direct product, metabelian, soluble, monomial, A-group
Aliases: C6xC42:C3, (C4xC12):6C6, (C2xC42):C32, C42:3(C3xC6), C23.5(C3xA4), C22.1(C6xA4), (C22xC6).11A4, (C2xC4xC12):C3, (C2xC6).9(C2xA4), SmallGroup(288,632)
Series: Derived ►Chief ►Lower central ►Upper central
| C42 — C6xC42:C3 |
Generators and relations for C6xC42:C3
G = < a,b,c,d | a6=b4=c4=d3=1, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=bc-1, dcd-1=b-1c2 >
Subgroups: 276 in 68 conjugacy classes, 20 normal (14 characteristic)
C1, C2, C2, C3, C3, C4, C22, C22, C6, C6, C2xC4, C23, C32, C12, A4, C2xC6, C2xC6, C42, C42, C22xC4, C3xC6, C2xC12, C2xA4, C22xC6, C2xC42, C3xA4, C42:C3, C4xC12, C4xC12, C22xC12, C6xA4, C2xC42:C3, C2xC4xC12, C3xC42:C3, C6xC42:C3
Quotients: C1, C2, C3, C6, C32, A4, C3xC6, C2xA4, C3xA4, C42:C3, C6xA4, C2xC42:C3, C3xC42:C3, C6xC42:C3
►On 36 points
(1 2 3 4 5 6)(7 8 9 10 11 12)(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 4)(2 5)(3 6)(7 35 10 32)(8 36 11 33)(9 31 12 34)(13 16)(14 17)(15 18)(19 30 22 27)(20 25 23 28)(21 26 24 29)
(1 16 4 13)(2 17 5 14)(3 18 6 15)(19 27 22 30)(20 28 23 25)(21 29 24 26)
(1 31 21)(2 32 22)(3 33 23)(4 34 24)(5 35 19)(6 36 20)(7 27 17)(8 28 18)(9 29 13)(10 30 14)(11 25 15)(12 26 16)
G:=sub<Sym(36)| (1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18)(19,20,21,22,23,24)(25,26,27,28,29,30)(31,32,33,34,35,36), (1,4)(2,5)(3,6)(7,35,10,32)(8,36,11,33)(9,31,12,34)(13,16)(14,17)(15,18)(19,30,22,27)(20,25,23,28)(21,26,24,29), (1,16,4,13)(2,17,5,14)(3,18,6,15)(19,27,22,30)(20,28,23,25)(21,29,24,26), (1,31,21)(2,32,22)(3,33,23)(4,34,24)(5,35,19)(6,36,20)(7,27,17)(8,28,18)(9,29,13)(10,30,14)(11,25,15)(12,26,16)>;
G:=Group( (1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18)(19,20,21,22,23,24)(25,26,27,28,29,30)(31,32,33,34,35,36), (1,4)(2,5)(3,6)(7,35,10,32)(8,36,11,33)(9,31,12,34)(13,16)(14,17)(15,18)(19,30,22,27)(20,25,23,28)(21,26,24,29), (1,16,4,13)(2,17,5,14)(3,18,6,15)(19,27,22,30)(20,28,23,25)(21,29,24,26), (1,31,21)(2,32,22)(3,33,23)(4,34,24)(5,35,19)(6,36,20)(7,27,17)(8,28,18)(9,29,13)(10,30,14)(11,25,15)(12,26,16) );
G=PermutationGroup([[(1,2,3,4,5,6),(7,8,9,10,11,12),(13,14,15,16,17,18),(19,20,21,22,23,24),(25,26,27,28,29,30),(31,32,33,34,35,36)], [(1,4),(2,5),(3,6),(7,35,10,32),(8,36,11,33),(9,31,12,34),(13,16),(14,17),(15,18),(19,30,22,27),(20,25,23,28),(21,26,24,29)], [(1,16,4,13),(2,17,5,14),(3,18,6,15),(19,27,22,30),(20,28,23,25),(21,29,24,26)], [(1,31,21),(2,32,22),(3,33,23),(4,34,24),(5,35,19),(6,36,20),(7,27,17),(8,28,18),(9,29,13),(10,30,14),(11,25,15),(12,26,16)]])
48 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | ··· | 3H | 4A | ··· | 4H | 6A | 6B | 6C | 6D | 6E | 6F | 6G | ··· | 6L | 12A | ··· | 12P |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | ··· | 3 | 4 | ··· | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | ··· | 6 | 12 | ··· | 12 |
| size | 1 | 1 | 3 | 3 | 1 | 1 | 16 | ··· | 16 | 3 | ··· | 3 | 1 | 1 | 3 | 3 | 3 | 3 | 16 | ··· | 16 | 3 | ··· | 3 |
48 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 | C2xA4 | C3xA4 | C42:C3 | C6xA4 | C2xC42:C3 | C3xC42:C3 | C6xC42:C3 |
| kernel | C6xC42:C3 | C3xC42:C3 | C2xC42:C3 | C2xC4xC12 | C42:C3 | C4xC12 | C22xC6 | C2xC6 | C23 | C6 | C22 | C3 | C2 | C1 |
| # reps | 1 | 1 | 6 | 2 | 6 | 2 | 1 | 1 | 2 | 4 | 2 | 4 | 8 | 8 |
Matrix representation of C6xC42:C3 ►in GL3(F13) generated by
| 4 | 0 | 0 |
| 0 | 4 | 0 |
| 0 | 0 | 4 |
| 0 | 8 | 3 |
| 8 | 2 | 1 |
| 3 | 2 | 12 |
| 5 | 0 | 0 |
| 12 | 0 | 2 |
| 7 | 9 | 4 |
| 3 | 2 | 0 |
| 0 | 11 | 6 |
| 0 | 1 | 12 |
G:=sub<GL(3,GF(13))| [4,0,0,0,4,0,0,0,4],[0,8,3,8,2,2,3,1,12],[5,12,7,0,0,9,0,2,4],[3,0,0,2,11,1,0,6,12] >;
C6xC42:C3 in GAP, Magma, Sage, TeX
C_6\times C_4^2\rtimes C_3
% in TeX
G:=Group("C6xC4^2:C3"); // GroupNames label
G:=SmallGroup(288,632);
// by ID
G=gap.SmallGroup(288,632);
# by ID
G:=PCGroup([7,-2,-3,-3,-2,2,-2,2,514,360,3476,102,3036,5305]);
// Polycyclic
G:=Group<a,b,c,d|a^6=b^4=c^4=d^3=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b*c^-1,d*c*d^-1=b^-1*c^2>;
// generators/relations