direct product, non-abelian, soluble, monomial
Aliases: C2×C62⋊S3, C62⋊14D6, C3⋊S4⋊C6, (C6×A4)⋊C6, (C3×C6)⋊1S4, C3.3(C6×S4), C6.12(C3×S4), C32⋊2(C2×S4), (C2×C62)⋊4S3, C23⋊(C32⋊C6), C32⋊A4⋊2C22, (C2×C3⋊S4)⋊C3, (C3×A4)⋊(C2×C6), (C2×C6).4(S3×C6), C22⋊(C2×C32⋊C6), (C2×C32⋊A4)⋊1C2, (C22×C6).9(C3×S3), SmallGroup(432,535)
Series: Derived ►Chief ►Lower central ►Upper central
| C3×A4 — C2×C62⋊S3 |
Generators and relations for C2×C62⋊S3
G = < a,b,c,d,e,f,g | a2=b3=c3=d2=e2=f3=g2=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, bc=cb, bd=db, be=eb, fbf-1=bc-1, bg=gb, cd=dc, ce=ec, cf=fc, gcg=c-1, fdf-1=gdg=de=ed, fef-1=d, eg=ge, gfg=f-1 >
Subgroups: 847 in 134 conjugacy classes, 22 normal (18 characteristic)
C1, C2, C2, C3, C3, C4, C22, C22, S3, C6, C6, C2×C4, D4, C23, C23, C32, C32, Dic3, C12, A4, D6, C2×C6, C2×C6, C2×D4, C3×S3, C3⋊S3, C3×C6, C3×C6, C2×Dic3, C3⋊D4, C2×C12, C3×D4, S4, C2×A4, C22×S3, C22×C6, C22×C6, He3, C3×Dic3, C3×A4, C3×A4, S3×C6, C2×C3⋊S3, C62, C62, C2×C3⋊D4, C6×D4, C2×S4, C32⋊C6, C2×He3, C6×Dic3, C3×C3⋊D4, C3⋊S4, C6×A4, C6×A4, S3×C2×C6, C2×C62, C32⋊A4, C2×C32⋊C6, C6×C3⋊D4, C2×C3⋊S4, C62⋊S3, C2×C32⋊A4, C2×C62⋊S3
Quotients: C1, C2, C3, C22, S3, C6, D6, C2×C6, C3×S3, S4, S3×C6, C2×S4, C32⋊C6, C3×S4, C2×C32⋊C6, C6×S4, C62⋊S3, C2×C62⋊S3
►On 18 points - transitive group 18T149
(1 2)(3 4)(5 6)(7 13)(8 14)(9 15)(10 18)(11 16)(12 17)
(7 8 9)(10 11 12)(13 14 15)(16 17 18)
(1 5 3)(2 6 4)(7 8 9)(10 12 11)(13 14 15)(16 18 17)
(1 2)(3 4)(5 6)(10 18)(11 16)(12 17)
(7 13)(8 14)(9 15)(10 18)(11 16)(12 17)
(1 16 9)(2 11 15)(3 17 8)(4 12 14)(5 18 7)(6 10 13)
(1 2)(3 6)(4 5)(7 12)(8 10)(9 11)(13 17)(14 18)(15 16)
G:=sub<Sym(18)| (1,2)(3,4)(5,6)(7,13)(8,14)(9,15)(10,18)(11,16)(12,17), (7,8,9)(10,11,12)(13,14,15)(16,17,18), (1,5,3)(2,6,4)(7,8,9)(10,12,11)(13,14,15)(16,18,17), (1,2)(3,4)(5,6)(10,18)(11,16)(12,17), (7,13)(8,14)(9,15)(10,18)(11,16)(12,17), (1,16,9)(2,11,15)(3,17,8)(4,12,14)(5,18,7)(6,10,13), (1,2)(3,6)(4,5)(7,12)(8,10)(9,11)(13,17)(14,18)(15,16)>;
G:=Group( (1,2)(3,4)(5,6)(7,13)(8,14)(9,15)(10,18)(11,16)(12,17), (7,8,9)(10,11,12)(13,14,15)(16,17,18), (1,5,3)(2,6,4)(7,8,9)(10,12,11)(13,14,15)(16,18,17), (1,2)(3,4)(5,6)(10,18)(11,16)(12,17), (7,13)(8,14)(9,15)(10,18)(11,16)(12,17), (1,16,9)(2,11,15)(3,17,8)(4,12,14)(5,18,7)(6,10,13), (1,2)(3,6)(4,5)(7,12)(8,10)(9,11)(13,17)(14,18)(15,16) );
G=PermutationGroup([[(1,2),(3,4),(5,6),(7,13),(8,14),(9,15),(10,18),(11,16),(12,17)], [(7,8,9),(10,11,12),(13,14,15),(16,17,18)], [(1,5,3),(2,6,4),(7,8,9),(10,12,11),(13,14,15),(16,18,17)], [(1,2),(3,4),(5,6),(10,18),(11,16),(12,17)], [(7,13),(8,14),(9,15),(10,18),(11,16),(12,17)], [(1,16,9),(2,11,15),(3,17,8),(4,12,14),(5,18,7),(6,10,13)], [(1,2),(3,6),(4,5),(7,12),(8,10),(9,11),(13,17),(14,18),(15,16)]])
G:=TransitiveGroup(18,149);
38 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | 3B | 3C | 3D | 3E | 3F | 4A | 4B | 6A | 6B | ··· | 6G | 6H | ··· | 6M | 6N | 6O | 6P | 6Q | 6R | 6S | 6T | 12A | 12B | 12C | 12D |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 3 | 4 | 4 | 6 | 6 | ··· | 6 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 12 | 12 | 12 | 12 |
| size | 1 | 1 | 3 | 3 | 18 | 18 | 2 | 3 | 3 | 24 | 24 | 24 | 18 | 18 | 2 | 3 | ··· | 3 | 6 | ··· | 6 | 18 | 18 | 18 | 18 | 24 | 24 | 24 | 18 | 18 | 18 | 18 |
38 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 6 | 6 | 6 | 6 | 6 | 6 |
| type | + | + | + | + | + | + | + | + | + | + | + | |||||||||
| image | C1 | C2 | C2 | C3 | C6 | C6 | S3 | D6 | C3×S3 | S3×C6 | S4 | C2×S4 | C3×S4 | C6×S4 | C32⋊C6 | C2×C32⋊C6 | C62⋊S3 | C62⋊S3 | C2×C62⋊S3 | C2×C62⋊S3 |
| kernel | C2×C62⋊S3 | C62⋊S3 | C2×C32⋊A4 | C2×C3⋊S4 | C3⋊S4 | C6×A4 | C2×C62 | C62 | C22×C6 | C2×C6 | C3×C6 | C32 | C6 | C3 | C23 | C22 | C2 | C2 | C1 | C1 |
| # reps | 1 | 2 | 1 | 2 | 4 | 2 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 1 | 1 | 1 | 2 | 1 | 2 |
Matrix representation of C2×C62⋊S3 ►in GL6(ℤ)
| -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 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | -1 | 1 | 0 | 0 |
| 0 | 0 | -1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | -1 |
| 0 | 0 | 0 | 0 | 1 | -1 |
| -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 |
| -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 |
| 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 | -1 | 1 | 0 | 0 |
| 0 | 0 | -1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | -1 | 1 |
| 0 | 0 | 0 | 0 | -1 | 0 |
| -1 | 1 | 0 | 0 | 0 | 0 |
| -1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
G:=sub<GL(6,Integers())| [-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],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,-1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,-1,-1],[-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],[-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],[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,-1,-1,0,0,0,0,1,0,-1,-1,0,0,0,0,1,0,0,0,0,0,0,0,-1,-1,0,0,0,0,1,0,0,0],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1,0,0,0] >;
C2×C62⋊S3 in GAP, Magma, Sage, TeX
C_2\times C_6^2\rtimes S_3
% in TeX
G:=Group("C2xC6^2:S3"); // GroupNames label
G:=SmallGroup(432,535);
// by ID
G=gap.SmallGroup(432,535);
# by ID
G:=PCGroup([7,-2,-2,-3,-3,-3,-2,2,675,353,2524,9077,782,5298,1350]);
// Polycyclic
G:=Group<a,b,c,d,e,f,g|a^2=b^3=c^3=d^2=e^2=f^3=g^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,a*g=g*a,b*c=c*b,b*d=d*b,b*e=e*b,f*b*f^-1=b*c^-1,b*g=g*b,c*d=d*c,c*e=e*c,c*f=f*c,g*c*g=c^-1,f*d*f^-1=g*d*g=d*e=e*d,f*e*f^-1=d,e*g=g*e,g*f*g=f^-1>;
// generators/relations