metabelian, supersoluble, monomial
Aliases: Dic6⋊12D6, C62.3C23, D4⋊6S32, (C4×S3)⋊8D6, (C3×D4)⋊9D6, C3⋊D4⋊3D6, D4⋊2S3⋊6S3, Dic3⋊D6⋊5C2, (S3×C12)⋊6C22, (C2×Dic3)⋊14D6, D6.3D6⋊4C2, D6.6D6⋊9C2, (C3×C6).19C24, C6.19(S3×C23), C3⋊D12⋊5C22, (S3×C6).10C23, C12.31(C22×S3), (C3×C12).31C23, (C6×Dic3)⋊6C22, D6.11(C22×S3), C32⋊2Q8⋊4C22, C6.D6⋊9C22, C32⋊7D4⋊3C22, C12⋊S3⋊10C22, Dic3.D6⋊12C2, (C3×Dic6)⋊14C22, (S3×Dic3)⋊14C22, (D4×C32)⋊11C22, C3⋊Dic3.21C23, Dic3.9(C22×S3), (C3×Dic3).13C23, (C4×S32)⋊6C2, C4.31(C2×S32), (D4×C3⋊S3)⋊7C2, C3⋊4(S3×C4○D4), C22.3(C2×S32), C32⋊8(C2×C4○D4), C3⋊S3⋊2(C4○D4), C2.21(C22×S32), (C3×D4⋊2S3)⋊9C2, (C2×S32).12C22, (C2×C6.D6)⋊4C2, (C3×C3⋊D4)⋊3C22, (C2×C6).4(C22×S3), (C4×C3⋊S3).43C22, (C2×C3⋊S3).24C23, (C22×C3⋊S3).58C22, SmallGroup(288,960)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic6⋊12D6
G = < a,b,c,d | a12=c6=d2=1, b2=a6, bab-1=a-1, cac-1=a7, dad=a5, cbc-1=a6b, bd=db, dcd=c-1 >
Subgroups: 1346 in 355 conjugacy classes, 110 normal (18 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, C22, S3, C6, C6, C2×C4, D4, D4, Q8, C23, C32, Dic3, Dic3, C12, C12, D6, D6, C2×C6, C2×C6, C22×C4, C2×D4, C2×Q8, C4○D4, C3×S3, C3⋊S3, C3⋊S3, C3×C6, C3×C6, Dic6, Dic6, C4×S3, C4×S3, D12, C2×Dic3, C2×Dic3, C3⋊D4, C3⋊D4, C2×C12, C3×D4, C3×D4, C3×Q8, C22×S3, C2×C4○D4, C3×Dic3, C3⋊Dic3, C3×C12, S32, S3×C6, C2×C3⋊S3, C2×C3⋊S3, C2×C3⋊S3, C62, S3×C2×C4, C4○D12, S3×D4, D4⋊2S3, D4⋊2S3, S3×Q8, Q8⋊3S3, C3×C4○D4, S3×Dic3, C6.D6, C6.D6, C3⋊D12, C32⋊2Q8, C3×Dic6, S3×C12, C6×Dic3, C3×C3⋊D4, C4×C3⋊S3, C12⋊S3, C32⋊7D4, D4×C32, C2×S32, C22×C3⋊S3, S3×C4○D4, Dic3.D6, D6.6D6, C4×S32, D6.3D6, C2×C6.D6, Dic3⋊D6, C3×D4⋊2S3, D4×C3⋊S3, Dic6⋊12D6
Quotients: C1, C2, C22, S3, C23, D6, C4○D4, C24, C22×S3, C2×C4○D4, S32, S3×C23, C2×S32, S3×C4○D4, C22×S32, Dic6⋊12D6
►On 24 points - transitive group 24T608
(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 16 7 22)(2 15 8 21)(3 14 9 20)(4 13 10 19)(5 24 11 18)(6 23 12 17)
(1 2 9 10 5 6)(3 4 11 12 7 8)(13 24 17 16 21 20)(14 19 18 23 22 15)
(1 3)(2 8)(4 6)(5 11)(7 9)(10 12)(13 23)(14 16)(15 21)(17 19)(18 24)(20 22)
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), (1,16,7,22)(2,15,8,21)(3,14,9,20)(4,13,10,19)(5,24,11,18)(6,23,12,17), (1,2,9,10,5,6)(3,4,11,12,7,8)(13,24,17,16,21,20)(14,19,18,23,22,15), (1,3)(2,8)(4,6)(5,11)(7,9)(10,12)(13,23)(14,16)(15,21)(17,19)(18,24)(20,22)>;
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), (1,16,7,22)(2,15,8,21)(3,14,9,20)(4,13,10,19)(5,24,11,18)(6,23,12,17), (1,2,9,10,5,6)(3,4,11,12,7,8)(13,24,17,16,21,20)(14,19,18,23,22,15), (1,3)(2,8)(4,6)(5,11)(7,9)(10,12)(13,23)(14,16)(15,21)(17,19)(18,24)(20,22) );
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)], [(1,16,7,22),(2,15,8,21),(3,14,9,20),(4,13,10,19),(5,24,11,18),(6,23,12,17)], [(1,2,9,10,5,6),(3,4,11,12,7,8),(13,24,17,16,21,20),(14,19,18,23,22,15)], [(1,3),(2,8),(4,6),(5,11),(7,9),(10,12),(13,23),(14,16),(15,21),(17,19),(18,24),(20,22)]])
G:=TransitiveGroup(24,608);
45 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 3A | 3B | 3C | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 6A | 6B | 6C | ··· | 6G | 6H | 6I | 6J | 6K | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 12I | 12J | 12K |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 |
| size | 1 | 1 | 2 | 2 | 6 | 6 | 9 | 9 | 18 | 18 | 2 | 2 | 4 | 2 | 3 | 3 | 3 | 3 | 6 | 6 | 6 | 6 | 18 | 2 | 2 | 4 | ··· | 4 | 8 | 8 | 12 | 12 | 4 | 4 | 6 | 6 | 6 | 6 | 8 | 12 | 12 | 12 | 12 |
45 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 8 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D6 | D6 | D6 | D6 | D6 | C4○D4 | S32 | C2×S32 | C2×S32 | S3×C4○D4 | Dic6⋊12D6 |
| kernel | Dic6⋊12D6 | Dic3.D6 | D6.6D6 | C4×S32 | D6.3D6 | C2×C6.D6 | Dic3⋊D6 | C3×D4⋊2S3 | D4×C3⋊S3 | D4⋊2S3 | Dic6 | C4×S3 | C2×Dic3 | C3⋊D4 | C3×D4 | C3⋊S3 | D4 | C4 | C22 | C3 | C1 |
| # reps | 1 | 1 | 2 | 1 | 4 | 2 | 2 | 2 | 1 | 2 | 2 | 2 | 4 | 4 | 2 | 4 | 1 | 1 | 2 | 4 | 1 |
Matrix representation of Dic6⋊12D6 ►in GL6(𝔽13)
| 12 | 5 | 0 | 0 | 0 | 0 |
| 10 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 12 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 5 | 0 | 0 | 0 | 0 | 0 |
| 2 | 8 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 8 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 1 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 12 | 1 |
G:=sub<GL(6,GF(13))| [12,10,0,0,0,0,5,1,0,0,0,0,0,0,0,12,0,0,0,0,1,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[5,2,0,0,0,0,0,8,0,0,0,0,0,0,0,12,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,8,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,12,0,0,0,0,1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,12,12,0,0,0,0,0,1] >;
Dic6⋊12D6 in GAP, Magma, Sage, TeX
{\rm Dic}_6\rtimes_{12}D_6 % in TeX
G:=Group("Dic6:12D6"); // GroupNames label
G:=SmallGroup(288,960);
// by ID
G=gap.SmallGroup(288,960);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,100,675,346,185,1356,9414]);
// Polycyclic
G:=Group<a,b,c,d|a^12=c^6=d^2=1,b^2=a^6,b*a*b^-1=a^-1,c*a*c^-1=a^7,d*a*d=a^5,c*b*c^-1=a^6*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations