direct product, metabelian, supersoluble, monomial, A-group
Aliases: S3xC2xC6, C62:4C2, C32:2C23, C6:(C2xC6), (C2xC6):5C6, C3:(C22xC6), (C3xC6):2C22, SmallGroup(72,48)
Series: Derived ►Chief ►Lower central ►Upper central
| C3 — S3xC2xC6 |
Generators and relations for S3xC2xC6
G = < a,b,c,d | a2=b6=c3=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >
Subgroups: 118 in 69 conjugacy classes, 42 normal (10 characteristic)
C1, C2, C2, C3, C3, C22, C22, S3, C6, C6, C23, C32, D6, C2xC6, C2xC6, C3xS3, C3xC6, C22xS3, C22xC6, S3xC6, C62, S3xC2xC6
Quotients: C1, C2, C3, C22, S3, C6, C23, D6, C2xC6, C3xS3, C22xS3, C22xC6, S3xC6, S3xC2xC6
►On 24 points - transitive group 24T68
(1 8)(2 9)(3 10)(4 11)(5 12)(6 7)(13 22)(14 23)(15 24)(16 19)(17 20)(18 21)
(1 2 3 4 5 6)(7 8 9 10 11 12)(13 14 15 16 17 18)(19 20 21 22 23 24)
(1 3 5)(2 4 6)(7 9 11)(8 10 12)(13 17 15)(14 18 16)(19 23 21)(20 24 22)
(1 17)(2 18)(3 13)(4 14)(5 15)(6 16)(7 19)(8 20)(9 21)(10 22)(11 23)(12 24)
G:=sub<Sym(24)| (1,8)(2,9)(3,10)(4,11)(5,12)(6,7)(13,22)(14,23)(15,24)(16,19)(17,20)(18,21), (1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18)(19,20,21,22,23,24), (1,3,5)(2,4,6)(7,9,11)(8,10,12)(13,17,15)(14,18,16)(19,23,21)(20,24,22), (1,17)(2,18)(3,13)(4,14)(5,15)(6,16)(7,19)(8,20)(9,21)(10,22)(11,23)(12,24)>;
G:=Group( (1,8)(2,9)(3,10)(4,11)(5,12)(6,7)(13,22)(14,23)(15,24)(16,19)(17,20)(18,21), (1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18)(19,20,21,22,23,24), (1,3,5)(2,4,6)(7,9,11)(8,10,12)(13,17,15)(14,18,16)(19,23,21)(20,24,22), (1,17)(2,18)(3,13)(4,14)(5,15)(6,16)(7,19)(8,20)(9,21)(10,22)(11,23)(12,24) );
G=PermutationGroup([[(1,8),(2,9),(3,10),(4,11),(5,12),(6,7),(13,22),(14,23),(15,24),(16,19),(17,20),(18,21)], [(1,2,3,4,5,6),(7,8,9,10,11,12),(13,14,15,16,17,18),(19,20,21,22,23,24)], [(1,3,5),(2,4,6),(7,9,11),(8,10,12),(13,17,15),(14,18,16),(19,23,21),(20,24,22)], [(1,17),(2,18),(3,13),(4,14),(5,15),(6,16),(7,19),(8,20),(9,21),(10,22),(11,23),(12,24)]])
G:=TransitiveGroup(24,68);
S3xC2xC6 is a maximal subgroup of
D6:Dic3
36 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3A | 3B | 3C | 3D | 3E | 6A | ··· | 6F | 6G | ··· | 6O | 6P | ··· | 6W |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 6 | ··· | 6 | 6 | ··· | 6 | 6 | ··· | 6 |
| size | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 | 1 | 1 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 3 | ··· | 3 |
36 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C3 | C6 | C6 | S3 | D6 | C3xS3 | S3xC6 |
| kernel | S3xC2xC6 | S3xC6 | C62 | C22xS3 | D6 | C2xC6 | C2xC6 | C6 | C22 | C2 |
| # reps | 1 | 6 | 1 | 2 | 12 | 2 | 1 | 3 | 2 | 6 |
Matrix representation of S3xC2xC6 ►in GL3(F7) generated by
| 1 | 0 | 0 |
| 0 | 6 | 0 |
| 0 | 0 | 6 |
| 3 | 0 | 0 |
| 0 | 2 | 0 |
| 0 | 0 | 2 |
| 1 | 0 | 0 |
| 0 | 4 | 0 |
| 0 | 0 | 2 |
| 6 | 0 | 0 |
| 0 | 0 | 1 |
| 0 | 1 | 0 |
G:=sub<GL(3,GF(7))| [1,0,0,0,6,0,0,0,6],[3,0,0,0,2,0,0,0,2],[1,0,0,0,4,0,0,0,2],[6,0,0,0,0,1,0,1,0] >;
S3xC2xC6 in GAP, Magma, Sage, TeX
S_3\times C_2\times C_6
% in TeX
G:=Group("S3xC2xC6"); // GroupNames label
G:=SmallGroup(72,48);
// by ID
G=gap.SmallGroup(72,48);
# by ID
G:=PCGroup([5,-2,-2,-2,-3,-3,1204]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^6=c^3=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations