metabelian, supersoluble, monomial
Aliases: D12.4D6, C24.31D6, Dic12⋊6S3, Dic6.4D6, C8.3S32, C24⋊C2⋊4S3, C6.34(S3×D4), C24⋊S3⋊2C2, (C3×Dic12)⋊1C2, C3⋊2(D4.D6), C3⋊2(Q16⋊S3), C3⋊Dic3.14D4, (C3×C24).1C22, C32⋊2Q16⋊4C2, Dic3.D6⋊2C2, Dic6⋊S3⋊4C2, D12⋊S3.1C2, (C3×C12).56C23, C12.76(C22×S3), (C3×D12).9C22, C32⋊5(C8.C22), C2.11(D6⋊D6), C32⋊4C8.4C22, (C3×Dic6).9C22, C4.71(C2×S32), (C3×C24⋊C2)⋊2C2, (C2×C3⋊S3).18D4, (C3×C6).40(C2×D4), (C4×C3⋊S3).10C22, SmallGroup(288,459)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D12.4D6
G = < a,b,c,d | a12=b2=1, c6=a3, d2=a6, bab=a-1, ac=ca, dad-1=a7, cbc-1=dbd-1=a3b, dcd-1=a9c5 >
Subgroups: 530 in 130 conjugacy classes, 38 normal (all characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, C6, C6, C8, C8, C2×C4, D4, Q8, C32, Dic3, C12, C12, D6, C2×C6, M4(2), SD16, Q16, C2×Q8, C4○D4, C3×S3, C3⋊S3, C3×C6, C3⋊C8, C24, C24, Dic6, Dic6, C4×S3, D12, D12, C2×Dic3, C3⋊D4, C3×D4, C3×Q8, C8.C22, C3×Dic3, C3⋊Dic3, C3×C12, S3×C6, C2×C3⋊S3, C8⋊S3, C24⋊C2, Dic12, D4.S3, Q8⋊2S3, C3⋊Q16, C3×SD16, C3×Q16, D4⋊2S3, S3×Q8, Q8⋊3S3, C32⋊4C8, C3×C24, S3×Dic3, C6.D6, C3⋊D12, C32⋊2Q8, C3×Dic6, C3×D12, C4×C3⋊S3, D4.D6, Q16⋊S3, Dic6⋊S3, C32⋊2Q16, C3×C24⋊C2, C3×Dic12, C24⋊S3, D12⋊S3, Dic3.D6, D12.4D6
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C22×S3, C8.C22, S32, S3×D4, C2×S32, D4.D6, Q16⋊S3, D6⋊D6, D12.4D6
►On 48 points
(1 3 5 7 9 11 13 15 17 19 21 23)(2 4 6 8 10 12 14 16 18 20 22 24)(25 35 45 31 41 27 37 47 33 43 29 39)(26 36 46 32 42 28 38 48 34 44 30 40)
(1 30)(2 25)(3 44)(4 39)(5 34)(6 29)(7 48)(8 43)(9 38)(10 33)(11 28)(12 47)(13 42)(14 37)(15 32)(16 27)(17 46)(18 41)(19 36)(20 31)(21 26)(22 45)(23 40)(24 35)
(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 27 13 39)(2 26 14 38)(3 25 15 37)(4 48 16 36)(5 47 17 35)(6 46 18 34)(7 45 19 33)(8 44 20 32)(9 43 21 31)(10 42 22 30)(11 41 23 29)(12 40 24 28)
G:=sub<Sym(48)| (1,3,5,7,9,11,13,15,17,19,21,23)(2,4,6,8,10,12,14,16,18,20,22,24)(25,35,45,31,41,27,37,47,33,43,29,39)(26,36,46,32,42,28,38,48,34,44,30,40), (1,30)(2,25)(3,44)(4,39)(5,34)(6,29)(7,48)(8,43)(9,38)(10,33)(11,28)(12,47)(13,42)(14,37)(15,32)(16,27)(17,46)(18,41)(19,36)(20,31)(21,26)(22,45)(23,40)(24,35), (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), (1,27,13,39)(2,26,14,38)(3,25,15,37)(4,48,16,36)(5,47,17,35)(6,46,18,34)(7,45,19,33)(8,44,20,32)(9,43,21,31)(10,42,22,30)(11,41,23,29)(12,40,24,28)>;
G:=Group( (1,3,5,7,9,11,13,15,17,19,21,23)(2,4,6,8,10,12,14,16,18,20,22,24)(25,35,45,31,41,27,37,47,33,43,29,39)(26,36,46,32,42,28,38,48,34,44,30,40), (1,30)(2,25)(3,44)(4,39)(5,34)(6,29)(7,48)(8,43)(9,38)(10,33)(11,28)(12,47)(13,42)(14,37)(15,32)(16,27)(17,46)(18,41)(19,36)(20,31)(21,26)(22,45)(23,40)(24,35), (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), (1,27,13,39)(2,26,14,38)(3,25,15,37)(4,48,16,36)(5,47,17,35)(6,46,18,34)(7,45,19,33)(8,44,20,32)(9,43,21,31)(10,42,22,30)(11,41,23,29)(12,40,24,28) );
G=PermutationGroup([[(1,3,5,7,9,11,13,15,17,19,21,23),(2,4,6,8,10,12,14,16,18,20,22,24),(25,35,45,31,41,27,37,47,33,43,29,39),(26,36,46,32,42,28,38,48,34,44,30,40)], [(1,30),(2,25),(3,44),(4,39),(5,34),(6,29),(7,48),(8,43),(9,38),(10,33),(11,28),(12,47),(13,42),(14,37),(15,32),(16,27),(17,46),(18,41),(19,36),(20,31),(21,26),(22,45),(23,40),(24,35)], [(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)], [(1,27,13,39),(2,26,14,38),(3,25,15,37),(4,48,16,36),(5,47,17,35),(6,46,18,34),(7,45,19,33),(8,44,20,32),(9,43,21,31),(10,42,22,30),(11,41,23,29),(12,40,24,28)]])
33 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 4A | 4B | 4C | 4D | 4E | 6A | 6B | 6C | 6D | 8A | 8B | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 24A | ··· | 24H |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 12 | 18 | 2 | 2 | 4 | 2 | 12 | 12 | 12 | 18 | 2 | 2 | 4 | 24 | 4 | 36 | 4 | 4 | 4 | 4 | 24 | 24 | 24 | 4 | ··· | 4 |
33 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | + | + | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | S3 | D4 | D4 | D6 | D6 | D6 | C8.C22 | S32 | S3×D4 | C2×S32 | D4.D6 | Q16⋊S3 | D6⋊D6 | D12.4D6 |
| kernel | D12.4D6 | Dic6⋊S3 | C32⋊2Q16 | C3×C24⋊C2 | C3×Dic12 | C24⋊S3 | D12⋊S3 | Dic3.D6 | C24⋊C2 | Dic12 | C3⋊Dic3 | C2×C3⋊S3 | C24 | Dic6 | D12 | C32 | C8 | C6 | C4 | C3 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 3 | 1 | 1 | 1 | 2 | 1 | 2 | 2 | 2 | 4 |
Matrix representation of D12.4D6 ►in GL4(𝔽5) generated by
| 0 | 0 | 2 | 0 |
| 1 | 2 | 0 | 0 |
| 4 | 4 | 4 | 1 |
| 4 | 4 | 0 | 4 |
| 2 | 3 | 3 | 2 |
| 1 | 1 | 0 | 2 |
| 4 | 3 | 4 | 3 |
| 1 | 1 | 1 | 3 |
| 4 | 4 | 0 | 1 |
| 4 | 3 | 1 | 0 |
| 4 | 1 | 4 | 2 |
| 2 | 4 | 4 | 0 |
| 3 | 3 | 0 | 2 |
| 0 | 0 | 0 | 3 |
| 2 | 3 | 2 | 1 |
| 0 | 3 | 0 | 0 |
G:=sub<GL(4,GF(5))| [0,1,4,4,0,2,4,4,2,0,4,0,0,0,1,4],[2,1,4,1,3,1,3,1,3,0,4,1,2,2,3,3],[4,4,4,2,4,3,1,4,0,1,4,4,1,0,2,0],[3,0,2,0,3,0,3,3,0,0,2,0,2,3,1,0] >;
D12.4D6 in GAP, Magma, Sage, TeX
D_{12}._4D_6 % in TeX
G:=Group("D12.4D6"); // GroupNames label
G:=SmallGroup(288,459);
// by ID
G=gap.SmallGroup(288,459);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,120,254,303,58,675,346,80,1356,9414]);
// Polycyclic
G:=Group<a,b,c,d|a^12=b^2=1,c^6=a^3,d^2=a^6,b*a*b=a^-1,a*c=c*a,d*a*d^-1=a^7,c*b*c^-1=d*b*d^-1=a^3*b,d*c*d^-1=a^9*c^5>;
// generators/relations