metabelian, supersoluble, monomial
Aliases: D12⋊13D6, Dic6⋊13D6, C62.5C23, C32⋊42+ 1+4, D4⋊8S32, (C4×S3)⋊4D6, (S3×D4)⋊5S3, C3⋊D4⋊5D6, (C3×D4)⋊11D6, Dic3⋊D6⋊6C2, D4⋊2S3⋊5S3, (S3×D12)⋊10C2, C3⋊3(D4○D12), (C2×Dic3)⋊7D6, (C22×S3)⋊7D6, C3⋊3(D4⋊6D6), (S3×C12)⋊8C22, D6.D6⋊7C2, D6.3D6⋊5C2, C6.21(S3×C23), (C3×C6).21C24, D12⋊S3⋊10C2, D6.6D6⋊10C2, (C3×D12)⋊15C22, (S3×C6).12C23, (C3×C12).33C23, C12.33(C22×S3), (C6×Dic3)⋊7C22, (S3×Dic3)⋊3C22, D6.12(C22×S3), C6.D6⋊3C22, C32⋊7D4⋊5C22, C12⋊S3⋊11C22, C3⋊D12⋊16C22, D6⋊S3⋊15C22, (C3×Dic6)⋊15C22, C32⋊2Q8⋊14C22, (D4×C32)⋊13C22, C3⋊Dic3.23C23, Dic3.22(C22×S3), (C3×Dic3).21C23, C4.33(C2×S32), (D4×C3⋊S3)⋊8C2, (C3×S3×D4)⋊10C2, (C2×S32)⋊5C22, (S3×C3⋊D4)⋊5C2, C22.5(C2×S32), (C4×C3⋊S3)⋊5C22, (S3×C2×C6)⋊11C22, C2.23(C22×S32), (C3×C3⋊D4)⋊5C22, (C2×C6).6(C22×S3), (C3×D4⋊2S3)⋊10C2, (C2×C3⋊D12)⋊20C2, (C2×C3⋊S3).26C23, (C22×C3⋊S3)⋊8C22, SmallGroup(288,962)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D12⋊13D6
G = < a,b,c,d | a12=b2=c6=d2=1, bab=a-1, cac-1=a7, dad=a5, cbc-1=a6b, dbd=a10b, dcd=c-1 >
Subgroups: 1506 in 359 conjugacy classes, 108 normal (50 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, C22, S3, C6, C6, C2×C4, D4, D4, Q8, C23, C32, Dic3, Dic3, Dic3, C12, C12, D6, D6, D6, C2×C6, C2×C6, C2×D4, C4○D4, C3×S3, C3⋊S3, C3×C6, C3×C6, Dic6, Dic6, C4×S3, C4×S3, D12, D12, C2×Dic3, C2×Dic3, C3⋊D4, C3⋊D4, C2×C12, C3×D4, C3×D4, C3×Q8, C22×S3, C22×S3, C22×C6, 2+ 1+4, C3×Dic3, C3×Dic3, C3⋊Dic3, C3×C12, S32, S3×C6, S3×C6, S3×C6, C2×C3⋊S3, C2×C3⋊S3, C2×C3⋊S3, C62, C2×D12, C4○D12, S3×D4, S3×D4, D4⋊2S3, D4⋊2S3, Q8⋊3S3, C2×C3⋊D4, C6×D4, C3×C4○D4, S3×Dic3, C6.D6, D6⋊S3, C3⋊D12, C3⋊D12, C32⋊2Q8, C3×Dic6, S3×C12, C3×D12, C6×Dic3, C3×C3⋊D4, C4×C3⋊S3, C12⋊S3, C32⋊7D4, D4×C32, C2×S32, S3×C2×C6, C22×C3⋊S3, D4⋊6D6, D4○D12, D12⋊S3, D6.D6, D6.6D6, S3×D12, D6.3D6, C2×C3⋊D12, S3×C3⋊D4, Dic3⋊D6, C3×S3×D4, C3×D4⋊2S3, D4×C3⋊S3, D12⋊13D6
Quotients: C1, C2, C22, S3, C23, D6, C24, C22×S3, 2+ 1+4, S32, S3×C23, C2×S32, D4⋊6D6, D4○D12, C22×S32, D12⋊13D6
►On 24 points - transitive group 24T607
(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 21)(2 20)(3 19)(4 18)(5 17)(6 16)(7 15)(8 14)(9 13)(10 24)(11 23)(12 22)
(1 5 9)(2 12 10 8 6 4)(3 7 11)(13 15 17 19 21 23)(14 22 18)(16 24 20)
(1 9)(3 7)(4 12)(6 10)(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), (1,21)(2,20)(3,19)(4,18)(5,17)(6,16)(7,15)(8,14)(9,13)(10,24)(11,23)(12,22), (1,5,9)(2,12,10,8,6,4)(3,7,11)(13,15,17,19,21,23)(14,22,18)(16,24,20), (1,9)(3,7)(4,12)(6,10)(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), (1,21)(2,20)(3,19)(4,18)(5,17)(6,16)(7,15)(8,14)(9,13)(10,24)(11,23)(12,22), (1,5,9)(2,12,10,8,6,4)(3,7,11)(13,15,17,19,21,23)(14,22,18)(16,24,20), (1,9)(3,7)(4,12)(6,10)(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)], [(1,21),(2,20),(3,19),(4,18),(5,17),(6,16),(7,15),(8,14),(9,13),(10,24),(11,23),(12,22)], [(1,5,9),(2,12,10,8,6,4),(3,7,11),(13,15,17,19,21,23),(14,22,18),(16,24,20)], [(1,9),(3,7),(4,12),(6,10),(13,23),(14,16),(15,21),(17,19),(18,24),(20,22)]])
G:=TransitiveGroup(24,607);
42 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 3A | 3B | 3C | 4A | 4B | 4C | 4D | 4E | 4F | 6A | 6B | 6C | ··· | 6G | 6H | 6I | 6J | 6K | 6L | 6M | 6N | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H |
| order | 1 | 2 | 2 | 2 | 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 | 6 | 6 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 |
| size | 1 | 1 | 2 | 2 | 6 | 6 | 6 | 6 | 18 | 18 | 18 | 2 | 2 | 4 | 2 | 6 | 6 | 6 | 6 | 18 | 2 | 2 | 4 | ··· | 4 | 6 | 6 | 8 | 8 | 12 | 12 | 12 | 4 | 4 | 6 | 6 | 8 | 12 | 12 | 12 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 8 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | S3 | D6 | D6 | D6 | D6 | D6 | D6 | D6 | 2+ 1+4 | S32 | C2×S32 | C2×S32 | D4⋊6D6 | D4○D12 | D12⋊13D6 |
| kernel | D12⋊13D6 | D12⋊S3 | D6.D6 | D6.6D6 | S3×D12 | D6.3D6 | C2×C3⋊D12 | S3×C3⋊D4 | Dic3⋊D6 | C3×S3×D4 | C3×D4⋊2S3 | D4×C3⋊S3 | S3×D4 | D4⋊2S3 | Dic6 | C4×S3 | D12 | C2×Dic3 | C3⋊D4 | C3×D4 | C22×S3 | C32 | D4 | C4 | C22 | C3 | C3 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 2 | 4 | 2 | 2 | 1 | 1 | 1 | 2 | 2 | 2 | 1 |
Matrix representation of D12⋊13D6 ►in GL8(ℤ)
| 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | -1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| -1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | -1 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | -1 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | -1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 |
| 0 | 0 | 0 | 0 | 1 | -1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | -1 | 0 | 0 |
| 0 | 0 | 1 | -1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 |
| 1 | -1 | 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 | -1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | -1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 | -1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| -1 | 1 | 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 | -1 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | -1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | -1 | 1 |
G:=sub<GL(8,Integers())| [0,0,0,-1,0,0,0,0,0,0,1,1,0,0,0,0,0,1,0,0,0,0,0,0,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,1,0,0,0,0,0,1,0,0,0,0,0,0,-1,-1,0,0],[0,0,0,0,0,0,1,0,0,0,0,0,0,0,-1,-1,0,0,0,0,1,0,0,0,0,0,0,0,-1,-1,0,0,0,0,1,0,0,0,0,0,0,0,-1,-1,0,0,0,0,1,0,0,0,0,0,0,0,-1,-1,0,0,0,0,0,0],[0,0,0,1,0,0,0,0,0,0,-1,-1,0,0,0,0,0,1,0,0,0,0,0,0,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,-1,0,0,0,0,0,1,1,0,0,0,0,0,0,-1,0,0,0],[-1,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,-1,-1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,-1,0,0,0,0,0,0,0,1] >;
D12⋊13D6 in GAP, Magma, Sage, TeX
D_{12}\rtimes_{13}D_6 % in TeX
G:=Group("D12:13D6"); // GroupNames label
G:=SmallGroup(288,962);
// by ID
G=gap.SmallGroup(288,962);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,219,675,185,80,1356,9414]);
// Polycyclic
G:=Group<a,b,c,d|a^12=b^2=c^6=d^2=1,b*a*b=a^-1,c*a*c^-1=a^7,d*a*d=a^5,c*b*c^-1=a^6*b,d*b*d=a^10*b,d*c*d=c^-1>;
// generators/relations