metabelian, supersoluble, monomial
Aliases: D12:4Dic3, C62.23D4, Dic6:4Dic3, C32:3C4wrC2, (C3xD12):3C4, (C2xC6).8D12, C12.18(C4xS3), (C3xDic6):3C4, C4oD12.2S3, (C2xC12).79D6, C4.Dic3:4S3, C4.8(S3xDic3), C6.38(D6:C4), (C3xC12).109D4, C12.9(C2xDic3), C3:3(D12:C4), C12.95(C3:D4), C2.9(D6:Dic3), (C6xC12).30C22, C3:1(Q8:3Dic3), C4.17(D6:S3), C6.8(C6.D4), C22.7(C3:D12), (C2xC4).101S32, (C4xC3:Dic3):1C2, (C3xC12).28(C2xC4), (C3xC4oD12).1C2, (C2xC6).10(C3:D4), (C3xC4.Dic3):15C2, (C3xC6).35(C22:C4), SmallGroup(288,216)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D12:4Dic3
G = < a,b,c,d | a12=b2=c6=1, d2=c3, bab=a-1, ac=ca, dad-1=a5, cbc-1=a6b, dbd-1=a7b, dcd-1=c-1 >
Subgroups: 362 in 103 conjugacy classes, 34 normal (all characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, C22, S3, C6, C6, C8, C2xC4, C2xC4, D4, Q8, C32, Dic3, C12, C12, D6, C2xC6, C2xC6, C42, M4(2), C4oD4, C3xS3, C3xC6, C3xC6, C3:C8, C24, Dic6, C4xS3, D12, C2xDic3, C3:D4, C2xC12, C2xC12, C3xD4, C3xQ8, C4wrC2, C3xDic3, C3:Dic3, C3xC12, S3xC6, C62, C4.Dic3, C4xDic3, C3xM4(2), C4oD12, C3xC4oD4, C3xC3:C8, C3xDic6, S3xC12, C3xD12, C3xC3:D4, C2xC3:Dic3, C6xC12, D12:C4, Q8:3Dic3, C3xC4.Dic3, C4xC3:Dic3, C3xC4oD12, D12:4Dic3
Quotients: C1, C2, C4, C22, S3, C2xC4, D4, Dic3, D6, C22:C4, C4xS3, D12, C2xDic3, C3:D4, C4wrC2, S32, D6:C4, C6.D4, S3xDic3, D6:S3, C3:D12, D12:C4, Q8:3Dic3, D6:Dic3, D12:4Dic3
(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 5 9)(2 6 10)(3 7 11)(4 8 12)(13 15 17 19 21 23)(14 16 18 20 22 24)
(2 6)(3 11)(5 9)(8 12)(13 16 19 22)(14 21 20 15)(17 24 23 18)
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), (1,5,9)(2,6,10)(3,7,11)(4,8,12)(13,15,17,19,21,23)(14,16,18,20,22,24), (2,6)(3,11)(5,9)(8,12)(13,16,19,22)(14,21,20,15)(17,24,23,18)>;
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), (1,5,9)(2,6,10)(3,7,11)(4,8,12)(13,15,17,19,21,23)(14,16,18,20,22,24), (2,6)(3,11)(5,9)(8,12)(13,16,19,22)(14,21,20,15)(17,24,23,18) );
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)], [(1,5,9),(2,6,10),(3,7,11),(4,8,12),(13,15,17,19,21,23),(14,16,18,20,22,24)], [(2,6),(3,11),(5,9),(8,12),(13,16,19,22),(14,21,20,15),(17,24,23,18)]])
G:=TransitiveGroup(24,612);
42 conjugacy classes
class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 6A | 6B | 6C | ··· | 6G | 6H | 6I | 8A | 8B | 12A | 12B | 12C | 12D | 12E | ··· | 12J | 12K | 12L | 24A | 24B | 24C | 24D |
order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | ··· | 6 | 6 | 6 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 12 | 12 | 24 | 24 | 24 | 24 |
size | 1 | 1 | 2 | 12 | 2 | 2 | 4 | 1 | 1 | 2 | 12 | 18 | 18 | 18 | 18 | 2 | 2 | 4 | ··· | 4 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 12 | 12 | 12 | 12 | 12 | 12 |
42 irreducible representations
dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 |
type | + | + | + | + | + | + | + | + | - | - | + | + | + | - | - | + | |||||||||
image | C1 | C2 | C2 | C2 | C4 | C4 | S3 | S3 | D4 | D4 | Dic3 | Dic3 | D6 | C4xS3 | C3:D4 | D12 | C3:D4 | C4wrC2 | S32 | S3xDic3 | D6:S3 | C3:D12 | D12:C4 | Q8:3Dic3 | D12:4Dic3 |
kernel | D12:4Dic3 | C3xC4.Dic3 | C4xC3:Dic3 | C3xC4oD12 | C3xDic6 | C3xD12 | C4.Dic3 | C4oD12 | C3xC12 | C62 | Dic6 | D12 | C2xC12 | C12 | C12 | C2xC6 | C2xC6 | C32 | C2xC4 | C4 | C4 | C22 | C3 | C3 | C1 |
# reps | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 2 | 2 | 4 | 1 | 1 | 1 | 1 | 2 | 2 | 4 |
Matrix representation of D12:4Dic3 ►in GL4(F5) generated by
0 | 0 | 4 | 0 |
0 | 0 | 0 | 1 |
4 | 0 | 2 | 0 |
0 | 1 | 0 | 3 |
0 | 3 | 0 | 0 |
2 | 0 | 0 | 0 |
0 | 4 | 0 | 2 |
1 | 0 | 3 | 0 |
1 | 0 | 3 | 0 |
0 | 0 | 0 | 3 |
3 | 0 | 0 | 0 |
0 | 3 | 0 | 4 |
2 | 0 | 1 | 0 |
0 | 1 | 0 | 3 |
0 | 0 | 3 | 0 |
0 | 0 | 0 | 4 |
G:=sub<GL(4,GF(5))| [0,0,4,0,0,0,0,1,4,0,2,0,0,1,0,3],[0,2,0,1,3,0,4,0,0,0,0,3,0,0,2,0],[1,0,3,0,0,0,0,3,3,0,0,0,0,3,0,4],[2,0,0,0,0,1,0,0,1,0,3,0,0,3,0,4] >;
D12:4Dic3 in GAP, Magma, Sage, TeX
D_{12}\rtimes_4{\rm Dic}_3
% in TeX
G:=Group("D12:4Dic3");
// GroupNames label
G:=SmallGroup(288,216);
// by ID
G=gap.SmallGroup(288,216);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,141,36,100,675,346,80,1356,9414]);
// Polycyclic
G:=Group<a,b,c,d|a^12=b^2=c^6=1,d^2=c^3,b*a*b=a^-1,a*c=c*a,d*a*d^-1=a^5,c*b*c^-1=a^6*b,d*b*d^-1=a^7*b,d*c*d^-1=c^-1>;
// generators/relations