direct product, metabelian, soluble, monomial
Aliases: C2×C32.A4, C62.7C6, C23⋊23- 1+2, C3.A4⋊3C6, C6.6(C3×A4), C3.4(C6×A4), (C3×C6).3A4, C32.(C2×A4), (C2×C62).2C3, (C22×C6).4C32, C22⋊2(C2×3- 1+2), (C2×C3.A4)⋊2C3, (C2×C6).4(C3×C6), SmallGroup(216,106)
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=1, f3=c, 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: 136 in 56 conjugacy classes, 20 normal (14 characteristic)
C1, C2, C2, C3, C3, C22, C22, C6, C6, C23, C9, C32, C2×C6, C2×C6, C18, C3×C6, C3×C6, C22×C6, C22×C6, 3- 1+2, C3.A4, C62, C62, C2×3- 1+2, C2×C3.A4, C2×C62, C32.A4, C2×C32.A4
Quotients: C1, C2, C3, C6, C32, A4, C3×C6, C2×A4, 3- 1+2, C3×A4, C2×3- 1+2, C6×A4, C32.A4, C2×C32.A4
►On 18 points - transitive group 18T92
(1 16)(2 17)(3 18)(4 10)(5 11)(6 12)(7 13)(8 14)(9 15)
(2 8 5)(3 6 9)(11 17 14)(12 15 18)
(1 4 7)(2 5 8)(3 6 9)(10 13 16)(11 14 17)(12 15 18)
(2 17)(3 18)(5 11)(6 12)(8 14)(9 15)
(1 16)(3 18)(4 10)(6 12)(7 13)(9 15)
(1 2 3 4 5 6 7 8 9)(10 11 12 13 14 15 16 17 18)
G:=sub<Sym(18)| (1,16)(2,17)(3,18)(4,10)(5,11)(6,12)(7,13)(8,14)(9,15), (2,8,5)(3,6,9)(11,17,14)(12,15,18), (1,4,7)(2,5,8)(3,6,9)(10,13,16)(11,14,17)(12,15,18), (2,17)(3,18)(5,11)(6,12)(8,14)(9,15), (1,16)(3,18)(4,10)(6,12)(7,13)(9,15), (1,2,3,4,5,6,7,8,9)(10,11,12,13,14,15,16,17,18)>;
G:=Group( (1,16)(2,17)(3,18)(4,10)(5,11)(6,12)(7,13)(8,14)(9,15), (2,8,5)(3,6,9)(11,17,14)(12,15,18), (1,4,7)(2,5,8)(3,6,9)(10,13,16)(11,14,17)(12,15,18), (2,17)(3,18)(5,11)(6,12)(8,14)(9,15), (1,16)(3,18)(4,10)(6,12)(7,13)(9,15), (1,2,3,4,5,6,7,8,9)(10,11,12,13,14,15,16,17,18) );
G=PermutationGroup([[(1,16),(2,17),(3,18),(4,10),(5,11),(6,12),(7,13),(8,14),(9,15)], [(2,8,5),(3,6,9),(11,17,14),(12,15,18)], [(1,4,7),(2,5,8),(3,6,9),(10,13,16),(11,14,17),(12,15,18)], [(2,17),(3,18),(5,11),(6,12),(8,14),(9,15)], [(1,16),(3,18),(4,10),(6,12),(7,13),(9,15)], [(1,2,3,4,5,6,7,8,9),(10,11,12,13,14,15,16,17,18)]])
G:=TransitiveGroup(18,92);
C2×C32.A4 is a maximal subgroup of
C62.Dic3
C2×C32.A4 is a maximal quotient of Q8⋊C9⋊4C6
40 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 3D | 6A | 6B | 6C | ··· | 6T | 9A | ··· | 9F | 18A | ··· | 18F |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 6 | 6 | 6 | ··· | 6 | 9 | ··· | 9 | 18 | ··· | 18 |
| size | 1 | 1 | 3 | 3 | 1 | 1 | 3 | 3 | 1 | 1 | 3 | ··· | 3 | 12 | ··· | 12 | 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 | 3- 1+2 | C3×A4 | C2×3- 1+2 | C6×A4 | C32.A4 | C2×C32.A4 |
| kernel | C2×C32.A4 | C32.A4 | C2×C3.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 |
| 4 | 0 | 0 |
| 0 | 2 | 0 |
| 0 | 0 | 1 |
| 4 | 0 | 0 |
| 0 | 4 | 0 |
| 0 | 0 | 4 |
| 1 | 0 | 0 |
| 0 | 6 | 0 |
| 0 | 0 | 6 |
| 6 | 0 | 0 |
| 0 | 6 | 0 |
| 0 | 0 | 1 |
| 0 | 6 | 0 |
| 0 | 0 | 3 |
| 1 | 0 | 0 |
G:=sub<GL(3,GF(7))| [6,0,0,0,6,0,0,0,6],[4,0,0,0,2,0,0,0,1],[4,0,0,0,4,0,0,0,4],[1,0,0,0,6,0,0,0,6],[6,0,0,0,6,0,0,0,1],[0,0,1,6,0,0,0,3,0] >;
C2×C32.A4 in GAP, Magma, Sage, TeX
C_2\times C_3^2.A_4
% in TeX
G:=Group("C2xC3^2.A4"); // GroupNames label
G:=SmallGroup(216,106);
// by ID
G=gap.SmallGroup(216,106);
# by ID
G:=PCGroup([6,-2,-3,-3,-3,-2,2,115,224,1630,2927]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^3=c^3=d^2=e^2=1,f^3=c,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