metabelian, supersoluble, monomial
Aliases: Dic3⋊2D12, C62.56C23, (S3×C6)⋊2D4, (C2×D12)⋊3S3, (C6×D12)⋊22C2, D6⋊2(C3⋊D4), D6⋊Dic3⋊9C2, (C3×Dic3)⋊1D4, C6.13(C2×D12), C2.17(S3×D12), C6.139(S3×D4), (C2×C12).19D6, C3⋊4(C12⋊D4), Dic3⋊C4⋊17S3, C32⋊4(C4⋊D4), C6.11D12⋊9C2, C6.8(D4⋊2S3), (C2×Dic3).22D6, (C22×S3).35D6, C3⋊1(C23.14D6), (C6×C12).181C22, C6.32(Q8⋊3S3), C2.15(D12⋊S3), (C6×Dic3).35C22, (C2×C4).22S32, (C2×S3×Dic3)⋊1C2, (C3×C6).45(C2×D4), C2.13(S3×C3⋊D4), C6.33(C2×C3⋊D4), (C2×C3⋊D12)⋊1C2, C22.103(C2×S32), (S3×C2×C6).17C22, (C3×Dic3⋊C4)⋊15C2, (C3×C6).33(C4○D4), (C2×C6).75(C22×S3), (C22×C3⋊S3).14C22, (C2×C3⋊Dic3).40C22, SmallGroup(288,534)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic3⋊D12
G = < a,b,c,d | a6=c12=d2=1, b2=a3, bab-1=a-1, ac=ca, ad=da, cbc-1=dbd=a3b, dcd=c-1 >
Subgroups: 922 in 205 conjugacy classes, 52 normal (44 characteristic)
C1, C2, C2, C3, C3, C4, C22, C22, S3, C6, C6, C2×C4, C2×C4, D4, C23, C32, Dic3, Dic3, C12, D6, D6, C2×C6, C2×C6, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C3×S3, C3⋊S3, C3×C6, C4×S3, D12, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C3×D4, C22×S3, C22×S3, C22×C6, C4⋊D4, C3×Dic3, C3×Dic3, C3⋊Dic3, C3×C12, S3×C6, S3×C6, C2×C3⋊S3, C62, Dic3⋊C4, D6⋊C4, C6.D4, C3×C4⋊C4, S3×C2×C4, C2×D12, C2×D12, C22×Dic3, C2×C3⋊D4, C6×D4, S3×Dic3, C3⋊D12, C3×D12, C6×Dic3, C2×C3⋊Dic3, C6×C12, S3×C2×C6, C22×C3⋊S3, C12⋊D4, C23.14D6, D6⋊Dic3, C3×Dic3⋊C4, C6.11D12, C2×S3×Dic3, C2×C3⋊D12, C6×D12, Dic3⋊D12
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C4○D4, D12, C3⋊D4, C22×S3, C4⋊D4, S32, C2×D12, S3×D4, D4⋊2S3, Q8⋊3S3, C2×C3⋊D4, C2×S32, C12⋊D4, C23.14D6, D12⋊S3, S3×D12, S3×C3⋊D4, Dic3⋊D12
►On 48 points
(1 32 5 36 9 28)(2 33 6 25 10 29)(3 34 7 26 11 30)(4 35 8 27 12 31)(13 48 21 44 17 40)(14 37 22 45 18 41)(15 38 23 46 19 42)(16 39 24 47 20 43)
(1 47 36 16)(2 17 25 48)(3 37 26 18)(4 19 27 38)(5 39 28 20)(6 21 29 40)(7 41 30 22)(8 23 31 42)(9 43 32 24)(10 13 33 44)(11 45 34 14)(12 15 35 46)
(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 43)(2 42)(3 41)(4 40)(5 39)(6 38)(7 37)(8 48)(9 47)(10 46)(11 45)(12 44)(13 35)(14 34)(15 33)(16 32)(17 31)(18 30)(19 29)(20 28)(21 27)(22 26)(23 25)(24 36)
G:=sub<Sym(48)| (1,32,5,36,9,28)(2,33,6,25,10,29)(3,34,7,26,11,30)(4,35,8,27,12,31)(13,48,21,44,17,40)(14,37,22,45,18,41)(15,38,23,46,19,42)(16,39,24,47,20,43), (1,47,36,16)(2,17,25,48)(3,37,26,18)(4,19,27,38)(5,39,28,20)(6,21,29,40)(7,41,30,22)(8,23,31,42)(9,43,32,24)(10,13,33,44)(11,45,34,14)(12,15,35,46), (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), (1,43)(2,42)(3,41)(4,40)(5,39)(6,38)(7,37)(8,48)(9,47)(10,46)(11,45)(12,44)(13,35)(14,34)(15,33)(16,32)(17,31)(18,30)(19,29)(20,28)(21,27)(22,26)(23,25)(24,36)>;
G:=Group( (1,32,5,36,9,28)(2,33,6,25,10,29)(3,34,7,26,11,30)(4,35,8,27,12,31)(13,48,21,44,17,40)(14,37,22,45,18,41)(15,38,23,46,19,42)(16,39,24,47,20,43), (1,47,36,16)(2,17,25,48)(3,37,26,18)(4,19,27,38)(5,39,28,20)(6,21,29,40)(7,41,30,22)(8,23,31,42)(9,43,32,24)(10,13,33,44)(11,45,34,14)(12,15,35,46), (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), (1,43)(2,42)(3,41)(4,40)(5,39)(6,38)(7,37)(8,48)(9,47)(10,46)(11,45)(12,44)(13,35)(14,34)(15,33)(16,32)(17,31)(18,30)(19,29)(20,28)(21,27)(22,26)(23,25)(24,36) );
G=PermutationGroup([[(1,32,5,36,9,28),(2,33,6,25,10,29),(3,34,7,26,11,30),(4,35,8,27,12,31),(13,48,21,44,17,40),(14,37,22,45,18,41),(15,38,23,46,19,42),(16,39,24,47,20,43)], [(1,47,36,16),(2,17,25,48),(3,37,26,18),(4,19,27,38),(5,39,28,20),(6,21,29,40),(7,41,30,22),(8,23,31,42),(9,43,32,24),(10,13,33,44),(11,45,34,14),(12,15,35,46)], [(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)], [(1,43),(2,42),(3,41),(4,40),(5,39),(6,38),(7,37),(8,48),(9,47),(10,46),(11,45),(12,44),(13,35),(14,34),(15,33),(16,32),(17,31),(18,30),(19,29),(20,28),(21,27),(22,26),(23,25),(24,36)]])
42 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3A | 3B | 3C | 4A | 4B | 4C | 4D | 4E | 4F | 6A | ··· | 6F | 6G | 6H | 6I | 6J | 6K | 6L | 6M | 12A | ··· | 12H | 12I | 12J | 12K | 12L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 12 | ··· | 12 | 12 | 12 | 12 | 12 |
| size | 1 | 1 | 1 | 1 | 6 | 6 | 12 | 36 | 2 | 2 | 4 | 4 | 6 | 6 | 12 | 18 | 18 | 2 | ··· | 2 | 4 | 4 | 4 | 12 | 12 | 12 | 12 | 4 | ··· | 4 | 12 | 12 | 12 | 12 |
42 irreducible representations
| dim | 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 | S3 | S3 | D4 | D4 | D6 | D6 | D6 | C4○D4 | D12 | C3⋊D4 | S32 | S3×D4 | D4⋊2S3 | Q8⋊3S3 | C2×S32 | D12⋊S3 | S3×D12 | S3×C3⋊D4 |
| kernel | Dic3⋊D12 | D6⋊Dic3 | C3×Dic3⋊C4 | C6.11D12 | C2×S3×Dic3 | C2×C3⋊D12 | C6×D12 | Dic3⋊C4 | C2×D12 | C3×Dic3 | S3×C6 | C2×Dic3 | C2×C12 | C22×S3 | C3×C6 | Dic3 | D6 | C2×C4 | C6 | C6 | C6 | C22 | C2 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 1 | 2 | 1 | 1 | 1 | 2 | 2 | 2 |
Matrix representation of Dic3⋊D12 ►in GL8(ℤ)
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | -1 | -1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 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 |
| 0 | 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 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | -1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | -1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | -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 | 1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
G:=sub<GL(8,Integers())| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,0],[0,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0],[-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0] >;
Dic3⋊D12 in GAP, Magma, Sage, TeX
{\rm Dic}_3\rtimes D_{12} % in TeX
G:=Group("Dic3:D12"); // GroupNames label
G:=SmallGroup(288,534);
// by ID
G=gap.SmallGroup(288,534);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,112,422,135,58,1356,9414]);
// Polycyclic
G:=Group<a,b,c,d|a^6=c^12=d^2=1,b^2=a^3,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d=a^3*b,d*c*d=c^-1>;
// generators/relations