metabelian, supersoluble, monomial
Aliases: D12⋊4Dic3, C62.23D4, Dic6⋊4Dic3, C32⋊3C4≀C2, (C3×D12)⋊3C4, (C2×C6).8D12, C12.18(C4×S3), (C3×Dic6)⋊3C4, C4○D12.2S3, (C2×C12).79D6, C4.Dic3⋊4S3, C4.8(S3×Dic3), C6.38(D6⋊C4), (C3×C12).109D4, C12.9(C2×Dic3), C3⋊3(D12⋊C4), C12.95(C3⋊D4), C2.9(D6⋊Dic3), (C6×C12).30C22, C3⋊1(Q8⋊3Dic3), C4.17(D6⋊S3), C6.8(C6.D4), C22.7(C3⋊D12), (C2×C4).101S32, (C4×C3⋊Dic3)⋊1C2, (C3×C12).28(C2×C4), (C3×C4○D12).1C2, (C2×C6).10(C3⋊D4), (C3×C4.Dic3)⋊15C2, (C3×C6).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, C2×C4, C2×C4, D4, Q8, C32, Dic3, C12, C12, D6, C2×C6, C2×C6, C42, M4(2), C4○D4, C3×S3, C3×C6, C3×C6, C3⋊C8, C24, Dic6, C4×S3, D12, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C3×D4, C3×Q8, C4≀C2, C3×Dic3, C3⋊Dic3, C3×C12, S3×C6, C62, C4.Dic3, C4×Dic3, C3×M4(2), C4○D12, C3×C4○D4, C3×C3⋊C8, C3×Dic6, S3×C12, C3×D12, C3×C3⋊D4, C2×C3⋊Dic3, C6×C12, D12⋊C4, Q8⋊3Dic3, C3×C4.Dic3, C4×C3⋊Dic3, C3×C4○D12, D12⋊4Dic3
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Dic3, D6, C22⋊C4, C4×S3, D12, C2×Dic3, C3⋊D4, C4≀C2, S32, D6⋊C4, C6.D4, S3×Dic3, D6⋊S3, C3⋊D12, D12⋊C4, Q8⋊3Dic3, D6⋊Dic3, D12⋊4Dic3
►On 24 points - transitive group 24T612
(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 | C4×S3 | C3⋊D4 | D12 | C3⋊D4 | C4≀C2 | S32 | S3×Dic3 | D6⋊S3 | C3⋊D12 | D12⋊C4 | Q8⋊3Dic3 | D12⋊4Dic3 |
| kernel | D12⋊4Dic3 | C3×C4.Dic3 | C4×C3⋊Dic3 | C3×C4○D12 | C3×Dic6 | C3×D12 | C4.Dic3 | C4○D12 | C3×C12 | C62 | Dic6 | D12 | C2×C12 | C12 | C12 | C2×C6 | C2×C6 | C32 | C2×C4 | 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