non-abelian, soluble, monomial
Aliases: C6.12S3≀C2, C3⋊S3.2D12, C6.D6⋊4S3, (C32×C6).6D4, C32⋊3(D6⋊C4), C32⋊4D6⋊1C4, C33⋊3(C22⋊C4), C2.1(C33⋊D4), C3⋊1(S32⋊C4), C3⋊S3.2(C4×S3), (C3×C3⋊S3).9D4, (C2×C3⋊S3).10D6, (C2×C33⋊C4)⋊2C2, (C3×C6.D6)⋊8C2, (C6×C3⋊S3).6C22, (C3×C6).12(C3⋊D4), (C2×C32⋊4D6).1C2, (C3×C3⋊S3).8(C2×C4), SmallGroup(432,579)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3⋊S3.2D12
G = < a,b,c,d,e | a3=b3=c2=d12=1, e2=c, ab=ba, cac=ebe-1=a-1, ad=da, eae-1=b, cbc=dbd-1=b-1, cd=dc, ce=ec, ede-1=cd-1 >
Subgroups: 1012 in 132 conjugacy classes, 21 normal (19 characteristic)
C1, C2, C2, C3, C3, C4, C22, S3, C6, C6, C2×C4, C23, C32, C32, Dic3, C12, D6, C2×C6, C22⋊C4, C3×S3, C3⋊S3, C3⋊S3, C3×C6, C3×C6, C4×S3, C2×Dic3, C2×C12, C22×S3, C33, C3×Dic3, C3×C12, C32⋊C4, S32, S3×C6, C2×C3⋊S3, C2×C3⋊S3, D6⋊C4, C3×C3⋊S3, C3×C3⋊S3, C32×C6, C6.D6, S3×C12, C2×C32⋊C4, C2×S32, C32×Dic3, C33⋊C4, C32⋊4D6, C32⋊4D6, C6×C3⋊S3, C6×C3⋊S3, S32⋊C4, C3×C6.D6, C2×C33⋊C4, C2×C32⋊4D6, C3⋊S3.2D12
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, D6, C22⋊C4, C4×S3, D12, C3⋊D4, D6⋊C4, S3≀C2, S32⋊C4, C33⋊D4, C3⋊S3.2D12
►On 24 points - transitive group 24T1313
(1 5 9)(2 6 10)(3 7 11)(4 8 12)(13 21 17)(14 22 18)(15 23 19)(16 24 20)
(1 5 9)(2 10 6)(3 7 11)(4 12 8)(13 17 21)(14 22 18)(15 19 23)(16 24 20)
(1 20)(2 21)(3 22)(4 23)(5 24)(6 13)(7 14)(8 15)(9 16)(10 17)(11 18)(12 19)
(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 13 20 6)(2 5 21 24)(3 23 22 4)(7 19 14 12)(8 11 15 18)(9 17 16 10)
G:=sub<Sym(24)| (1,5,9)(2,6,10)(3,7,11)(4,8,12)(13,21,17)(14,22,18)(15,23,19)(16,24,20), (1,5,9)(2,10,6)(3,7,11)(4,12,8)(13,17,21)(14,22,18)(15,19,23)(16,24,20), (1,20)(2,21)(3,22)(4,23)(5,24)(6,13)(7,14)(8,15)(9,16)(10,17)(11,18)(12,19), (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24), (1,13,20,6)(2,5,21,24)(3,23,22,4)(7,19,14,12)(8,11,15,18)(9,17,16,10)>;
G:=Group( (1,5,9)(2,6,10)(3,7,11)(4,8,12)(13,21,17)(14,22,18)(15,23,19)(16,24,20), (1,5,9)(2,10,6)(3,7,11)(4,12,8)(13,17,21)(14,22,18)(15,19,23)(16,24,20), (1,20)(2,21)(3,22)(4,23)(5,24)(6,13)(7,14)(8,15)(9,16)(10,17)(11,18)(12,19), (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24), (1,13,20,6)(2,5,21,24)(3,23,22,4)(7,19,14,12)(8,11,15,18)(9,17,16,10) );
G=PermutationGroup([[(1,5,9),(2,6,10),(3,7,11),(4,8,12),(13,21,17),(14,22,18),(15,23,19),(16,24,20)], [(1,5,9),(2,10,6),(3,7,11),(4,12,8),(13,17,21),(14,22,18),(15,19,23),(16,24,20)], [(1,20),(2,21),(3,22),(4,23),(5,24),(6,13),(7,14),(8,15),(9,16),(10,17),(11,18),(12,19)], [(1,2,3,4,5,6,7,8,9,10,11,12),(13,14,15,16,17,18,19,20,21,22,23,24)], [(1,13,20,6),(2,5,21,24),(3,23,22,4),(7,19,14,12),(8,11,15,18),(9,17,16,10)]])
G:=TransitiveGroup(24,1313);
36 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | 3B | 3C | 3D | 3E | 3F | 4A | 4B | 4C | 4D | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 6H | 6I | 6J | 12A | 12B | 12C | 12D | 12E | ··· | 12J |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 12 | 12 | 12 | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 9 | 9 | 18 | 18 | 2 | 4 | 4 | 4 | 4 | 8 | 6 | 6 | 54 | 54 | 2 | 4 | 4 | 4 | 4 | 8 | 18 | 18 | 36 | 36 | 6 | 6 | 6 | 6 | 12 | ··· | 12 |
36 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 8 | 8 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | ||||||
| image | C1 | C2 | C2 | C2 | C4 | S3 | D4 | D4 | D6 | C4×S3 | D12 | C3⋊D4 | S3≀C2 | S32⋊C4 | S32⋊C4 | C33⋊D4 | C3⋊S3.2D12 | C33⋊D4 | C3⋊S3.2D12 |
| kernel | C3⋊S3.2D12 | C3×C6.D6 | C2×C33⋊C4 | C2×C32⋊4D6 | C32⋊4D6 | C6.D6 | C3×C3⋊S3 | C32×C6 | C2×C3⋊S3 | C3⋊S3 | C3⋊S3 | C3×C6 | C6 | C3 | C3 | C2 | C1 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 4 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 2 | 2 | 4 | 4 | 1 | 1 |
Matrix representation of C3⋊S3.2D12 ►in GL6(𝔽13)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 12 |
| 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 | 12 | 0 | 12 | 0 |
| 0 | 0 | 0 | 12 | 0 | 12 |
| 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 | 12 | 0 | 12 | 0 |
| 0 | 0 | 0 | 12 | 0 | 12 |
| 10 | 10 | 0 | 0 | 0 | 0 |
| 3 | 7 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 1 |
| 0 | 0 | 1 | 0 | 1 | 0 |
| 10 | 6 | 0 | 0 | 0 | 0 |
| 3 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 12 |
| 0 | 0 | 0 | 0 | 1 | 0 |
G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,1,0,0,0,0,0,0,12,0,0,12,0,0,0,0,0,0,1,0,12],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,1,0,12,0,0,0,0,1,0,12],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,12,0,0,0,0,1,0,12,0,0,0,0,12,0,0,0,0,0,0,12],[10,3,0,0,0,0,10,7,0,0,0,0,0,0,0,12,0,1,0,0,12,0,1,0,0,0,0,0,0,1,0,0,0,0,1,0],[10,3,0,0,0,0,6,3,0,0,0,0,0,0,0,1,0,0,0,0,1,0,12,0,0,0,0,0,0,1,0,0,0,0,12,0] >;
C3⋊S3.2D12 in GAP, Magma, Sage, TeX
C_3\rtimes S_3._2D_{12} % in TeX
G:=Group("C3:S3.2D12"); // GroupNames label
G:=SmallGroup(432,579);
// by ID
G=gap.SmallGroup(432,579);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,3,-3,85,64,1684,571,298,677,1027,14118]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^3=c^2=d^12=1,e^2=c,a*b=b*a,c*a*c=e*b*e^-1=a^-1,a*d=d*a,e*a*e^-1=b,c*b*c=d*b*d^-1=b^-1,c*d=d*c,c*e=e*c,e*d*e^-1=c*d^-1>;
// generators/relations