Aliases: C62⋊2A4, C24⋊3He3, C32⋊(C22⋊A4), (C22×C62)⋊2C3, C22⋊2(C32⋊A4), (C23×C6).15C32, (C3×C22⋊A4)⋊2C3, (C2×C6).16(C3×A4), C3.5(C3×C22⋊A4), SmallGroup(432,555)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C62⋊A4
G = < a,b,c,d,e | a6=b6=c2=d2=e3=1, ab=ba, ac=ca, ad=da, eae-1=ab-1, bc=cb, bd=db, ebe-1=a3b4, ece-1=cd=dc, ede-1=c >
Subgroups: 883 in 197 conjugacy classes, 25 normal (7 characteristic)
C1, C2, C3, C3, C22, C22, C6, C23, C32, C32, A4, C2×C6, C2×C6, C24, C3×C6, C22×C6, He3, C3×A4, C62, C62, C22⋊A4, C23×C6, C23×C6, C2×C62, C32⋊A4, C3×C22⋊A4, C22×C62, C62⋊A4
Quotients: C1, C3, C32, A4, He3, C3×A4, C22⋊A4, C32⋊A4, C3×C22⋊A4, C62⋊A4
►On 36 points
Generators in S36
(13 14 15 16 17 18)(19 20 21 22 23 24)(25 26 27 28 29 30)(31 32 33 34 35 36)
(1 6 9 2 5 10)(3 8 11 4 7 12)(13 14 15 16 17 18)(19 20 21 22 23 24)(25 27 29)(26 28 30)(31 33 35)(32 34 36)
(1 4)(2 3)(5 8)(6 7)(9 12)(10 11)(13 23)(14 24)(15 19)(16 20)(17 21)(18 22)(25 28)(26 29)(27 30)(31 34)(32 35)(33 36)
(1 3)(2 4)(5 7)(6 8)(9 11)(10 12)(13 16)(14 17)(15 18)(19 22)(20 23)(21 24)(25 31)(26 32)(27 33)(28 34)(29 35)(30 36)
(1 29 13)(2 26 16)(3 32 23)(4 35 20)(5 25 17)(6 28 14)(7 34 21)(8 31 24)(9 27 15)(10 30 18)(11 36 19)(12 33 22)
G:=sub<Sym(36)| (13,14,15,16,17,18)(19,20,21,22,23,24)(25,26,27,28,29,30)(31,32,33,34,35,36), (1,6,9,2,5,10)(3,8,11,4,7,12)(13,14,15,16,17,18)(19,20,21,22,23,24)(25,27,29)(26,28,30)(31,33,35)(32,34,36), (1,4)(2,3)(5,8)(6,7)(9,12)(10,11)(13,23)(14,24)(15,19)(16,20)(17,21)(18,22)(25,28)(26,29)(27,30)(31,34)(32,35)(33,36), (1,3)(2,4)(5,7)(6,8)(9,11)(10,12)(13,16)(14,17)(15,18)(19,22)(20,23)(21,24)(25,31)(26,32)(27,33)(28,34)(29,35)(30,36), (1,29,13)(2,26,16)(3,32,23)(4,35,20)(5,25,17)(6,28,14)(7,34,21)(8,31,24)(9,27,15)(10,30,18)(11,36,19)(12,33,22)>;
G:=Group( (13,14,15,16,17,18)(19,20,21,22,23,24)(25,26,27,28,29,30)(31,32,33,34,35,36), (1,6,9,2,5,10)(3,8,11,4,7,12)(13,14,15,16,17,18)(19,20,21,22,23,24)(25,27,29)(26,28,30)(31,33,35)(32,34,36), (1,4)(2,3)(5,8)(6,7)(9,12)(10,11)(13,23)(14,24)(15,19)(16,20)(17,21)(18,22)(25,28)(26,29)(27,30)(31,34)(32,35)(33,36), (1,3)(2,4)(5,7)(6,8)(9,11)(10,12)(13,16)(14,17)(15,18)(19,22)(20,23)(21,24)(25,31)(26,32)(27,33)(28,34)(29,35)(30,36), (1,29,13)(2,26,16)(3,32,23)(4,35,20)(5,25,17)(6,28,14)(7,34,21)(8,31,24)(9,27,15)(10,30,18)(11,36,19)(12,33,22) );
G=PermutationGroup([[(13,14,15,16,17,18),(19,20,21,22,23,24),(25,26,27,28,29,30),(31,32,33,34,35,36)], [(1,6,9,2,5,10),(3,8,11,4,7,12),(13,14,15,16,17,18),(19,20,21,22,23,24),(25,27,29),(26,28,30),(31,33,35),(32,34,36)], [(1,4),(2,3),(5,8),(6,7),(9,12),(10,11),(13,23),(14,24),(15,19),(16,20),(17,21),(18,22),(25,28),(26,29),(27,30),(31,34),(32,35),(33,36)], [(1,3),(2,4),(5,7),(6,8),(9,11),(10,12),(13,16),(14,17),(15,18),(19,22),(20,23),(21,24),(25,31),(26,32),(27,33),(28,34),(29,35),(30,36)], [(1,29,13),(2,26,16),(3,32,23),(4,35,20),(5,25,17),(6,28,14),(7,34,21),(8,31,24),(9,27,15),(10,30,18),(11,36,19),(12,33,22)]])
56 conjugacy classes
| class | 1 | 2A | ··· | 2E | 3A | 3B | 3C | 3D | 3E | ··· | 3J | 6A | ··· | 6AN |
| order | 1 | 2 | ··· | 2 | 3 | 3 | 3 | 3 | 3 | ··· | 3 | 6 | ··· | 6 |
| size | 1 | 3 | ··· | 3 | 1 | 1 | 3 | 3 | 48 | ··· | 48 | 3 | ··· | 3 |
56 irreducible representations
| dim | 1 | 1 | 1 | 3 | 3 | 3 | 3 |
| type | + | + | |||||
| image | C1 | C3 | C3 | A4 | He3 | C3×A4 | C32⋊A4 |
| kernel | C62⋊A4 | C3×C22⋊A4 | C22×C62 | C62 | C24 | C2×C6 | C22 |
| # reps | 1 | 6 | 2 | 5 | 2 | 10 | 30 |
Matrix representation of C62⋊A4 ►in GL6(𝔽7)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 5 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 0 |
| 0 | 0 | 0 | 0 | 0 | 5 |
| 5 | 0 | 0 | 0 | 0 | 0 |
| 0 | 5 | 0 | 0 | 0 | 0 |
| 0 | 0 | 2 | 0 | 0 | 0 |
| 0 | 0 | 0 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 |
| 6 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 6 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 6 | 0 |
| 0 | 0 | 0 | 0 | 0 | 6 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 6 | 0 | 0 | 0 | 0 |
| 0 | 0 | 6 | 0 | 0 | 0 |
| 0 | 0 | 0 | 6 | 0 | 0 |
| 0 | 0 | 0 | 0 | 6 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 1 | 0 | 0 |
G:=sub<GL(6,GF(7))| [1,0,0,0,0,0,0,5,0,0,0,0,0,0,3,0,0,0,0,0,0,1,0,0,0,0,0,0,3,0,0,0,0,0,0,5],[5,0,0,0,0,0,0,5,0,0,0,0,0,0,2,0,0,0,0,0,0,3,0,0,0,0,0,0,3,0,0,0,0,0,0,4],[6,0,0,0,0,0,0,1,0,0,0,0,0,0,6,0,0,0,0,0,0,1,0,0,0,0,0,0,6,0,0,0,0,0,0,6],[1,0,0,0,0,0,0,6,0,0,0,0,0,0,6,0,0,0,0,0,0,6,0,0,0,0,0,0,6,0,0,0,0,0,0,1],[0,0,1,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,1,0] >;
C62⋊A4 in GAP, Magma, Sage, TeX
C_6^2\rtimes A_4
% in TeX
G:=Group("C6^2:A4"); // GroupNames label
G:=SmallGroup(432,555);
// by ID
G=gap.SmallGroup(432,555);
# by ID
G:=PCGroup([7,-3,-3,-3,-2,2,-2,2,169,1515,2839,9077,15882]);
// Polycyclic
G:=Group<a,b,c,d,e|a^6=b^6=c^2=d^2=e^3=1,a*b=b*a,a*c=c*a,a*d=d*a,e*a*e^-1=a*b^-1,b*c=c*b,b*d=d*b,e*b*e^-1=a^3*b^4,e*c*e^-1=c*d=d*c,e*d*e^-1=c>;
// generators/relations