metabelian, supersoluble, monomial
Aliases: D12⋊27D6, Dic6⋊26D6, C32⋊22+ 1+4, C62.140C23, (C4×S3)⋊2D6, (C2×C12)⋊6D6, C3⋊D4⋊7D6, C4○D12⋊9S3, (S3×D12)⋊8C2, Dic3⋊D6⋊3C2, C3⋊2(D4○D12), (C6×C12)⋊8C22, (S3×C12)⋊3C22, D6.6D6⋊8C2, (S3×C6).7C23, C6.15(S3×C23), (C3×C6).15C24, D6.8(C22×S3), (C3×D12)⋊32C22, C3⋊D12⋊2C22, C6.D6⋊2C22, C12⋊S3⋊24C22, C12.133(C22×S3), (C3×C12).120C23, (C3×Dic6)⋊31C22, Dic3.7(C22×S3), (C3×Dic3).10C23, (C2×C4)⋊4S32, C4.64(C2×S32), (C2×S32)⋊3C22, C2.17(C22×S32), C22.13(C2×S32), (C3×C4○D12)⋊14C2, (C2×C12⋊S3)⋊19C2, (C3×C3⋊D4)⋊9C22, (C2×C3⋊S3).21C23, (C22×C3⋊S3)⋊6C22, (C2×C6).15(C22×S3), SmallGroup(288,956)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D12⋊27D6
G = < a,b,c,d | a12=b2=c6=d2=1, bab=dad=a-1, ac=ca, cbc-1=a6b, dbd=a4b, dcd=c-1 >
Subgroups: 1602 in 359 conjugacy classes, 108 normal (10 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, C22, S3, C6, C6, C2×C4, C2×C4, D4, Q8, C23, C32, Dic3, C12, C12, D6, D6, C2×C6, C2×C6, C2×D4, C4○D4, C3×S3, C3⋊S3, C3×C6, C3×C6, Dic6, C4×S3, C4×S3, D12, D12, C3⋊D4, C3⋊D4, C2×C12, C2×C12, C3×D4, C3×Q8, C22×S3, 2+ 1+4, C3×Dic3, C3×C12, S32, S3×C6, C2×C3⋊S3, C2×C3⋊S3, C62, C2×D12, C4○D12, C4○D12, S3×D4, Q8⋊3S3, C3×C4○D4, C6.D6, C3⋊D12, C3×Dic6, S3×C12, C3×D12, C3×C3⋊D4, C12⋊S3, C6×C12, C2×S32, C22×C3⋊S3, D4○D12, D6.6D6, S3×D12, Dic3⋊D6, C3×C4○D12, C2×C12⋊S3, D12⋊27D6
Quotients: C1, C2, C22, S3, C23, D6, C24, C22×S3, 2+ 1+4, S32, S3×C23, C2×S32, D4○D12, C22×S32, D12⋊27D6
►On 24 points - transitive group 24T611
(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 24)(2 23)(3 22)(4 21)(5 20)(6 19)(7 18)(8 17)(9 16)(10 15)(11 14)(12 13)
(1 9 5)(2 10 6)(3 11 7)(4 12 8)(13 23 21 19 17 15)(14 24 22 20 18 16)
(1 5)(2 4)(6 12)(7 11)(8 10)(13 15)(16 24)(17 23)(18 22)(19 21)
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), (1,24)(2,23)(3,22)(4,21)(5,20)(6,19)(7,18)(8,17)(9,16)(10,15)(11,14)(12,13), (1,9,5)(2,10,6)(3,11,7)(4,12,8)(13,23,21,19,17,15)(14,24,22,20,18,16), (1,5)(2,4)(6,12)(7,11)(8,10)(13,15)(16,24)(17,23)(18,22)(19,21)>;
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), (1,24)(2,23)(3,22)(4,21)(5,20)(6,19)(7,18)(8,17)(9,16)(10,15)(11,14)(12,13), (1,9,5)(2,10,6)(3,11,7)(4,12,8)(13,23,21,19,17,15)(14,24,22,20,18,16), (1,5)(2,4)(6,12)(7,11)(8,10)(13,15)(16,24)(17,23)(18,22)(19,21) );
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)], [(1,24),(2,23),(3,22),(4,21),(5,20),(6,19),(7,18),(8,17),(9,16),(10,15),(11,14),(12,13)], [(1,9,5),(2,10,6),(3,11,7),(4,12,8),(13,23,21,19,17,15),(14,24,22,20,18,16)], [(1,5),(2,4),(6,12),(7,11),(8,10),(13,15),(16,24),(17,23),(18,22),(19,21)]])
G:=TransitiveGroup(24,611);
45 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 | 12A | 12B | 12C | 12D | 12E | ··· | 12J | 12K | 12L | 12M | 12N |
| 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 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 12 | 12 | 12 | 12 |
| size | 1 | 1 | 2 | 6 | 6 | 6 | 6 | 18 | 18 | 18 | 18 | 2 | 2 | 4 | 2 | 2 | 6 | 6 | 6 | 6 | 2 | 2 | 4 | ··· | 4 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 12 | 12 | 12 | 12 |
45 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | D6 | D6 | D6 | D6 | D6 | 2+ 1+4 | S32 | C2×S32 | C2×S32 | D4○D12 | D12⋊27D6 |
| kernel | D12⋊27D6 | D6.6D6 | S3×D12 | Dic3⋊D6 | C3×C4○D12 | C2×C12⋊S3 | C4○D12 | Dic6 | C4×S3 | D12 | C3⋊D4 | C2×C12 | C32 | C2×C4 | C4 | C22 | C3 | C1 |
| # reps | 1 | 4 | 4 | 4 | 2 | 1 | 2 | 2 | 4 | 2 | 4 | 2 | 1 | 1 | 2 | 1 | 4 | 4 |
Matrix representation of D12⋊27D6 ►in GL4(𝔽13) generated by
| 7 | 3 | 0 | 0 |
| 10 | 10 | 0 | 0 |
| 9 | 8 | 10 | 3 |
| 8 | 9 | 10 | 7 |
| 4 | 9 | 12 | 11 |
| 4 | 9 | 11 | 12 |
| 9 | 8 | 10 | 3 |
| 8 | 9 | 10 | 3 |
| 0 | 12 | 0 | 0 |
| 1 | 12 | 0 | 0 |
| 0 | 3 | 0 | 12 |
| 9 | 7 | 1 | 1 |
| 10 | 10 | 0 | 0 |
| 7 | 3 | 0 | 0 |
| 3 | 6 | 7 | 10 |
| 5 | 4 | 3 | 6 |
G:=sub<GL(4,GF(13))| [7,10,9,8,3,10,8,9,0,0,10,10,0,0,3,7],[4,4,9,8,9,9,8,9,12,11,10,10,11,12,3,3],[0,1,0,9,12,12,3,7,0,0,0,1,0,0,12,1],[10,7,3,5,10,3,6,4,0,0,7,3,0,0,10,6] >;
D12⋊27D6 in GAP, Magma, Sage, TeX
D_{12}\rtimes_{27}D_6 % in TeX
G:=Group("D12:27D6"); // GroupNames label
G:=SmallGroup(288,956);
// by ID
G=gap.SmallGroup(288,956);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,219,100,675,80,1356,9414]);
// Polycyclic
G:=Group<a,b,c,d|a^12=b^2=c^6=d^2=1,b*a*b=d*a*d=a^-1,a*c=c*a,c*b*c^-1=a^6*b,d*b*d=a^4*b,d*c*d=c^-1>;
// generators/relations