Aliases: C33⋊2A4, C62.18C32, C22⋊3C3≀C3, C32⋊A4⋊4C3, (C3×C62)⋊1C3, C32.A4⋊6C3, (C2×C6).12He3, C32.13(C3×A4), C3.12(C32⋊A4), SmallGroup(324,60)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C33⋊2A4
G = < a,b,c,d,e,f | a3=b3=c3=d2=e2=f3=1, ab=ba, ac=ca, ad=da, ae=ea, faf-1=ab-1c, bc=cb, bd=db, be=eb, fbf-1=bc-1, cd=dc, ce=ec, cf=fc, fdf-1=de=ed, fef-1=d >
Subgroups: 250 in 68 conjugacy classes, 12 normal (10 characteristic)
C1, C2, C3, C3, C22, C6, C9, C32, C32, A4, C2×C6, C2×C6, C3×C6, He3, 3- 1+2, C33, C3.A4, C3×A4, C62, C62, C32×C6, C3≀C3, C32.A4, C32⋊A4, C3×C62, C33⋊2A4
Quotients: C1, C3, C32, A4, He3, C3×A4, C3≀C3, C32⋊A4, C33⋊2A4
►On 18 points - transitive group 18T127
(13 14 15)(16 17 18)
(7 11 9)(8 12 10)(13 15 14)(16 18 17)
(1 5 3)(2 6 4)(7 11 9)(8 12 10)(13 14 15)(16 17 18)
(1 2)(3 4)(5 6)(13 16)(14 17)(15 18)
(7 8)(9 10)(11 12)(13 16)(14 17)(15 18)
(1 13 7)(2 16 8)(3 15 9)(4 18 10)(5 14 11)(6 17 12)
G:=sub<Sym(18)| (13,14,15)(16,17,18), (7,11,9)(8,12,10)(13,15,14)(16,18,17), (1,5,3)(2,6,4)(7,11,9)(8,12,10)(13,14,15)(16,17,18), (1,2)(3,4)(5,6)(13,16)(14,17)(15,18), (7,8)(9,10)(11,12)(13,16)(14,17)(15,18), (1,13,7)(2,16,8)(3,15,9)(4,18,10)(5,14,11)(6,17,12)>;
G:=Group( (13,14,15)(16,17,18), (7,11,9)(8,12,10)(13,15,14)(16,18,17), (1,5,3)(2,6,4)(7,11,9)(8,12,10)(13,14,15)(16,17,18), (1,2)(3,4)(5,6)(13,16)(14,17)(15,18), (7,8)(9,10)(11,12)(13,16)(14,17)(15,18), (1,13,7)(2,16,8)(3,15,9)(4,18,10)(5,14,11)(6,17,12) );
G=PermutationGroup([[(13,14,15),(16,17,18)], [(7,11,9),(8,12,10),(13,15,14),(16,18,17)], [(1,5,3),(2,6,4),(7,11,9),(8,12,10),(13,14,15),(16,17,18)], [(1,2),(3,4),(5,6),(13,16),(14,17),(15,18)], [(7,8),(9,10),(11,12),(13,16),(14,17),(15,18)], [(1,13,7),(2,16,8),(3,15,9),(4,18,10),(5,14,11),(6,17,12)]])
G:=TransitiveGroup(18,127);
44 conjugacy classes
| class | 1 | 2 | 3A | 3B | 3C | ··· | 3J | 3K | 3L | 6A | ··· | 6Z | 9A | 9B | 9C | 9D |
| order | 1 | 2 | 3 | 3 | 3 | ··· | 3 | 3 | 3 | 6 | ··· | 6 | 9 | 9 | 9 | 9 |
| size | 1 | 3 | 1 | 1 | 3 | ··· | 3 | 36 | 36 | 3 | ··· | 3 | 36 | 36 | 36 | 36 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 | 3 | 3 |
| type | + | + | ||||||||
| image | C1 | C3 | C3 | C3 | A4 | He3 | C3×A4 | C3≀C3 | C32⋊A4 | C33⋊2A4 |
| kernel | C33⋊2A4 | C32.A4 | C32⋊A4 | C3×C62 | C33 | C2×C6 | C32 | C22 | C3 | C1 |
| # reps | 1 | 4 | 2 | 2 | 1 | 2 | 2 | 6 | 6 | 18 |
Matrix representation of C33⋊2A4 ►in GL3(𝔽7) generated by
| 1 | 0 | 0 |
| 0 | 2 | 1 |
| 0 | 2 | 3 |
| 1 | 0 | 0 |
| 5 | 1 | 4 |
| 2 | 1 | 5 |
| 4 | 0 | 0 |
| 0 | 4 | 0 |
| 0 | 0 | 4 |
| 6 | 0 | 0 |
| 0 | 2 | 3 |
| 0 | 6 | 5 |
| 6 | 0 | 0 |
| 1 | 5 | 4 |
| 6 | 1 | 2 |
| 3 | 4 | 5 |
| 3 | 0 | 3 |
| 0 | 0 | 4 |
G:=sub<GL(3,GF(7))| [1,0,0,0,2,2,0,1,3],[1,5,2,0,1,1,0,4,5],[4,0,0,0,4,0,0,0,4],[6,0,0,0,2,6,0,3,5],[6,1,6,0,5,1,0,4,2],[3,3,0,4,0,0,5,3,4] >;
C33⋊2A4 in GAP, Magma, Sage, TeX
C_3^3\rtimes_2A_4
% in TeX
G:=Group("C3^3:2A4"); // GroupNames label
G:=SmallGroup(324,60);
// by ID
G=gap.SmallGroup(324,60);
# by ID
G:=PCGroup([6,-3,-3,-3,-3,-2,2,145,650,4864,8753]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^3=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,f*a*f^-1=a*b^-1*c,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