direct product, metabelian, soluble, monomial
Aliases: C22xC32:A4, C62:1A4, C24:2He3, (C6xA4):2C6, C23:(C2xHe3), C6.17(C6xA4), (C2xC62):7C6, C62:14(C2xC6), (C2xC6).6C62, C22:(C22xHe3), (C22xC62):1C3, C32:3(C22xA4), (C23xC6).10C32, (A4xC2xC6):2C3, C3.5(A4xC2xC6), (C3xC6):2(C2xA4), (C3xA4):3(C2xC6), (C2xC6).26(C3xA4), (C22xC6).11(C3xC6), SmallGroup(432,550)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C22xC32:A4
G = < a,b,c,d,e,f,g | a2=b2=c3=d3=e2=f2=g3=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, bc=cb, bd=db, be=eb, bf=fb, bg=gb, cd=dc, ce=ec, cf=fc, gcg-1=cd-1, de=ed, df=fd, dg=gd, geg-1=ef=fe, gfg-1=e >
Subgroups: 727 in 224 conjugacy classes, 50 normal (14 characteristic)
C1, C2, C2, C3, C3, C22, C22, C6, C6, C23, C23, C32, C32, A4, C2xC6, C2xC6, C24, C3xC6, C3xC6, C2xA4, C22xC6, C22xC6, He3, C3xA4, C62, C62, C22xA4, C23xC6, C23xC6, C2xHe3, C6xA4, C2xC62, C2xC62, C32:A4, C22xHe3, A4xC2xC6, C22xC62, C2xC32:A4, C22xC32:A4
Quotients: C1, C2, C3, C22, C6, C32, A4, C2xC6, C3xC6, C2xA4, He3, C3xA4, C62, C22xA4, C2xHe3, C6xA4, C32:A4, C22xHe3, A4xC2xC6, C2xC32:A4, C22xC32:A4
►On 36 points
(1 11)(2 10)(3 12)(4 7)(5 9)(6 8)(13 36)(14 34)(15 35)(16 32)(17 33)(18 31)(19 25)(20 26)(21 27)(22 30)(23 28)(24 29)
(1 10)(2 11)(3 7)(4 12)(5 8)(6 9)(13 21)(14 19)(15 20)(16 28)(17 29)(18 30)(22 31)(23 32)(24 33)(25 34)(26 35)(27 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 5 3)(2 6 4)(7 10 8)(9 12 11)(13 14 15)(16 18 17)(19 20 21)(22 24 23)(25 26 27)(28 30 29)(31 33 32)(34 35 36)
(1 2)(3 4)(5 6)(7 12)(8 9)(10 11)(16 23)(17 24)(18 22)(28 32)(29 33)(30 31)
(13 27)(14 25)(15 26)(16 23)(17 24)(18 22)(19 34)(20 35)(21 36)(28 32)(29 33)(30 31)
(1 17 19)(2 24 34)(3 18 21)(4 22 36)(5 16 20)(6 23 35)(7 30 13)(8 28 15)(9 32 26)(10 29 14)(11 33 25)(12 31 27)
G:=sub<Sym(36)| (1,11)(2,10)(3,12)(4,7)(5,9)(6,8)(13,36)(14,34)(15,35)(16,32)(17,33)(18,31)(19,25)(20,26)(21,27)(22,30)(23,28)(24,29), (1,10)(2,11)(3,7)(4,12)(5,8)(6,9)(13,21)(14,19)(15,20)(16,28)(17,29)(18,30)(22,31)(23,32)(24,33)(25,34)(26,35)(27,36), (13,14,15)(16,17,18)(19,20,21)(22,23,24)(25,26,27)(28,29,30)(31,32,33)(34,35,36), (1,5,3)(2,6,4)(7,10,8)(9,12,11)(13,14,15)(16,18,17)(19,20,21)(22,24,23)(25,26,27)(28,30,29)(31,33,32)(34,35,36), (1,2)(3,4)(5,6)(7,12)(8,9)(10,11)(16,23)(17,24)(18,22)(28,32)(29,33)(30,31), (13,27)(14,25)(15,26)(16,23)(17,24)(18,22)(19,34)(20,35)(21,36)(28,32)(29,33)(30,31), (1,17,19)(2,24,34)(3,18,21)(4,22,36)(5,16,20)(6,23,35)(7,30,13)(8,28,15)(9,32,26)(10,29,14)(11,33,25)(12,31,27)>;
G:=Group( (1,11)(2,10)(3,12)(4,7)(5,9)(6,8)(13,36)(14,34)(15,35)(16,32)(17,33)(18,31)(19,25)(20,26)(21,27)(22,30)(23,28)(24,29), (1,10)(2,11)(3,7)(4,12)(5,8)(6,9)(13,21)(14,19)(15,20)(16,28)(17,29)(18,30)(22,31)(23,32)(24,33)(25,34)(26,35)(27,36), (13,14,15)(16,17,18)(19,20,21)(22,23,24)(25,26,27)(28,29,30)(31,32,33)(34,35,36), (1,5,3)(2,6,4)(7,10,8)(9,12,11)(13,14,15)(16,18,17)(19,20,21)(22,24,23)(25,26,27)(28,30,29)(31,33,32)(34,35,36), (1,2)(3,4)(5,6)(7,12)(8,9)(10,11)(16,23)(17,24)(18,22)(28,32)(29,33)(30,31), (13,27)(14,25)(15,26)(16,23)(17,24)(18,22)(19,34)(20,35)(21,36)(28,32)(29,33)(30,31), (1,17,19)(2,24,34)(3,18,21)(4,22,36)(5,16,20)(6,23,35)(7,30,13)(8,28,15)(9,32,26)(10,29,14)(11,33,25)(12,31,27) );
G=PermutationGroup([[(1,11),(2,10),(3,12),(4,7),(5,9),(6,8),(13,36),(14,34),(15,35),(16,32),(17,33),(18,31),(19,25),(20,26),(21,27),(22,30),(23,28),(24,29)], [(1,10),(2,11),(3,7),(4,12),(5,8),(6,9),(13,21),(14,19),(15,20),(16,28),(17,29),(18,30),(22,31),(23,32),(24,33),(25,34),(26,35),(27,36)], [(13,14,15),(16,17,18),(19,20,21),(22,23,24),(25,26,27),(28,29,30),(31,32,33),(34,35,36)], [(1,5,3),(2,6,4),(7,10,8),(9,12,11),(13,14,15),(16,18,17),(19,20,21),(22,24,23),(25,26,27),(28,30,29),(31,33,32),(34,35,36)], [(1,2),(3,4),(5,6),(7,12),(8,9),(10,11),(16,23),(17,24),(18,22),(28,32),(29,33),(30,31)], [(13,27),(14,25),(15,26),(16,23),(17,24),(18,22),(19,34),(20,35),(21,36),(28,32),(29,33),(30,31)], [(1,17,19),(2,24,34),(3,18,21),(4,22,36),(5,16,20),(6,23,35),(7,30,13),(8,28,15),(9,32,26),(10,29,14),(11,33,25),(12,31,27)]])
80 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3A | 3B | 3C | 3D | 3E | ··· | 3J | 6A | ··· | 6F | 6G | ··· | 6AR | 6AS | ··· | 6BJ |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | ··· | 3 | 6 | ··· | 6 | 6 | ··· | 6 | 6 | ··· | 6 |
| size | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 | 1 | 1 | 3 | 3 | 12 | ··· | 12 | 1 | ··· | 1 | 3 | ··· | 3 | 12 | ··· | 12 |
80 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 | He3 | C3xA4 | C2xHe3 | C6xA4 | C32:A4 | C2xC32:A4 |
| kernel | C22xC32:A4 | C2xC32:A4 | A4xC2xC6 | C22xC62 | C6xA4 | C2xC62 | C62 | C3xC6 | C24 | C2xC6 | C23 | C6 | C22 | C2 |
| # reps | 1 | 3 | 6 | 2 | 18 | 6 | 1 | 3 | 2 | 2 | 6 | 6 | 6 | 18 |
Matrix representation of C22xC32:A4 ►in GL4(F7) generated by
| 6 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 6 | 0 | 0 |
| 0 | 0 | 6 | 0 |
| 0 | 0 | 0 | 6 |
| 2 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 2 | 0 |
| 0 | 0 | 0 | 4 |
| 1 | 0 | 0 | 0 |
| 0 | 2 | 0 | 0 |
| 0 | 0 | 2 | 0 |
| 0 | 0 | 0 | 2 |
| 1 | 0 | 0 | 0 |
| 0 | 6 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 6 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 6 | 0 |
| 0 | 0 | 0 | 6 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 |
G:=sub<GL(4,GF(7))| [6,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,6,0,0,0,0,6,0,0,0,0,6],[2,0,0,0,0,1,0,0,0,0,2,0,0,0,0,4],[1,0,0,0,0,2,0,0,0,0,2,0,0,0,0,2],[1,0,0,0,0,6,0,0,0,0,1,0,0,0,0,6],[1,0,0,0,0,1,0,0,0,0,6,0,0,0,0,6],[1,0,0,0,0,0,0,1,0,1,0,0,0,0,1,0] >;
C22xC32:A4 in GAP, Magma, Sage, TeX
C_2^2\times C_3^2\rtimes A_4
% in TeX
G:=Group("C2^2xC3^2:A4"); // GroupNames label
G:=SmallGroup(432,550);
// by ID
G=gap.SmallGroup(432,550);
# by ID
G:=PCGroup([7,-2,-2,-3,-3,-3,-2,2,353,2287,3989]);
// Polycyclic
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^3=d^3=e^2=f^2=g^3=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,a*g=g*a,b*c=c*b,b*d=d*b,b*e=e*b,b*f=f*b,b*g=g*b,c*d=d*c,c*e=e*c,c*f=f*c,g*c*g^-1=c*d^-1,d*e=e*d,d*f=f*d,d*g=g*d,g*e*g^-1=e*f=f*e,g*f*g^-1=e>;
// generators/relations