metabelian, supersoluble, monomial
Aliases: D6.3S32, Dic3.3S32, C3⋊D12⋊5S3, (S3×Dic3)⋊3S3, (S3×C6).21D6, C32⋊2Q8⋊4S3, C33⋊4(C4○D4), C33⋊8D4⋊5C2, C33⋊7D4⋊6C2, C3⋊Dic3.30D6, C3⋊1(D12⋊S3), C3⋊4(D6.3D6), C3⋊2(D6.6D6), C32⋊8(C4○D12), (C3×Dic3).11D6, C32⋊9(Q8⋊3S3), (C32×C6).16C23, C32⋊10(D4⋊2S3), (C32×Dic3).5C22, C2.16S33, C6.16(C2×S32), (C3×S3×Dic3)⋊5C2, C33⋊8(C2×C4)⋊3C2, C33⋊9(C2×C4)⋊3C2, (C2×C3⋊S3).18D6, (S3×C3×C6).7C22, (C3×C3⋊D12)⋊7C2, (C3×C32⋊2Q8)⋊5C2, (C6×C3⋊S3).21C22, (C3×C6).65(C22×S3), (C3×C3⋊Dic3).13C22, (C2×C33⋊C2).5C22, SmallGroup(432,609)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D6.3S32
G = < a,b,c,d,e,f | a3=b2=c6=e3=f2=1, d2=c3, bab=faf=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, fbf=bc3, dcd-1=c-1, ce=ec, cf=fc, de=ed, fdf=c3d, fef=e-1 >
Subgroups: 1436 in 218 conjugacy classes, 46 normal (all characteristic)
C1, C2, C2, C3, C3, C4, C22, S3, C6, C6, C2×C4, D4, Q8, C32, C32, Dic3, Dic3, C12, D6, D6, C2×C6, C4○D4, C3×S3, C3⋊S3, C3×C6, C3×C6, Dic6, C4×S3, D12, C2×Dic3, C3⋊D4, C2×C12, C3×D4, C3×Q8, C33, C3×Dic3, C3×Dic3, C3⋊Dic3, C3×C12, S3×C6, S3×C6, C2×C3⋊S3, C2×C3⋊S3, C62, C4○D12, D4⋊2S3, Q8⋊3S3, S3×C32, C3×C3⋊S3, C33⋊C2, C32×C6, S3×Dic3, S3×Dic3, C6.D6, C3⋊D12, C3⋊D12, C32⋊2Q8, C3×Dic6, S3×C12, C3×D12, C6×Dic3, C3×C3⋊D4, C4×C3⋊S3, C12⋊S3, C32⋊7D4, C32×Dic3, C3×C3⋊Dic3, S3×C3×C6, C6×C3⋊S3, C2×C33⋊C2, D12⋊S3, D6.6D6, D6.3D6, C3×S3×Dic3, C3×C3⋊D12, C3×C32⋊2Q8, C33⋊8(C2×C4), C33⋊7D4, C33⋊8D4, C33⋊9(C2×C4), D6.3S32
Quotients: C1, C2, C22, S3, C23, D6, C4○D4, C22×S3, S32, C4○D12, D4⋊2S3, Q8⋊3S3, C2×S32, D12⋊S3, D6.6D6, D6.3D6, S33, D6.3S32
►On 24 points - transitive group 24T1301
(1 5 3)(2 6 4)(7 11 9)(8 12 10)(13 15 17)(14 16 18)(19 21 23)(20 22 24)
(1 19)(2 20)(3 21)(4 22)(5 23)(6 24)(7 18)(8 13)(9 14)(10 15)(11 16)(12 17)
(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 4 13)(2 15 5 18)(3 14 6 17)(7 20 10 23)(8 19 11 22)(9 24 12 21)
(1 5 3)(2 6 4)(7 9 11)(8 10 12)(13 15 17)(14 16 18)(19 23 21)(20 24 22)
(1 16)(2 17)(3 18)(4 13)(5 14)(6 15)(7 24)(8 19)(9 20)(10 21)(11 22)(12 23)
G:=sub<Sym(24)| (1,5,3)(2,6,4)(7,11,9)(8,12,10)(13,15,17)(14,16,18)(19,21,23)(20,22,24), (1,19)(2,20)(3,21)(4,22)(5,23)(6,24)(7,18)(8,13)(9,14)(10,15)(11,16)(12,17), (1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18)(19,20,21,22,23,24), (1,16,4,13)(2,15,5,18)(3,14,6,17)(7,20,10,23)(8,19,11,22)(9,24,12,21), (1,5,3)(2,6,4)(7,9,11)(8,10,12)(13,15,17)(14,16,18)(19,23,21)(20,24,22), (1,16)(2,17)(3,18)(4,13)(5,14)(6,15)(7,24)(8,19)(9,20)(10,21)(11,22)(12,23)>;
G:=Group( (1,5,3)(2,6,4)(7,11,9)(8,12,10)(13,15,17)(14,16,18)(19,21,23)(20,22,24), (1,19)(2,20)(3,21)(4,22)(5,23)(6,24)(7,18)(8,13)(9,14)(10,15)(11,16)(12,17), (1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18)(19,20,21,22,23,24), (1,16,4,13)(2,15,5,18)(3,14,6,17)(7,20,10,23)(8,19,11,22)(9,24,12,21), (1,5,3)(2,6,4)(7,9,11)(8,10,12)(13,15,17)(14,16,18)(19,23,21)(20,24,22), (1,16)(2,17)(3,18)(4,13)(5,14)(6,15)(7,24)(8,19)(9,20)(10,21)(11,22)(12,23) );
G=PermutationGroup([[(1,5,3),(2,6,4),(7,11,9),(8,12,10),(13,15,17),(14,16,18),(19,21,23),(20,22,24)], [(1,19),(2,20),(3,21),(4,22),(5,23),(6,24),(7,18),(8,13),(9,14),(10,15),(11,16),(12,17)], [(1,2,3,4,5,6),(7,8,9,10,11,12),(13,14,15,16,17,18),(19,20,21,22,23,24)], [(1,16,4,13),(2,15,5,18),(3,14,6,17),(7,20,10,23),(8,19,11,22),(9,24,12,21)], [(1,5,3),(2,6,4),(7,9,11),(8,10,12),(13,15,17),(14,16,18),(19,23,21),(20,24,22)], [(1,16),(2,17),(3,18),(4,13),(5,14),(6,15),(7,24),(8,19),(9,20),(10,21),(11,22),(12,23)]])
G:=TransitiveGroup(24,1301);
42 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3A | 3B | 3C | 3D | 3E | 3F | 3G | 4A | 4B | 4C | 4D | 4E | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 6H | 6I | 6J | 6K | 6L | 6M | 12A | 12B | 12C | ··· | 12I | 12J | 12K | 12L |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 12 | 12 | 12 | ··· | 12 | 12 | 12 | 12 |
| size | 1 | 1 | 6 | 18 | 54 | 2 | 2 | 2 | 4 | 4 | 4 | 8 | 6 | 6 | 9 | 9 | 18 | 2 | 2 | 2 | 4 | 4 | 4 | 6 | 6 | 8 | 12 | 12 | 12 | 36 | 6 | 6 | 12 | ··· | 12 | 18 | 18 | 36 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | 8 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | + | + | + | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | S3 | S3 | D6 | D6 | D6 | D6 | C4○D4 | C4○D12 | S32 | S32 | D4⋊2S3 | Q8⋊3S3 | C2×S32 | D12⋊S3 | D6.6D6 | D6.3D6 | S33 | D6.3S32 |
| kernel | D6.3S32 | C3×S3×Dic3 | C3×C3⋊D12 | C3×C32⋊2Q8 | C33⋊8(C2×C4) | C33⋊7D4 | C33⋊8D4 | C33⋊9(C2×C4) | S3×Dic3 | C3⋊D12 | C32⋊2Q8 | C3×Dic3 | C3⋊Dic3 | S3×C6 | C2×C3⋊S3 | C33 | C32 | Dic3 | D6 | C32 | C32 | C6 | C3 | C3 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 2 | 2 | 1 | 2 | 4 | 2 | 1 | 1 | 1 | 3 | 2 | 2 | 2 | 1 | 1 |
Matrix representation of D6.3S32 ►in GL8(ℤ)
| -1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | -1 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | -1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | -1 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | -1 | 1 |
| 0 | 0 | 0 | 0 | 0 | 0 | -1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | -1 | 0 | 0 |
| 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 |
| 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 |
| 0 | 0 | -1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 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 | 0 | 0 | 0 | 1 | -1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | -1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 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 |
| 0 | -1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | -1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 |
| -1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| -1 | 0 | 0 | 0 | 0 | 0 | 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 | -1 | 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 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 |
| 0 | 0 | 0 | 0 | 0 | 0 | -1 | 0 |
| 0 | -1 | 0 | 0 | 0 | 0 | 0 | 0 |
| -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 |
| 0 | 0 | -1 | 0 | 0 | 0 | 0 | 0 |
G:=sub<GL(8,Integers())| [-1,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,-1,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,-1,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,-1,-1,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0],[0,-1,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,-1,0],[0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0],[-1,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,-1,0,0,0,0,0,0,0,0,-1,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,-1],[0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0] >;
D6.3S32 in GAP, Magma, Sage, TeX
D_6._3S_3^2
% in TeX
G:=Group("D6.3S3^2"); // GroupNames label
G:=SmallGroup(432,609);
// by ID
G=gap.SmallGroup(432,609);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,64,254,135,58,298,2028,14118]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^3=b^2=c^6=e^3=f^2=1,d^2=c^3,b*a*b=f*a*f=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,f*b*f=b*c^3,d*c*d^-1=c^-1,c*e=e*c,c*f=f*c,d*e=e*d,f*d*f=c^3*d,f*e*f=e^-1>;
// generators/relations