direct product, metacyclic, supersoluble, monomial, A-group
Aliases: S3×C12, D6.C6, C12⋊2C6, Dic3○C12, C6.18D6, Dic3⋊2C6, (C3×C12)⋊3C2, C3⋊1(C2×C12), C2.1(S3×C6), C6.2(C2×C6), C32⋊4(C2×C4), C4○(C3×Dic3), (S3×C6).2C2, C12○(C3×Dic3), (C3×Dic3)⋊5C2, (C3×C6).7C22, SmallGroup(72,27)
Series: Derived ►Chief ►Lower central ►Upper central
| C3 — S3×C12 |
Generators and relations for S3×C12
G = < a,b,c | a12=b3=c2=1, ab=ba, ac=ca, cbc=b-1 >
Permutation representations of S3×C12
►On 24 points - transitive group 24T65
Generators in S24
(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 5 9)(2 6 10)(3 7 11)(4 8 12)(13 21 17)(14 22 18)(15 23 19)(16 24 20)
(1 19)(2 20)(3 21)(4 22)(5 23)(6 24)(7 13)(8 14)(9 15)(10 16)(11 17)(12 18)
G:=sub<Sym(24)| (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24), (1,5,9)(2,6,10)(3,7,11)(4,8,12)(13,21,17)(14,22,18)(15,23,19)(16,24,20), (1,19)(2,20)(3,21)(4,22)(5,23)(6,24)(7,13)(8,14)(9,15)(10,16)(11,17)(12,18)>;
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), (1,5,9)(2,6,10)(3,7,11)(4,8,12)(13,21,17)(14,22,18)(15,23,19)(16,24,20), (1,19)(2,20)(3,21)(4,22)(5,23)(6,24)(7,13)(8,14)(9,15)(10,16)(11,17)(12,18) );
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)], [(1,5,9),(2,6,10),(3,7,11),(4,8,12),(13,21,17),(14,22,18),(15,23,19),(16,24,20)], [(1,19),(2,20),(3,21),(4,22),(5,23),(6,24),(7,13),(8,14),(9,15),(10,16),(11,17),(12,18)]])
G:=TransitiveGroup(24,65);
S3×C12 is a maximal subgroup of
D6.Dic3 D12⋊5S3 D6.D6 D6.6D6
36 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 3D | 3E | 4A | 4B | 4C | 4D | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 6H | 6I | 12A | 12B | 12C | 12D | 12E | ··· | 12J | 12K | 12L | 12M | 12N |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 12 | 12 | 12 | 12 |
| size | 1 | 1 | 3 | 3 | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 3 | 3 | 1 | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 3 | 3 | 3 | 3 |
36 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | ||||||||||
| image | C1 | C2 | C2 | C2 | C3 | C4 | C6 | C6 | C6 | C12 | S3 | D6 | C3×S3 | C4×S3 | S3×C6 | S3×C12 |
| kernel | S3×C12 | C3×Dic3 | C3×C12 | S3×C6 | C4×S3 | C3×S3 | Dic3 | C12 | D6 | S3 | C12 | C6 | C4 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 4 | 2 | 2 | 2 | 8 | 1 | 1 | 2 | 2 | 2 | 4 |
Matrix representation of S3×C12 ►in GL2(𝔽13) generated by
| 11 | 0 |
| 0 | 11 |
| 3 | 0 |
| 0 | 9 |
| 0 | 1 |
| 1 | 0 |
G:=sub<GL(2,GF(13))| [11,0,0,11],[3,0,0,9],[0,1,1,0] >;
S3×C12 in GAP, Magma, Sage, TeX
S_3\times C_{12} % in TeX
G:=Group("S3xC12"); // GroupNames label
G:=SmallGroup(72,27);
// by ID
G=gap.SmallGroup(72,27);
# by ID
G:=PCGroup([5,-2,-2,-3,-2,-3,66,1204]);
// Polycyclic
G:=Group<a,b,c|a^12=b^3=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
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