metabelian, supersoluble, monomial
Aliases: D12⋊18D6, Dic6⋊16D6, C12.74D12, C62.44D4, C3⋊C8⋊3D6, C4○D12⋊4S3, C3⋊D24⋊5C2, C3⋊5(C8⋊D6), C6.66(C2×D12), (C2×C6).12D12, (C3×C12).63D4, C4.Dic3⋊7S3, C3⋊1(D4⋊D6), (C2×C12).114D6, C32⋊7(C8⋊C22), C32⋊5SD16⋊3C2, (C3×D12)⋊22C22, C12.44(C3⋊D4), (C6×C12).74C22, (C3×C12).61C23, C4.16(C3⋊D12), C12.124(C22×S3), (C3×Dic6)⋊19C22, C12⋊S3.25C22, C22.9(C3⋊D12), (C2×C4).6S32, C4.51(C2×S32), (C3×C3⋊C8)⋊3C22, C6.2(C2×C3⋊D4), (C3×C4○D12)⋊3C2, (C3×C6).65(C2×D4), C2.6(C2×C3⋊D12), (C2×C12⋊S3)⋊10C2, (C3×C4.Dic3)⋊1C2, (C2×C6).17(C3⋊D4), SmallGroup(288,473)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D12⋊18D6
G = < a,b,c,d | a12=b2=c6=d2=1, bab=dad=a-1, ac=ca, cbc-1=a6b, dbd=ab, dcd=c-1 >
Subgroups: 866 in 163 conjugacy classes, 44 normal (34 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, C22, S3, C6, C6, C8, C2×C4, C2×C4, D4, Q8, C23, C32, Dic3, C12, C12, D6, C2×C6, C2×C6, M4(2), D8, SD16, C2×D4, C4○D4, C3×S3, C3⋊S3, C3×C6, C3×C6, C3⋊C8, C24, Dic6, C4×S3, D12, D12, C3⋊D4, C2×C12, C2×C12, C3×D4, C3×Q8, C22×S3, C8⋊C22, C3×Dic3, C3×C12, S3×C6, C2×C3⋊S3, C62, C24⋊C2, D24, C4.Dic3, D4⋊S3, Q8⋊2S3, C3×M4(2), C2×D12, C4○D12, C3×C4○D4, C3×C3⋊C8, C3×Dic6, S3×C12, C3×D12, C3×C3⋊D4, C12⋊S3, C12⋊S3, C6×C12, C22×C3⋊S3, C8⋊D6, D4⋊D6, C3⋊D24, C32⋊5SD16, C3×C4.Dic3, C3×C4○D12, C2×C12⋊S3, D12⋊18D6
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, D12, C3⋊D4, C22×S3, C8⋊C22, S32, C2×D12, C2×C3⋊D4, C3⋊D12, C2×S32, C8⋊D6, D4⋊D6, C2×C3⋊D12, D12⋊18D6
►On 24 points - transitive group 24T613
(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 15)(2 14)(3 13)(4 24)(5 23)(6 22)(7 21)(8 20)(9 19)(10 18)(11 17)(12 16)
(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 24)(14 23)(15 22)(16 21)(17 20)(18 19)
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), (1,15)(2,14)(3,13)(4,24)(5,23)(6,22)(7,21)(8,20)(9,19)(10,18)(11,17)(12,16), (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)(13,24)(14,23)(15,22)(16,21)(17,20)(18,19)>;
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), (1,15)(2,14)(3,13)(4,24)(5,23)(6,22)(7,21)(8,20)(9,19)(10,18)(11,17)(12,16), (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)(13,24)(14,23)(15,22)(16,21)(17,20)(18,19) );
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)], [(1,15),(2,14),(3,13),(4,24),(5,23),(6,22),(7,21),(8,20),(9,19),(10,18),(11,17),(12,16)], [(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),(13,24),(14,23),(15,22),(16,21),(17,20),(18,19)]])
G:=TransitiveGroup(24,613);
39 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | 3B | 3C | 4A | 4B | 4C | 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 | 2 | 2 | 3 | 3 | 3 | 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 | 36 | 36 | 2 | 2 | 4 | 2 | 2 | 12 | 2 | 2 | 4 | ··· | 4 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 12 | 12 | 12 | 12 | 12 | 12 |
39 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 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | S3 | D4 | D4 | D6 | D6 | D6 | D6 | D12 | C3⋊D4 | D12 | C3⋊D4 | C8⋊C22 | S32 | C3⋊D12 | C2×S32 | C3⋊D12 | C8⋊D6 | D4⋊D6 | D12⋊18D6 |
| kernel | D12⋊18D6 | C3⋊D24 | C32⋊5SD16 | C3×C4.Dic3 | C3×C4○D12 | C2×C12⋊S3 | C4.Dic3 | C4○D12 | C3×C12 | C62 | C3⋊C8 | Dic6 | D12 | C2×C12 | C12 | C12 | C2×C6 | C2×C6 | C32 | C2×C4 | C4 | C4 | C22 | C3 | C3 | C1 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 4 |
Matrix representation of D12⋊18D6 ►in GL4(𝔽73) generated by
| 14 | 66 | 0 | 0 |
| 7 | 7 | 0 | 0 |
| 0 | 0 | 14 | 66 |
| 0 | 0 | 7 | 7 |
| 0 | 0 | 66 | 66 |
| 0 | 0 | 59 | 7 |
| 66 | 66 | 0 | 0 |
| 59 | 7 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 72 | 1 | 0 | 0 |
| 0 | 0 | 72 | 1 |
| 0 | 0 | 72 | 0 |
| 1 | 72 | 0 | 0 |
| 0 | 72 | 0 | 0 |
| 0 | 0 | 66 | 14 |
| 0 | 0 | 7 | 7 |
G:=sub<GL(4,GF(73))| [14,7,0,0,66,7,0,0,0,0,14,7,0,0,66,7],[0,0,66,59,0,0,66,7,66,59,0,0,66,7,0,0],[0,72,0,0,1,1,0,0,0,0,72,72,0,0,1,0],[1,0,0,0,72,72,0,0,0,0,66,7,0,0,14,7] >;
D12⋊18D6 in GAP, Magma, Sage, TeX
D_{12}\rtimes_{18}D_6 % in TeX
G:=Group("D12:18D6"); // GroupNames label
G:=SmallGroup(288,473);
// by ID
G=gap.SmallGroup(288,473);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,141,176,219,100,675,80,1356,9414]);
// 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*b,d*c*d=c^-1>;
// generators/relations