direct product, metabelian, supersoluble, monomial, A-group
Aliases: S3×C2×C6, C62⋊4C2, C32⋊2C23, C6⋊(C2×C6), (C2×C6)⋊5C6, C3⋊(C22×C6), (C3×C6)⋊2C22, SmallGroup(72,48)
Series: Derived ►Chief ►Lower central ►Upper central
| C3 — S3×C2×C6 |
Generators and relations for S3×C2×C6
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, C2×C6, C2×C6, C3×S3, C3×C6, C22×S3, C22×C6, S3×C6, C62, S3×C2×C6
Quotients: C1, C2, C3, C22, S3, C6, C23, D6, C2×C6, C3×S3, C22×S3, C22×C6, S3×C6, S3×C2×C6
►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);
S3×C2×C6 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 | C3×S3 | S3×C6 |
| kernel | S3×C2×C6 | S3×C6 | C62 | C22×S3 | D6 | C2×C6 | C2×C6 | C6 | C22 | C2 |
| # reps | 1 | 6 | 1 | 2 | 12 | 2 | 1 | 3 | 2 | 6 |
Matrix representation of S3×C2×C6 ►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] >;
S3×C2×C6 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