direct product, metabelian, soluble, monomial
Aliases: C4xC32:A4, C62:10C12, (C3xC12):A4, (C12xA4):C3, (C22xC4):He3, (C3xA4):2C12, C22:(C4xHe3), (C6xA4).3C6, C6.10(C6xA4), C12.7(C3xA4), C3.5(C12xA4), C32:3(C4xA4), C23.(C2xHe3), (C2xC62).13C6, (C22xC12).5C32, (C2xC6xC12):1C3, (C3xC6).5(C2xA4), (C2xC6).5(C3xC12), C2.1(C2xC32:A4), (C2xC32:A4).4C2, (C22xC6).6(C3xC6), SmallGroup(432,333)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C4xC32:A4
G = < a,b,c,d,e,f | a4=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: 357 in 108 conjugacy classes, 30 normal (21 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, C22, C6, C6, C2xC4, C23, C32, C32, C12, C12, A4, C2xC6, C2xC6, C22xC4, C3xC6, C3xC6, C2xC12, C2xA4, C22xC6, C22xC6, He3, C3xC12, C3xC12, C3xA4, C62, C62, C4xA4, C22xC12, C22xC12, C2xHe3, C6xC12, C6xA4, C2xC62, C4xHe3, C32:A4, C12xA4, C2xC6xC12, C2xC32:A4, C4xC32:A4
Quotients: C1, C2, C3, C4, C6, C32, C12, A4, C3xC6, C2xA4, He3, C3xC12, C3xA4, C4xA4, C2xHe3, C6xA4, C4xHe3, C32:A4, C12xA4, C2xC32:A4, C4xC32:A4
►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)
(5 26 32)(6 27 29)(7 28 30)(8 25 31)(9 20 36)(10 17 33)(11 18 34)(12 19 35)
(1 13 21)(2 14 22)(3 15 23)(4 16 24)(5 26 32)(6 27 29)(7 28 30)(8 25 31)(9 36 20)(10 33 17)(11 34 18)(12 35 19)
(5 7)(6 8)(9 11)(10 12)(17 19)(18 20)(25 27)(26 28)(29 31)(30 32)(33 35)(34 36)
(1 3)(2 4)(5 7)(6 8)(13 15)(14 16)(21 23)(22 24)(25 27)(26 28)(29 31)(30 32)
(1 35 8)(2 36 5)(3 33 6)(4 34 7)(9 32 22)(10 29 23)(11 30 24)(12 31 21)(13 19 25)(14 20 26)(15 17 27)(16 18 28)
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), (5,26,32)(6,27,29)(7,28,30)(8,25,31)(9,20,36)(10,17,33)(11,18,34)(12,19,35), (1,13,21)(2,14,22)(3,15,23)(4,16,24)(5,26,32)(6,27,29)(7,28,30)(8,25,31)(9,36,20)(10,33,17)(11,34,18)(12,35,19), (5,7)(6,8)(9,11)(10,12)(17,19)(18,20)(25,27)(26,28)(29,31)(30,32)(33,35)(34,36), (1,3)(2,4)(5,7)(6,8)(13,15)(14,16)(21,23)(22,24)(25,27)(26,28)(29,31)(30,32), (1,35,8)(2,36,5)(3,33,6)(4,34,7)(9,32,22)(10,29,23)(11,30,24)(12,31,21)(13,19,25)(14,20,26)(15,17,27)(16,18,28)>;
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), (5,26,32)(6,27,29)(7,28,30)(8,25,31)(9,20,36)(10,17,33)(11,18,34)(12,19,35), (1,13,21)(2,14,22)(3,15,23)(4,16,24)(5,26,32)(6,27,29)(7,28,30)(8,25,31)(9,36,20)(10,33,17)(11,34,18)(12,35,19), (5,7)(6,8)(9,11)(10,12)(17,19)(18,20)(25,27)(26,28)(29,31)(30,32)(33,35)(34,36), (1,3)(2,4)(5,7)(6,8)(13,15)(14,16)(21,23)(22,24)(25,27)(26,28)(29,31)(30,32), (1,35,8)(2,36,5)(3,33,6)(4,34,7)(9,32,22)(10,29,23)(11,30,24)(12,31,21)(13,19,25)(14,20,26)(15,17,27)(16,18,28) );
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)], [(5,26,32),(6,27,29),(7,28,30),(8,25,31),(9,20,36),(10,17,33),(11,18,34),(12,19,35)], [(1,13,21),(2,14,22),(3,15,23),(4,16,24),(5,26,32),(6,27,29),(7,28,30),(8,25,31),(9,36,20),(10,33,17),(11,34,18),(12,35,19)], [(5,7),(6,8),(9,11),(10,12),(17,19),(18,20),(25,27),(26,28),(29,31),(30,32),(33,35),(34,36)], [(1,3),(2,4),(5,7),(6,8),(13,15),(14,16),(21,23),(22,24),(25,27),(26,28),(29,31),(30,32)], [(1,35,8),(2,36,5),(3,33,6),(4,34,7),(9,32,22),(10,29,23),(11,30,24),(12,31,21),(13,19,25),(14,20,26),(15,17,27),(16,18,28)]])
80 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 3D | 3E | ··· | 3J | 4A | 4B | 4C | 4D | 6A | 6B | 6C | ··· | 6T | 6U | ··· | 6Z | 12A | 12B | 12C | 12D | 12E | ··· | 12X | 12Y | ··· | 12AJ |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | ··· | 3 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | ··· | 6 | 6 | ··· | 6 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 3 | 3 | 1 | 1 | 3 | 3 | 12 | ··· | 12 | 1 | 1 | 3 | 3 | 1 | 1 | 3 | ··· | 3 | 12 | ··· | 12 | 1 | 1 | 1 | 1 | 3 | ··· | 3 | 12 | ··· | 12 |
80 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 |
| type | + | + | + | + | |||||||||||||||||
| image | C1 | C2 | C3 | C3 | C4 | C6 | C6 | C12 | C12 | A4 | C2xA4 | He3 | C3xA4 | C4xA4 | C2xHe3 | C6xA4 | C32:A4 | C4xHe3 | C12xA4 | C2xC32:A4 | C4xC32:A4 |
| kernel | C4xC32:A4 | C2xC32:A4 | C12xA4 | C2xC6xC12 | C32:A4 | C6xA4 | C2xC62 | C3xA4 | C62 | C3xC12 | C3xC6 | C22xC4 | C12 | C32 | C23 | C6 | C4 | C22 | C3 | C2 | C1 |
| # reps | 1 | 1 | 6 | 2 | 2 | 6 | 2 | 12 | 4 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 6 | 4 | 4 | 6 | 12 |
Matrix representation of C4xC32:A4 ►in GL3(F13) generated by
| 5 | 0 | 0 |
| 0 | 5 | 0 |
| 0 | 0 | 5 |
| 9 | 0 | 0 |
| 0 | 3 | 0 |
| 0 | 0 | 1 |
| 3 | 0 | 0 |
| 0 | 3 | 0 |
| 0 | 0 | 3 |
| 1 | 0 | 0 |
| 0 | 12 | 0 |
| 0 | 0 | 12 |
| 12 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 12 |
| 0 | 0 | 12 |
| 8 | 0 | 0 |
| 0 | 8 | 0 |
G:=sub<GL(3,GF(13))| [5,0,0,0,5,0,0,0,5],[9,0,0,0,3,0,0,0,1],[3,0,0,0,3,0,0,0,3],[1,0,0,0,12,0,0,0,12],[12,0,0,0,1,0,0,0,12],[0,8,0,0,0,8,12,0,0] >;
C4xC32:A4 in GAP, Magma, Sage, TeX
C_4\times C_3^2\rtimes A_4
% in TeX
G:=Group("C4xC3^2:A4"); // GroupNames label
G:=SmallGroup(432,333);
// by ID
G=gap.SmallGroup(432,333);
# by ID
G:=PCGroup([7,-2,-3,-3,-2,-3,-2,2,126,450,4548,7951]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^4=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