direct product, metacyclic, supersoluble, monomial, A-group
Aliases: C6×Dic3, C6⋊C12, C6.20D6, C62.1C2, (C3×C6)⋊2C4, C3⋊2(C2×C12), C2.2(S3×C6), C6.4(C2×C6), (C2×C6).6S3, (C2×C6).3C6, C32⋊6(C2×C4), C22.(C3×S3), (C3×C6).9C22, SmallGroup(72,29)
Series: Derived ►Chief ►Lower central ►Upper central
| C3 — C6×Dic3 |
Generators and relations for C6×Dic3
G = < a,b,c | a6=b6=1, c2=b3, ab=ba, ac=ca, cbc-1=b-1 >
Permutation representations of C6×Dic3
►On 24 points - transitive group 24T66
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 14 3 16 5 18)(2 15 4 17 6 13)(7 19 11 23 9 21)(8 20 12 24 10 22)
(1 11 16 21)(2 12 17 22)(3 7 18 23)(4 8 13 24)(5 9 14 19)(6 10 15 20)
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), (1,14,3,16,5,18)(2,15,4,17,6,13)(7,19,11,23,9,21)(8,20,12,24,10,22), (1,11,16,21)(2,12,17,22)(3,7,18,23)(4,8,13,24)(5,9,14,19)(6,10,15,20)>;
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), (1,14,3,16,5,18)(2,15,4,17,6,13)(7,19,11,23,9,21)(8,20,12,24,10,22), (1,11,16,21)(2,12,17,22)(3,7,18,23)(4,8,13,24)(5,9,14,19)(6,10,15,20) );
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)], [(1,14,3,16,5,18),(2,15,4,17,6,13),(7,19,11,23,9,21),(8,20,12,24,10,22)], [(1,11,16,21),(2,12,17,22),(3,7,18,23),(4,8,13,24),(5,9,14,19),(6,10,15,20)]])
G:=TransitiveGroup(24,66);
C6×Dic3 is a maximal subgroup of
D6⋊Dic3 C6.D12 Dic3⋊Dic3 C62.C22 D6.3D6 S3×C2×C12
36 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 3D | 3E | 4A | 4B | 4C | 4D | 6A | ··· | 6F | 6G | ··· | 6O | 12A | ··· | 12H |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 1 | ··· | 1 | 2 | ··· | 2 | 3 | ··· | 3 |
36 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | + | ||||||||
| image | C1 | C2 | C2 | C3 | C4 | C6 | C6 | C12 | S3 | Dic3 | D6 | C3×S3 | C3×Dic3 | S3×C6 |
| kernel | C6×Dic3 | C3×Dic3 | C62 | C2×Dic3 | C3×C6 | Dic3 | C2×C6 | C6 | C2×C6 | C6 | C6 | C22 | C2 | C2 |
| # reps | 1 | 2 | 1 | 2 | 4 | 4 | 2 | 8 | 1 | 2 | 1 | 2 | 4 | 2 |
Matrix representation of C6×Dic3 ►in GL4(𝔽13) generated by
| 4 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 3 | 0 |
| 0 | 0 | 0 | 3 |
| 12 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 3 | 0 |
| 0 | 0 | 0 | 9 |
| 8 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
G:=sub<GL(4,GF(13))| [4,0,0,0,0,4,0,0,0,0,3,0,0,0,0,3],[12,0,0,0,0,1,0,0,0,0,3,0,0,0,0,9],[8,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0] >;
C6×Dic3 in GAP, Magma, Sage, TeX
C_6\times {\rm Dic}_3 % in TeX
G:=Group("C6xDic3"); // GroupNames label
G:=SmallGroup(72,29);
// by ID
G=gap.SmallGroup(72,29);
# by ID
G:=PCGroup([5,-2,-2,-3,-2,-3,60,1204]);
// Polycyclic
G:=Group<a,b,c|a^6=b^6=1,c^2=b^3,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
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