metabelian, supersoluble, monomial
Aliases: D12:5D6, Dic6:5D6, C3:C8:9D6, D4:S3:6S3, D4.10S32, D4.S3:6S3, C6.60(S3xD4), C3:3(Q8:3D6), C3:4(D8:S3), (C3xD4).14D6, D12:S3:5C2, C3:D24:12C2, (C3xD12):8C22, C3:Dic3.21D4, C32:5SD16:8C2, C12.31D6:1C2, C32:13(C8:C22), (C3xC12).14C23, C12.14(C22xS3), (C3xDic6):9C22, C2.20(Dic3:D6), C12:S3.9C22, (D4xC32).10C22, C4.14(C2xS32), (D4xC3:S3):2C2, (C3xD4:S3):8C2, (C3xC3:C8):13C22, (C3xD4.S3):4C2, (C2xC3:S3).58D4, (C3xC6).129(C2xD4), (C4xC3:S3).16C22, SmallGroup(288,585)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D12:5D6
G = < a,b,c,d | a12=b2=c6=d2=1, bab=dad=a-1, cac-1=a7, cbc-1=a3b, dbd=ab, dcd=c-1 >
Subgroups: 866 in 163 conjugacy classes, 38 normal (all characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, C6, C6, C8, C2xC4, D4, D4, Q8, C23, C32, Dic3, C12, C12, D6, C2xC6, M4(2), D8, SD16, C2xD4, C4oD4, C3xS3, C3:S3, C3xC6, C3xC6, C3:C8, C24, Dic6, C4xS3, D12, D12, C2xDic3, C3:D4, C3xD4, C3xD4, C3xQ8, C22xS3, C8:C22, C3xDic3, C3:Dic3, C3xC12, S3xC6, C2xC3:S3, C2xC3:S3, C62, C8:S3, C24:C2, D24, D4:S3, D4:S3, D4.S3, Q8:2S3, C3xD8, C3xSD16, S3xD4, D4:2S3, Q8:3S3, C3xC3:C8, S3xDic3, C3:D12, C3xDic6, C3xD12, C4xC3:S3, C12:S3, C32:7D4, D4xC32, C22xC3:S3, D8:S3, Q8:3D6, C12.31D6, C3:D24, C32:5SD16, C3xD4:S3, C3xD4.S3, D12:S3, D4xC3:S3, D12:5D6
Quotients: C1, C2, C22, S3, D4, C23, D6, C2xD4, C22xS3, C8:C22, S32, S3xD4, C2xS32, D8:S3, Q8:3D6, Dic3:D6, D12:5D6
Character table of D12:5D6
class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | 3B | 3C | 4A | 4B | 4C | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 6H | 8A | 8B | 12A | 12B | 12C | 12D | 24A | 24B | 24C | 24D | |
size | 1 | 1 | 4 | 12 | 18 | 36 | 2 | 2 | 4 | 2 | 12 | 18 | 2 | 2 | 4 | 8 | 8 | 8 | 8 | 24 | 12 | 12 | 4 | 4 | 8 | 24 | 12 | 12 | 12 | 12 | |
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 | 0 | 0 | 0 | 2 | -1 | -1 | 2 | 2 | 0 | 2 | -1 | -1 | 2 | -1 | -1 | -1 | 0 | 2 | 0 | 2 | -1 | -1 | -1 | 0 | -1 | -1 | 0 | orthogonal lifted from S3 |
ρ10 | 2 | 2 | 2 | -2 | 0 | 0 | -1 | 2 | -1 | 2 | 0 | 0 | -1 | 2 | -1 | -1 | -1 | -1 | 2 | 1 | 0 | -2 | -1 | 2 | -1 | 0 | 1 | 0 | 0 | 1 | orthogonal lifted from D6 |
ρ11 | 2 | 2 | -2 | 0 | 0 | 0 | 2 | -1 | -1 | 2 | -2 | 0 | 2 | -1 | -1 | -2 | 1 | 1 | 1 | 0 | 2 | 0 | 2 | -1 | -1 | 1 | 0 | -1 | -1 | 0 | orthogonal lifted from D6 |
ρ12 | 2 | 2 | -2 | -2 | 0 | 0 | -1 | 2 | -1 | 2 | 0 | 0 | -1 | 2 | -1 | 1 | 1 | 1 | -2 | 1 | 0 | 2 | -1 | 2 | -1 | 0 | -1 | 0 | 0 | -1 | orthogonal lifted from D6 |
ρ13 | 2 | 2 | 0 | 0 | -2 | 0 | 2 | 2 | 2 | -2 | 0 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
ρ14 | 2 | 2 | -2 | 0 | 0 | 0 | 2 | -1 | -1 | 2 | 2 | 0 | 2 | -1 | -1 | -2 | 1 | 1 | 1 | 0 | -2 | 0 | 2 | -1 | -1 | -1 | 0 | 1 | 1 | 0 | orthogonal lifted from D6 |
ρ15 | 2 | 2 | 2 | 2 | 0 | 0 | -1 | 2 | -1 | 2 | 0 | 0 | -1 | 2 | -1 | -1 | -1 | -1 | 2 | -1 | 0 | 2 | -1 | 2 | -1 | 0 | -1 | 0 | 0 | -1 | orthogonal lifted from S3 |
ρ16 | 2 | 2 | 0 | 0 | 2 | 0 | 2 | 2 | 2 | -2 | 0 | -2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
ρ17 | 2 | 2 | -2 | 2 | 0 | 0 | -1 | 2 | -1 | 2 | 0 | 0 | -1 | 2 | -1 | 1 | 1 | 1 | -2 | -1 | 0 | -2 | -1 | 2 | -1 | 0 | 1 | 0 | 0 | 1 | orthogonal lifted from D6 |
ρ18 | 2 | 2 | 2 | 0 | 0 | 0 | 2 | -1 | -1 | 2 | -2 | 0 | 2 | -1 | -1 | 2 | -1 | -1 | -1 | 0 | -2 | 0 | 2 | -1 | -1 | 1 | 0 | 1 | 1 | 0 | orthogonal lifted from D6 |
ρ19 | 4 | 4 | 0 | 0 | 0 | 0 | -2 | -2 | 1 | -4 | 0 | 0 | -2 | -2 | 1 | 0 | 3 | -3 | 0 | 0 | 0 | 0 | 2 | 2 | -1 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from Dic3:D6 |
ρ20 | 4 | 4 | 0 | 0 | 0 | 0 | -2 | -2 | 1 | -4 | 0 | 0 | -2 | -2 | 1 | 0 | -3 | 3 | 0 | 0 | 0 | 0 | 2 | 2 | -1 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from Dic3:D6 |
ρ21 | 4 | 4 | 0 | 0 | 0 | 0 | -2 | 4 | -2 | -4 | 0 | 0 | -2 | 4 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -4 | 2 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from S3xD4 |
ρ22 | 4 | -4 | 0 | 0 | 0 | 0 | 4 | 4 | 4 | 0 | 0 | 0 | -4 | -4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C8:C22 |
ρ23 | 4 | 4 | 0 | 0 | 0 | 0 | 4 | -2 | -2 | -4 | 0 | 0 | 4 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -4 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from S3xD4 |
ρ24 | 4 | 4 | -4 | 0 | 0 | 0 | -2 | -2 | 1 | 4 | 0 | 0 | -2 | -2 | 1 | 2 | -1 | -1 | 2 | 0 | 0 | 0 | -2 | -2 | 1 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C2xS32 |
ρ25 | 4 | 4 | 4 | 0 | 0 | 0 | -2 | -2 | 1 | 4 | 0 | 0 | -2 | -2 | 1 | -2 | 1 | 1 | -2 | 0 | 0 | 0 | -2 | -2 | 1 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from S32 |
ρ26 | 4 | -4 | 0 | 0 | 0 | 0 | 4 | -2 | -2 | 0 | 0 | 0 | -4 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -√6 | √6 | 0 | orthogonal lifted from Q8:3D6 |
ρ27 | 4 | -4 | 0 | 0 | 0 | 0 | 4 | -2 | -2 | 0 | 0 | 0 | -4 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | √6 | -√6 | 0 | orthogonal lifted from Q8:3D6 |
ρ28 | 4 | -4 | 0 | 0 | 0 | 0 | -2 | 4 | -2 | 0 | 0 | 0 | 2 | -4 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -√-6 | 0 | 0 | √-6 | complex lifted from D8:S3 |
ρ29 | 4 | -4 | 0 | 0 | 0 | 0 | -2 | 4 | -2 | 0 | 0 | 0 | 2 | -4 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | √-6 | 0 | 0 | -√-6 | complex lifted from D8:S3 |
ρ30 | 8 | -8 | 0 | 0 | 0 | 0 | -4 | -4 | 2 | 0 | 0 | 0 | 4 | 4 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal faithful |
(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 18)(2 17)(3 16)(4 15)(5 14)(6 13)(7 24)(8 23)(9 22)(10 21)(11 20)(12 19)
(1 9 5)(2 4 6 8 10 12)(3 11 7)(13 20 21 16 17 24)(14 15 22 23 18 19)
(1 5)(2 4)(6 12)(7 11)(8 10)(13 18)(14 17)(15 16)(19 24)(20 23)(21 22)
G:=sub<Sym(24)| (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24), (1,18)(2,17)(3,16)(4,15)(5,14)(6,13)(7,24)(8,23)(9,22)(10,21)(11,20)(12,19), (1,9,5)(2,4,6,8,10,12)(3,11,7)(13,20,21,16,17,24)(14,15,22,23,18,19), (1,5)(2,4)(6,12)(7,11)(8,10)(13,18)(14,17)(15,16)(19,24)(20,23)(21,22)>;
G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24), (1,18)(2,17)(3,16)(4,15)(5,14)(6,13)(7,24)(8,23)(9,22)(10,21)(11,20)(12,19), (1,9,5)(2,4,6,8,10,12)(3,11,7)(13,20,21,16,17,24)(14,15,22,23,18,19), (1,5)(2,4)(6,12)(7,11)(8,10)(13,18)(14,17)(15,16)(19,24)(20,23)(21,22) );
G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10,11,12),(13,14,15,16,17,18,19,20,21,22,23,24)], [(1,18),(2,17),(3,16),(4,15),(5,14),(6,13),(7,24),(8,23),(9,22),(10,21),(11,20),(12,19)], [(1,9,5),(2,4,6,8,10,12),(3,11,7),(13,20,21,16,17,24),(14,15,22,23,18,19)], [(1,5),(2,4),(6,12),(7,11),(8,10),(13,18),(14,17),(15,16),(19,24),(20,23),(21,22)]])
G:=TransitiveGroup(24,669);
Matrix representation of D12:5D6 ►in GL8(Z)
0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 |
0 | 0 | 1 | -1 | 0 | 0 | 0 | 0 |
0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
-1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
0 | 0 | 0 | 0 | 0 | 0 | -1 | 1 |
0 | 0 | 0 | 0 | 0 | -1 | 0 | 0 |
0 | 0 | 0 | 0 | 1 | -1 | 0 | 0 |
0 | 0 | 0 | 0 | -1 | 1 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 0 | -1 | 1 |
0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
-1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | -1 | 1 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
0 | -1 | 0 | 0 | 0 | 0 | 0 | 0 |
1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
0 | 0 | -1 | 1 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 0 | -1 | 1 |
0 | 0 | 0 | 0 | 0 | 0 | -1 | 0 |
0 | 0 | 0 | 0 | -1 | 1 | 0 | 0 |
0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 |
-1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 1 | -1 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 0 | -1 | 0 |
0 | 0 | 0 | 0 | 0 | 0 | -1 | 1 |
0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | -1 | 1 | 0 | 0 |
G:=sub<GL(8,Integers())| [0,0,0,-1,0,0,0,0,0,0,1,1,0,0,0,0,0,1,0,0,0,0,0,0,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,-1,0,0,0,0,0,-1,0,0,0,0,0,0,1,1,0,0],[0,0,0,0,-1,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,1,1,-1,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,1,1,0,0,0,0],[0,1,0,0,0,0,0,0,-1,-1,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,0,0,-1,-1,0,0,0,0,0,0,1,0,0,0,0,0,-1,-1,0,0,0,0,0,0,1,0,0,0],[-1,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,-1,-1,0,0,0,0,0,0,0,0,0,0,-1,-1,0,0,0,0,0,0,0,1,0,0,0,0,-1,-1,0,0,0,0,0,0,0,1,0,0] >;
D12:5D6 in GAP, Magma, Sage, TeX
D_{12}\rtimes_5D_6
% in TeX
G:=Group("D12:5D6");
// GroupNames label
G:=SmallGroup(288,585);
// by ID
G=gap.SmallGroup(288,585);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,120,254,303,675,346,185,80,1356,9414]);
// Polycyclic
G:=Group<a,b,c,d|a^12=b^2=c^6=d^2=1,b*a*b=d*a*d=a^-1,c*a*c^-1=a^7,c*b*c^-1=a^3*b,d*b*d=a*b,d*c*d=c^-1>;
// generators/relations
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