metabelian, supersoluble, monomial
Aliases: C24.4D6, D6.7D12, Dic12⋊2S3, Dic6.16D6, Dic3.9D12, C8.2S32, C3⋊C8.2D6, C6.7(S3×D4), C8⋊S3⋊4S3, (S3×C6).4D4, (C4×S3).4D6, C6.7(C2×D12), C24⋊2S3⋊2C2, (S3×Dic6)⋊3C2, C2.12(S3×D12), (C3×Dic12)⋊2C2, C3⋊1(Q16⋊S3), C3⋊2(C8.D6), (C3×C24).2C22, (C3×Dic3).4D4, C32⋊3Q16⋊4C2, C32⋊5SD16⋊2C2, (S3×C12).6C22, (C3×C12).49C23, D6.6D6.1C2, C32⋊4(C8.C22), C12⋊S3.3C22, C12.123(C22×S3), (C3×Dic6).3C22, C32⋊4Q8.3C22, C4.46(C2×S32), (C3×C8⋊S3)⋊2C2, (C3×C6).33(C2×D4), (C3×C3⋊C8).2C22, SmallGroup(288,449)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic12⋊S3
G = < a,b,c,d | a24=c3=d2=1, b2=a12, bab-1=a-1, ac=ca, dad=a13, bc=cb, bd=db, dcd=c-1 >
Subgroups: 586 in 130 conjugacy classes, 40 normal (all characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, C6, C6, C8, C8, C2×C4, D4, Q8, C32, Dic3, Dic3, C12, C12, D6, 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, C4×S3, D12, C2×Dic3, C3⋊D4, C2×C12, C3×Q8, C8.C22, C3×Dic3, C3×Dic3, C3⋊Dic3, C3×C12, S3×C6, C2×C3⋊S3, C8⋊S3, C24⋊C2, Dic12, Dic12, Q8⋊2S3, C3⋊Q16, C3×M4(2), C3×Q16, C2×Dic6, C4○D12, S3×Q8, Q8⋊3S3, C3×C3⋊C8, C3×C24, S3×Dic3, C6.D6, C3⋊D12, C32⋊2Q8, C3×Dic6, S3×C12, C32⋊4Q8, C12⋊S3, C8.D6, Q16⋊S3, C32⋊5SD16, C32⋊3Q16, C3×C8⋊S3, C3×Dic12, C24⋊2S3, S3×Dic6, D6.6D6, Dic12⋊S3
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, D12, C22×S3, C8.C22, S32, C2×D12, S3×D4, C2×S32, C8.D6, Q16⋊S3, S3×D12, Dic12⋊S3
►On 48 points
(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 31 13 43)(2 30 14 42)(3 29 15 41)(4 28 16 40)(5 27 17 39)(6 26 18 38)(7 25 19 37)(8 48 20 36)(9 47 21 35)(10 46 22 34)(11 45 23 33)(12 44 24 32)
(1 17 9)(2 18 10)(3 19 11)(4 20 12)(5 21 13)(6 22 14)(7 23 15)(8 24 16)(25 33 41)(26 34 42)(27 35 43)(28 36 44)(29 37 45)(30 38 46)(31 39 47)(32 40 48)
(1 28)(2 41)(3 30)(4 43)(5 32)(6 45)(7 34)(8 47)(9 36)(10 25)(11 38)(12 27)(13 40)(14 29)(15 42)(16 31)(17 44)(18 33)(19 46)(20 35)(21 48)(22 37)(23 26)(24 39)
G:=sub<Sym(48)| (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), (1,31,13,43)(2,30,14,42)(3,29,15,41)(4,28,16,40)(5,27,17,39)(6,26,18,38)(7,25,19,37)(8,48,20,36)(9,47,21,35)(10,46,22,34)(11,45,23,33)(12,44,24,32), (1,17,9)(2,18,10)(3,19,11)(4,20,12)(5,21,13)(6,22,14)(7,23,15)(8,24,16)(25,33,41)(26,34,42)(27,35,43)(28,36,44)(29,37,45)(30,38,46)(31,39,47)(32,40,48), (1,28)(2,41)(3,30)(4,43)(5,32)(6,45)(7,34)(8,47)(9,36)(10,25)(11,38)(12,27)(13,40)(14,29)(15,42)(16,31)(17,44)(18,33)(19,46)(20,35)(21,48)(22,37)(23,26)(24,39)>;
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)(25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48), (1,31,13,43)(2,30,14,42)(3,29,15,41)(4,28,16,40)(5,27,17,39)(6,26,18,38)(7,25,19,37)(8,48,20,36)(9,47,21,35)(10,46,22,34)(11,45,23,33)(12,44,24,32), (1,17,9)(2,18,10)(3,19,11)(4,20,12)(5,21,13)(6,22,14)(7,23,15)(8,24,16)(25,33,41)(26,34,42)(27,35,43)(28,36,44)(29,37,45)(30,38,46)(31,39,47)(32,40,48), (1,28)(2,41)(3,30)(4,43)(5,32)(6,45)(7,34)(8,47)(9,36)(10,25)(11,38)(12,27)(13,40)(14,29)(15,42)(16,31)(17,44)(18,33)(19,46)(20,35)(21,48)(22,37)(23,26)(24,39) );
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),(25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48)], [(1,31,13,43),(2,30,14,42),(3,29,15,41),(4,28,16,40),(5,27,17,39),(6,26,18,38),(7,25,19,37),(8,48,20,36),(9,47,21,35),(10,46,22,34),(11,45,23,33),(12,44,24,32)], [(1,17,9),(2,18,10),(3,19,11),(4,20,12),(5,21,13),(6,22,14),(7,23,15),(8,24,16),(25,33,41),(26,34,42),(27,35,43),(28,36,44),(29,37,45),(30,38,46),(31,39,47),(32,40,48)], [(1,28),(2,41),(3,30),(4,43),(5,32),(6,45),(7,34),(8,47),(9,36),(10,25),(11,38),(12,27),(13,40),(14,29),(15,42),(16,31),(17,44),(18,33),(19,46),(20,35),(21,48),(22,37),(23,26),(24,39)]])
36 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 | 12H | 24A | ··· | 24H | 24I | 24J |
| 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 | 12 | 24 | ··· | 24 | 24 | 24 |
| size | 1 | 1 | 6 | 36 | 2 | 2 | 4 | 2 | 6 | 12 | 12 | 36 | 2 | 2 | 4 | 12 | 4 | 12 | 2 | 2 | 4 | 4 | 4 | 12 | 24 | 24 | 4 | ··· | 4 | 12 | 12 |
36 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 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 | D6 | D12 | D12 | C8.C22 | S32 | S3×D4 | C2×S32 | C8.D6 | Q16⋊S3 | S3×D12 | Dic12⋊S3 |
| kernel | Dic12⋊S3 | C32⋊5SD16 | C32⋊3Q16 | C3×C8⋊S3 | C3×Dic12 | C24⋊2S3 | S3×Dic6 | D6.6D6 | C8⋊S3 | Dic12 | C3×Dic3 | S3×C6 | C3⋊C8 | C24 | Dic6 | C4×S3 | Dic3 | D6 | C32 | C8 | C6 | C4 | C3 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 |
Matrix representation of Dic12⋊S3 ►in GL4(𝔽73) generated by
| 36 | 25 | 0 | 0 |
| 48 | 11 | 0 | 0 |
| 0 | 0 | 62 | 25 |
| 0 | 0 | 48 | 37 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 72 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 |
| 72 | 72 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 72 | 72 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 48 | 11 |
| 0 | 0 | 36 | 25 |
| 25 | 62 | 0 | 0 |
| 37 | 48 | 0 | 0 |
G:=sub<GL(4,GF(73))| [36,48,0,0,25,11,0,0,0,0,62,48,0,0,25,37],[0,0,72,0,0,0,0,72,1,0,0,0,0,1,0,0],[72,1,0,0,72,0,0,0,0,0,72,1,0,0,72,0],[0,0,25,37,0,0,62,48,48,36,0,0,11,25,0,0] >;
Dic12⋊S3 in GAP, Magma, Sage, TeX
{\rm Dic}_{12}\rtimes S_3 % in TeX
G:=Group("Dic12:S3"); // GroupNames label
G:=SmallGroup(288,449);
// by ID
G=gap.SmallGroup(288,449);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,120,422,135,142,346,80,1356,9414]);
// Polycyclic
G:=Group<a,b,c,d|a^24=c^3=d^2=1,b^2=a^12,b*a*b^-1=a^-1,a*c=c*a,d*a*d=a^13,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations