metabelian, supersoluble, monomial
Aliases: D6.3D12, C24.51D6, Dic12⋊7S3, Dic6.18D6, Dic3.14D12, C8.10S32, (S3×C8)⋊3S3, C3⋊C8.26D6, (S3×C24)⋊2C2, C6.10(S3×D4), C32⋊5D8⋊7C2, C3⋊3(C4○D24), (C4×S3).38D6, (S3×C6).22D4, C6.10(C2×D12), C2.15(S3×D12), C32⋊5(C4○D8), D6.6D6⋊3C2, C3⋊1(D24⋊C2), (C3×Dic12)⋊10C2, C32⋊5SD16⋊6C2, (C3×C12).53C23, C12.73(C22×S3), (C3×C24).24C22, (C3×Dic3).27D4, (S3×C12).46C22, C12⋊S3.5C22, (C3×Dic6).6C22, C4.49(C2×S32), (C3×C6).37(C2×D4), (C3×C3⋊C8).33C22, SmallGroup(288,456)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D6.3D12
G = < a,b,c,d | a6=b2=1, c12=d2=a3, bab=a-1, ac=ca, ad=da, bc=cb, dbd-1=a3b, dcd-1=a3c11 >
Subgroups: 674 in 135 conjugacy classes, 40 normal (30 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, C2×C8, D8, SD16, Q16, C4○D4, C3×S3, C3⋊S3, C3×C6, C3⋊C8, C24, C24, Dic6, C4×S3, C4×S3, D12, C3⋊D4, C2×C12, C3×Q8, C4○D8, C3×Dic3, C3×Dic3, C3×C12, S3×C6, C2×C3⋊S3, S3×C8, C24⋊C2, D24, Dic12, Q8⋊2S3, C2×C24, C3×Q16, C4○D12, Q8⋊3S3, C3×C3⋊C8, C3×C24, C6.D6, C3⋊D12, C3×Dic6, S3×C12, C12⋊S3, C4○D24, D24⋊C2, C32⋊5SD16, S3×C24, C3×Dic12, C32⋊5D8, D6.6D6, D6.3D12
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, D12, C22×S3, C4○D8, S32, C2×D12, S3×D4, C2×S32, C4○D24, D24⋊C2, S3×D12, D6.3D12
►On 48 points
(1 21 17 13 9 5)(2 22 18 14 10 6)(3 23 19 15 11 7)(4 24 20 16 12 8)(25 29 33 37 41 45)(26 30 34 38 42 46)(27 31 35 39 43 47)(28 32 36 40 44 48)
(1 42)(2 43)(3 44)(4 45)(5 46)(6 47)(7 48)(8 25)(9 26)(10 27)(11 28)(12 29)(13 30)(14 31)(15 32)(16 33)(17 34)(18 35)(19 36)(20 37)(21 38)(22 39)(23 40)(24 41)
(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 33 13 45)(2 32 14 44)(3 31 15 43)(4 30 16 42)(5 29 17 41)(6 28 18 40)(7 27 19 39)(8 26 20 38)(9 25 21 37)(10 48 22 36)(11 47 23 35)(12 46 24 34)
G:=sub<Sym(48)| (1,21,17,13,9,5)(2,22,18,14,10,6)(3,23,19,15,11,7)(4,24,20,16,12,8)(25,29,33,37,41,45)(26,30,34,38,42,46)(27,31,35,39,43,47)(28,32,36,40,44,48), (1,42)(2,43)(3,44)(4,45)(5,46)(6,47)(7,48)(8,25)(9,26)(10,27)(11,28)(12,29)(13,30)(14,31)(15,32)(16,33)(17,34)(18,35)(19,36)(20,37)(21,38)(22,39)(23,40)(24,41), (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), (1,33,13,45)(2,32,14,44)(3,31,15,43)(4,30,16,42)(5,29,17,41)(6,28,18,40)(7,27,19,39)(8,26,20,38)(9,25,21,37)(10,48,22,36)(11,47,23,35)(12,46,24,34)>;
G:=Group( (1,21,17,13,9,5)(2,22,18,14,10,6)(3,23,19,15,11,7)(4,24,20,16,12,8)(25,29,33,37,41,45)(26,30,34,38,42,46)(27,31,35,39,43,47)(28,32,36,40,44,48), (1,42)(2,43)(3,44)(4,45)(5,46)(6,47)(7,48)(8,25)(9,26)(10,27)(11,28)(12,29)(13,30)(14,31)(15,32)(16,33)(17,34)(18,35)(19,36)(20,37)(21,38)(22,39)(23,40)(24,41), (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), (1,33,13,45)(2,32,14,44)(3,31,15,43)(4,30,16,42)(5,29,17,41)(6,28,18,40)(7,27,19,39)(8,26,20,38)(9,25,21,37)(10,48,22,36)(11,47,23,35)(12,46,24,34) );
G=PermutationGroup([[(1,21,17,13,9,5),(2,22,18,14,10,6),(3,23,19,15,11,7),(4,24,20,16,12,8),(25,29,33,37,41,45),(26,30,34,38,42,46),(27,31,35,39,43,47),(28,32,36,40,44,48)], [(1,42),(2,43),(3,44),(4,45),(5,46),(6,47),(7,48),(8,25),(9,26),(10,27),(11,28),(12,29),(13,30),(14,31),(15,32),(16,33),(17,34),(18,35),(19,36),(20,37),(21,38),(22,39),(23,40),(24,41)], [(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)], [(1,33,13,45),(2,32,14,44),(3,31,15,43),(4,30,16,42),(5,29,17,41),(6,28,18,40),(7,27,19,39),(8,26,20,38),(9,25,21,37),(10,48,22,36),(11,47,23,35),(12,46,24,34)]])
45 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3A | 3B | 3C | 4A | 4B | 4C | 4D | 4E | 6A | 6B | 6C | 6D | 6E | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 12I | 24A | 24B | 24C | 24D | 24E | ··· | 24J | 24K | 24L | 24M | 24N |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 24 | 24 | 24 | 24 | 24 | ··· | 24 | 24 | 24 | 24 | 24 |
| size | 1 | 1 | 6 | 36 | 36 | 2 | 2 | 4 | 2 | 3 | 3 | 12 | 12 | 2 | 2 | 4 | 6 | 6 | 2 | 2 | 6 | 6 | 2 | 2 | 4 | 4 | 4 | 6 | 6 | 24 | 24 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 6 | 6 | 6 | 6 |
45 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 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | S3 | D4 | D4 | D6 | D6 | D6 | D6 | D12 | D12 | C4○D8 | C4○D24 | S32 | S3×D4 | C2×S32 | D24⋊C2 | S3×D12 | D6.3D12 |
| kernel | D6.3D12 | C32⋊5SD16 | S3×C24 | C3×Dic12 | C32⋊5D8 | D6.6D6 | S3×C8 | Dic12 | C3×Dic3 | S3×C6 | C3⋊C8 | C24 | Dic6 | C4×S3 | Dic3 | D6 | C32 | C3 | C8 | C6 | C4 | C3 | C2 | C1 |
| # reps | 1 | 2 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 2 | 2 | 4 | 8 | 1 | 1 | 1 | 2 | 2 | 4 |
Matrix representation of D6.3D12 ►in GL6(𝔽73)
| 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 1 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 27 | 0 | 0 | 0 | 0 |
| 46 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 57 | 16 | 0 | 0 | 0 | 0 |
| 57 | 57 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 1 |
| 0 | 0 | 0 | 0 | 72 | 0 |
| 67 | 67 | 0 | 0 | 0 | 0 |
| 67 | 6 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 0 | 72 | 1 |
G:=sub<GL(6,GF(73))| [72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,72,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,46,0,0,0,0,27,0,0,0,0,0,0,0,1,1,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[57,57,0,0,0,0,16,57,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,72,0,0,0,0,1,0],[67,67,0,0,0,0,67,6,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,72,0,0,0,0,0,1] >;
D6.3D12 in GAP, Magma, Sage, TeX
D_6._3D_{12} % in TeX
G:=Group("D6.3D12"); // GroupNames label
G:=SmallGroup(288,456);
// by ID
G=gap.SmallGroup(288,456);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,253,120,135,142,346,80,1356,9414]);
// Polycyclic
G:=Group<a,b,c,d|a^6=b^2=1,c^12=d^2=a^3,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=a^3*b,d*c*d^-1=a^3*c^11>;
// generators/relations