3), (C22xS3):4C22, C22.3(C22xS3), (C3xC4oD4):4C2, SmallGroup(96,216)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D4oD12
G = < a,b,c,d | a4=b2=d2=1, c6=a2, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=a2c5 >
Subgroups: 394 in 166 conjugacy classes, 85 normal (12 characteristic)
C1, C2, C2, C3, C4, C4, C4, C22, C22, S3, C6, C6, C2xC4, C2xC4, D4, D4, Q8, Q8, C23, Dic3, C12, C12, D6, D6, C2xC6, C2xD4, C4oD4, C4oD4, Dic6, C4xS3, D12, C3:D4, C2xC12, C3xD4, C3xQ8, C22xS3, 2+ 1+4, C2xD12, C4oD12, S3xD4, Q8:3S3, C3xC4oD4, D4oD12
Quotients: C1, C2, C22, S3, C23, D6, C24, C22xS3, 2+ 1+4, S3xC23, D4oD12
Character table of D4oD12
class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 6A | 6B | 6C | 6D | 12A | 12B | 12C | 12D | 12E | |
size | 1 | 1 | 2 | 2 | 2 | 6 | 6 | 6 | 6 | 6 | 6 | 2 | 2 | 2 | 2 | 2 | 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 | 0 | 0 | 0 | 0 | -1 | -2 | 2 | -2 | 2 | 0 | 0 | -1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | orthogonal lifted from D6 |
ρ18 | 2 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -2 | -2 | 2 | 2 | 0 | 0 | -1 | 1 | -1 | 1 | -1 | -1 | 1 | 1 | -1 | orthogonal lifted from D6 |
ρ19 | 2 | 2 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -2 | 2 | -2 | -2 | 0 | 0 | -1 | -1 | -1 | 1 | 1 | 1 | -1 | 1 | 1 | orthogonal lifted from D6 |
ρ20 | 2 | 2 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -2 | -2 | 2 | -2 | 0 | 0 | -1 | -1 | 1 | -1 | 1 | 1 | 1 | 1 | -1 | orthogonal lifted from D6 |
ρ21 | 2 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 2 | -2 | -2 | 2 | 0 | 0 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | orthogonal lifted from D6 |
ρ22 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 2 | 2 | 2 | 2 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
ρ23 | 2 | 2 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 2 | -2 | -2 | -2 | 0 | 0 | -1 | 1 | -1 | -1 | 1 | 1 | 1 | -1 | 1 | orthogonal lifted from D6 |
ρ24 | 2 | 2 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 2 | 2 | 2 | -2 | 0 | 0 | -1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | orthogonal lifted from D6 |
ρ25 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from 2+ 1+4 |
ρ26 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 2√3 | -2√3 | 0 | 0 | 0 | orthogonal faithful |
ρ27 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | -2√3 | 2√3 | 0 | 0 | 0 | orthogonal faithful |
(1 13 7 19)(2 14 8 20)(3 15 9 21)(4 16 10 22)(5 17 11 23)(6 18 12 24)
(1 19)(2 20)(3 21)(4 22)(5 23)(6 24)(7 13)(8 14)(9 15)(10 16)(11 17)(12 18)
(1 2 3 4 5 6 7 8 9 10 11 12)(13 14 15 16 17 18 19 20 21 22 23 24)
(1 12)(2 11)(3 10)(4 9)(5 8)(6 7)(13 24)(14 23)(15 22)(16 21)(17 20)(18 19)
G:=sub<Sym(24)| (1,13,7,19)(2,14,8,20)(3,15,9,21)(4,16,10,22)(5,17,11,23)(6,18,12,24), (1,19)(2,20)(3,21)(4,22)(5,23)(6,24)(7,13)(8,14)(9,15)(10,16)(11,17)(12,18), (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24), (1,12)(2,11)(3,10)(4,9)(5,8)(6,7)(13,24)(14,23)(15,22)(16,21)(17,20)(18,19)>;
G:=Group( (1,13,7,19)(2,14,8,20)(3,15,9,21)(4,16,10,22)(5,17,11,23)(6,18,12,24), (1,19)(2,20)(3,21)(4,22)(5,23)(6,24)(7,13)(8,14)(9,15)(10,16)(11,17)(12,18), (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24), (1,12)(2,11)(3,10)(4,9)(5,8)(6,7)(13,24)(14,23)(15,22)(16,21)(17,20)(18,19) );
G=PermutationGroup([[(1,13,7,19),(2,14,8,20),(3,15,9,21),(4,16,10,22),(5,17,11,23),(6,18,12,24)], [(1,19),(2,20),(3,21),(4,22),(5,23),(6,24),(7,13),(8,14),(9,15),(10,16),(11,17),(12,18)], [(1,2,3,4,5,6,7,8,9,10,11,12),(13,14,15,16,17,18,19,20,21,22,23,24)], [(1,12),(2,11),(3,10),(4,9),(5,8),(6,7),(13,24),(14,23),(15,22),(16,21),(17,20),(18,19)]])
G:=TransitiveGroup(24,102);
D4oD12 is a maximal subgroup of
Q8:5D12 C42:5D6 D12:18D4 D12.39D4 D4.11D12 D4.12D12 D8:15D6 D8:11D6 D8:5D6 C24.C23 D12.32C23 D12.34C23 C6.C25 S3x2+ 1+4 D12.39C23 D4:8D18 Dic6.A4 D12:24D6 D12:27D6 D12:13D6 D12:16D6 C62.154C23 D20:26D6 D20:29D6 D12:14D10 D20:17D6 D4:8D30
D4oD12 is a maximal quotient of
C42.90D6 C42:9D6 C42.91D6 C42:11D6 C42:12D6 C42.95D6 C42.97D6 C42.99D6 C42.100D6 D4:6Dic6 C42:13D6 D4xD12 D12:23D4 Dic6:24D4 C42:19D6 C42.116D6 C42.117D6 C42.119D6 Q8xDic6 C42.126D6 Q8:7D12 D12:10Q8 C42.133D6 C42.136D6 C6.372+ 1+4 C6.382+ 1+4 D12:19D4 C6.462+ 1+4 C6.1152+ 1+4 C6.472+ 1+4 C6.482+ 1+4 C6.172- 1+4 D12:21D4 C6.512+ 1+4 C6.1182+ 1+4 C6.242- 1+4 C6.562+ 1+4 C6.592+ 1+4 C6.1202+ 1+4 C6.1212+ 1+4 C6.612+ 1+4 C6.1222+ 1+4 C6.662+ 1+4 C6.852- 1+4 C6.682+ 1+4 C6.692+ 1+4 C42:20D6 D12:10D4 C42:22D6 C42.143D6 C42.144D6 C42:24D6 C42.145D6 C42.148D6 D12:7Q8 C42.150D6 C42.153D6 C42.155D6 C42.157D6 C42.158D6 C42:25D6 C42:26D6 C42.161D6 C42.163D6 C42.164D6 C42:27D6 C42.165D6 C6.1442+ 1+4 C6.1452+ 1+4 C6.1462+ 1+4 C6.1082- 1+4 C6.1482+ 1+4 D4:8D18 D12:24D6 D12:27D6 D12:13D6 D12:16D6 C62.154C23 D20:26D6 D20:29D6 D12:14D10 D20:17D6 D4:8D30
Matrix representation of D4oD12 ►in GL4(F13) generated by
0 | 0 | 1 | 0 |
0 | 0 | 0 | 1 |
12 | 0 | 0 | 0 |
0 | 12 | 0 | 0 |
0 | 0 | 1 | 0 |
0 | 0 | 0 | 1 |
1 | 0 | 0 | 0 |
0 | 1 | 0 | 0 |
10 | 3 | 0 | 0 |
10 | 7 | 0 | 0 |
0 | 0 | 10 | 3 |
0 | 0 | 10 | 7 |
1 | 1 | 0 | 0 |
0 | 12 | 0 | 0 |
0 | 0 | 1 | 1 |
0 | 0 | 0 | 12 |
G:=sub<GL(4,GF(13))| [0,0,12,0,0,0,0,12,1,0,0,0,0,1,0,0],[0,0,1,0,0,0,0,1,1,0,0,0,0,1,0,0],[10,10,0,0,3,7,0,0,0,0,10,10,0,0,3,7],[1,0,0,0,1,12,0,0,0,0,1,0,0,0,1,12] >;
D4oD12 in GAP, Magma, Sage, TeX
D_4\circ D_{12}
% in TeX
G:=Group("D4oD12");
// GroupNames label
G:=SmallGroup(96,216);
// by ID
G=gap.SmallGroup(96,216);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-3,188,579,69,2309]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^2=d^2=1,c^6=a^2,b*a*b=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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