direct product, non-abelian, soluble, monomial
Aliases: C2xC32:S4, C62:15D6, (C3xC6):2S4, (C6xA4):1S3, (C3xA4):2D6, (C2xC62):5S3, C32:3(C2xS4), C6.13(C3:S4), C32:A4:3C22, C23:(He3:C2), C3.3(C2xC3:S4), (C2xC32:A4):2C2, C22:(C2xHe3:C2), (C22xC6).2(C3:S3), (C2xC6).1(C2xC3:S3), SmallGroup(432,538)
Series: Derived ►Chief ►Lower central ►Upper central
| C32:A4 — C2xC32:S4 |
Generators and relations for C2xC32:S4
G = < a,b,c,d,e | a2=b6=c6=d3=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, dbd-1=b4c, ebe=b2c3, dcd-1=b3c, ece=b3c4, ede=d-1 >
Subgroups: 967 in 166 conjugacy classes, 23 normal (15 characteristic)
C1, C2, C2, C3, C3, C4, C22, C22, S3, C6, C6, C2xC4, D4, C23, C23, C32, C32, Dic3, C12, A4, D6, C2xC6, C2xC6, C2xD4, C3xS3, C3xC6, C3xC6, C2xDic3, C3:D4, C2xC12, C3xD4, S4, C2xA4, C22xS3, C22xC6, C22xC6, He3, C3xDic3, C3xA4, S3xC6, C62, C62, C2xC3:D4, C6xD4, C2xS4, He3:C2, C2xHe3, C6xDic3, C3xC3:D4, C3xS4, C6xA4, S3xC2xC6, C2xC62, C32:A4, C2xHe3:C2, C6xC3:D4, C6xS4, C32:S4, C2xC32:A4, C2xC32:S4
Quotients: C1, C2, C22, S3, D6, C3:S3, S4, C2xC3:S3, C2xS4, He3:C2, C3:S4, C2xHe3:C2, C2xC3:S4, C32:S4, C2xC32:S4
►On 18 points - transitive group 18T156
(1 2)(3 4)(5 6)(7 12)(8 10)(9 11)(13 16)(14 17)(15 18)
(1 2)(3 4)(5 6)(7 8 9)(10 11 12)(13 14 15 16 17 18)
(1 4 5 2 3 6)(7 11 8 12 9 10)(13 17 15)(14 18 16)
(1 13 7)(2 16 12)(3 17 9)(4 14 11)(5 15 8)(6 18 10)
(7 13)(8 15)(9 17)(10 18)(11 14)(12 16)
G:=sub<Sym(18)| (1,2)(3,4)(5,6)(7,12)(8,10)(9,11)(13,16)(14,17)(15,18), (1,2)(3,4)(5,6)(7,8,9)(10,11,12)(13,14,15,16,17,18), (1,4,5,2,3,6)(7,11,8,12,9,10)(13,17,15)(14,18,16), (1,13,7)(2,16,12)(3,17,9)(4,14,11)(5,15,8)(6,18,10), (7,13)(8,15)(9,17)(10,18)(11,14)(12,16)>;
G:=Group( (1,2)(3,4)(5,6)(7,12)(8,10)(9,11)(13,16)(14,17)(15,18), (1,2)(3,4)(5,6)(7,8,9)(10,11,12)(13,14,15,16,17,18), (1,4,5,2,3,6)(7,11,8,12,9,10)(13,17,15)(14,18,16), (1,13,7)(2,16,12)(3,17,9)(4,14,11)(5,15,8)(6,18,10), (7,13)(8,15)(9,17)(10,18)(11,14)(12,16) );
G=PermutationGroup([[(1,2),(3,4),(5,6),(7,12),(8,10),(9,11),(13,16),(14,17),(15,18)], [(1,2),(3,4),(5,6),(7,8,9),(10,11,12),(13,14,15,16,17,18)], [(1,4,5,2,3,6),(7,11,8,12,9,10),(13,17,15),(14,18,16)], [(1,13,7),(2,16,12),(3,17,9),(4,14,11),(5,15,8),(6,18,10)], [(7,13),(8,15),(9,17),(10,18),(11,14),(12,16)]])
G:=TransitiveGroup(18,156);
38 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | 3B | 3C | 3D | 3E | 3F | 4A | 4B | 6A | 6B | 6C | 6D | 6E | 6F | 6G | ··· | 6M | 6N | 6O | 6P | 6Q | 6R | 6S | 6T | 12A | 12B | 12C | 12D |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 3 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 12 | 12 | 12 | 12 |
| size | 1 | 1 | 3 | 3 | 18 | 18 | 1 | 1 | 6 | 24 | 24 | 24 | 18 | 18 | 1 | 1 | 3 | 3 | 3 | 3 | 6 | ··· | 6 | 18 | 18 | 18 | 18 | 24 | 24 | 24 | 18 | 18 | 18 | 18 |
38 irreducible representations
| dim | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 3 | 6 | 6 | 6 | 6 |
| type | + | + | + | + | + | + | + | + | + | + | + | ||||||
| image | C1 | C2 | C2 | S3 | S3 | D6 | D6 | S4 | C2xS4 | He3:C2 | C2xHe3:C2 | C32:S4 | C2xC32:S4 | C3:S4 | C2xC3:S4 | C32:S4 | C2xC32:S4 |
| kernel | C2xC32:S4 | C32:S4 | C2xC32:A4 | C6xA4 | C2xC62 | C3xA4 | C62 | C3xC6 | C32 | C23 | C22 | C2 | C1 | C6 | C3 | C2 | C1 |
| # reps | 1 | 2 | 1 | 3 | 1 | 3 | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 1 | 1 | 2 | 2 |
Matrix representation of C2xC32:S4 ►in GL3(F7) generated by
| 6 | 0 | 0 |
| 0 | 6 | 0 |
| 0 | 0 | 6 |
| 4 | 3 | 3 |
| 1 | 1 | 0 |
| 4 | 1 | 3 |
| 1 | 6 | 1 |
| 6 | 3 | 2 |
| 2 | 4 | 1 |
| 5 | 4 | 1 |
| 6 | 0 | 6 |
| 4 | 4 | 2 |
| 1 | 0 | 5 |
| 0 | 1 | 5 |
| 0 | 0 | 6 |
G:=sub<GL(3,GF(7))| [6,0,0,0,6,0,0,0,6],[4,1,4,3,1,1,3,0,3],[1,6,2,6,3,4,1,2,1],[5,6,4,4,0,4,1,6,2],[1,0,0,0,1,0,5,5,6] >;
C2xC32:S4 in GAP, Magma, Sage, TeX
C_2\times C_3^2\rtimes S_4
% in TeX
G:=Group("C2xC3^2:S4"); // GroupNames label
G:=SmallGroup(432,538);
// by ID
G=gap.SmallGroup(432,538);
# by ID
G:=PCGroup([7,-2,-2,-3,-3,-3,-2,2,170,675,353,9077,2287,5298,3989]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^6=c^6=d^3=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,d*b*d^-1=b^4*c,e*b*e=b^2*c^3,d*c*d^-1=b^3*c,e*c*e=b^3*c^4,e*d*e=d^-1>;
// generators/relations