direct product, metabelian, soluble, monomial
Aliases: C2×C32⋊A4, C23⋊He3, C62⋊7C6, (C3×C6)⋊A4, (C6×A4)⋊C3, (C3×A4)⋊2C6, C6.7(C3×A4), C3.5(C6×A4), C22⋊(C2×He3), (C2×C62)⋊1C3, C32⋊2(C2×A4), (C22×C6).5C32, (C2×C6).5(C3×C6), SmallGroup(216,107)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×C32⋊A4
G = < a,b,c,d,e,f | a2=b3=c3=d2=e2=f3=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, fbf-1=bc-1, cd=dc, ce=ec, cf=fc, fdf-1=de=ed, fef-1=d >
Subgroups: 226 in 68 conjugacy classes, 20 normal (14 characteristic)
C1, C2, C2, C3, C3, C22, C22, C6, C6, C23, C32, C32, A4, C2×C6, C2×C6, C3×C6, C3×C6, C2×A4, C22×C6, C22×C6, He3, C3×A4, C62, C62, C2×He3, C6×A4, C2×C62, C32⋊A4, C2×C32⋊A4
Quotients: C1, C2, C3, C6, C32, A4, C3×C6, C2×A4, He3, C3×A4, C2×He3, C6×A4, C32⋊A4, C2×C32⋊A4
►On 18 points - transitive group 18T91
(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)
(7 13)(8 14)(9 15)(10 18)(11 16)(12 17)
(1 2)(3 4)(5 6)(7 13)(8 14)(9 15)
(1 17 7)(2 12 13)(3 18 9)(4 10 15)(5 16 8)(6 11 14)
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), (7,13)(8,14)(9,15)(10,18)(11,16)(12,17), (1,2)(3,4)(5,6)(7,13)(8,14)(9,15), (1,17,7)(2,12,13)(3,18,9)(4,10,15)(5,16,8)(6,11,14)>;
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), (7,13)(8,14)(9,15)(10,18)(11,16)(12,17), (1,2)(3,4)(5,6)(7,13)(8,14)(9,15), (1,17,7)(2,12,13)(3,18,9)(4,10,15)(5,16,8)(6,11,14) );
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)], [(7,13),(8,14),(9,15),(10,18),(11,16),(12,17)], [(1,2),(3,4),(5,6),(7,13),(8,14),(9,15)], [(1,17,7),(2,12,13),(3,18,9),(4,10,15),(5,16,8),(6,11,14)]])
G:=TransitiveGroup(18,91);
C2×C32⋊A4 is a maximal subgroup of
C62⋊5Dic3 C62⋊6Dic3 C62⋊4C12
C2×C32⋊A4 is a maximal quotient of C4○D4⋊He3
40 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 3D | 3E | ··· | 3J | 6A | 6B | 6C | ··· | 6T | 6U | ··· | 6Z |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | ··· | 3 | 6 | 6 | 6 | ··· | 6 | 6 | ··· | 6 |
| size | 1 | 1 | 3 | 3 | 1 | 1 | 3 | 3 | 12 | ··· | 12 | 1 | 1 | 3 | ··· | 3 | 12 | ··· | 12 |
40 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 |
| type | + | + | + | + | ||||||||||
| image | C1 | C2 | C3 | C3 | C6 | C6 | A4 | C2×A4 | He3 | C3×A4 | C2×He3 | C6×A4 | C32⋊A4 | C2×C32⋊A4 |
| kernel | C2×C32⋊A4 | C32⋊A4 | C6×A4 | C2×C62 | C3×A4 | C62 | C3×C6 | C32 | C23 | C6 | C22 | C3 | C2 | C1 |
| # reps | 1 | 1 | 6 | 2 | 6 | 2 | 1 | 1 | 2 | 2 | 2 | 2 | 6 | 6 |
Matrix representation of C2×C32⋊A4 ►in GL3(𝔽7) generated by
| 6 | 0 | 0 |
| 0 | 6 | 0 |
| 0 | 0 | 6 |
| 1 | 0 | 0 |
| 0 | 4 | 0 |
| 0 | 0 | 2 |
| 4 | 0 | 0 |
| 0 | 4 | 0 |
| 0 | 0 | 4 |
| 6 | 0 | 0 |
| 0 | 6 | 0 |
| 0 | 0 | 1 |
| 6 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 6 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 1 | 0 | 0 |
G:=sub<GL(3,GF(7))| [6,0,0,0,6,0,0,0,6],[1,0,0,0,4,0,0,0,2],[4,0,0,0,4,0,0,0,4],[6,0,0,0,6,0,0,0,1],[6,0,0,0,1,0,0,0,6],[0,0,1,1,0,0,0,1,0] >;
C2×C32⋊A4 in GAP, Magma, Sage, TeX
C_2\times C_3^2\rtimes A_4
% in TeX
G:=Group("C2xC3^2:A4"); // GroupNames label
G:=SmallGroup(216,107);
// by ID
G=gap.SmallGroup(216,107);
# by ID
G:=PCGroup([6,-2,-3,-3,-3,-2,2,224,1630,2927]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^3=c^3=d^2=e^2=f^3=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,b*d=d*b,b*e=e*b,f*b*f^-1=b*c^-1,c*d=d*c,c*e=e*c,c*f=f*c,f*d*f^-1=d*e=e*d,f*e*f^-1=d>;
// generators/relations