direct product, metabelian, supersoluble, monomial, A-group
Aliases: C3×C6.D6, C6.26S32, C3⋊S3⋊2C12, C3⋊1(S3×C12), C6.2(S3×C6), C33⋊5(C2×C4), C32⋊9(C4×S3), (C3×C6).39D6, C32⋊5(C2×C12), (C3×Dic3)⋊5S3, Dic3⋊2(C3×S3), (C3×Dic3)⋊3C6, (C32×Dic3)⋊4C2, (C32×C6).2C22, C2.2(C3×S32), (C3×C3⋊S3)⋊1C4, (C2×C3⋊S3).2C6, (C6×C3⋊S3).1C2, (C3×C6).7(C2×C6), (C3×Dic3)○(C3×Dic3), SmallGroup(216,120)
Series: Derived ►Chief ►Lower central ►Upper central
| C32 — C3×C6.D6 |
Generators and relations for C3×C6.D6
G = < a,b,c,d | a3=b6=d2=1, c6=b3, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=c5 >
Subgroups: 268 in 90 conjugacy classes, 32 normal (12 characteristic)
C1, C2, C2, C3, C3, C3, C4, C22, S3, C6, C6, C6, C2×C4, C32, C32, C32, Dic3, C12, D6, C2×C6, C3×S3, C3⋊S3, C3×C6, C3×C6, C3×C6, C4×S3, C2×C12, C33, C3×Dic3, C3×Dic3, C3×C12, S3×C6, C2×C3⋊S3, C3×C3⋊S3, C32×C6, C6.D6, S3×C12, C32×Dic3, C6×C3⋊S3, C3×C6.D6
Quotients: C1, C2, C3, C4, C22, S3, C6, C2×C4, C12, D6, C2×C6, C3×S3, C4×S3, C2×C12, S32, S3×C6, C6.D6, S3×C12, C3×S32, C3×C6.D6
►On 24 points - transitive group 24T544
(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 11 9 7 5 3)(2 4 6 8 10 12)(13 23 21 19 17 15)(14 16 18 20 22 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 23)(2 16)(3 21)(4 14)(5 19)(6 24)(7 17)(8 22)(9 15)(10 20)(11 13)(12 18)
G:=sub<Sym(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), (1,11,9,7,5,3)(2,4,6,8,10,12)(13,23,21,19,17,15)(14,16,18,20,22,24), (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24), (1,23)(2,16)(3,21)(4,14)(5,19)(6,24)(7,17)(8,22)(9,15)(10,20)(11,13)(12,18)>;
G:=Group( (1,5,9)(2,6,10)(3,7,11)(4,8,12)(13,21,17)(14,22,18)(15,23,19)(16,24,20), (1,11,9,7,5,3)(2,4,6,8,10,12)(13,23,21,19,17,15)(14,16,18,20,22,24), (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24), (1,23)(2,16)(3,21)(4,14)(5,19)(6,24)(7,17)(8,22)(9,15)(10,20)(11,13)(12,18) );
G=PermutationGroup([[(1,5,9),(2,6,10),(3,7,11),(4,8,12),(13,21,17),(14,22,18),(15,23,19),(16,24,20)], [(1,11,9,7,5,3),(2,4,6,8,10,12),(13,23,21,19,17,15),(14,16,18,20,22,24)], [(1,2,3,4,5,6,7,8,9,10,11,12),(13,14,15,16,17,18,19,20,21,22,23,24)], [(1,23),(2,16),(3,21),(4,14),(5,19),(6,24),(7,17),(8,22),(9,15),(10,20),(11,13),(12,18)]])
G:=TransitiveGroup(24,544);
C3×C6.D6 is a maximal subgroup of
C3⋊S3.2D12 C33⋊C4⋊C4 Dic3⋊6S32 C3⋊S3⋊4D12 C33⋊5(C2×Q8) D6.S32 Dic3.S32 S32×C12
C3×C6.D6 is a maximal quotient of
C3×Dic32
54 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | ··· | 3H | 3I | 3J | 3K | 4A | 4B | 4C | 4D | 6A | 6B | 6C | ··· | 6H | 6I | 6J | 6K | 6L | 6M | 6N | 6O | 12A | ··· | 12H | 12I | ··· | 12T |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | ··· | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 12 | ··· | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 9 | 9 | 1 | 1 | 2 | ··· | 2 | 4 | 4 | 4 | 3 | 3 | 3 | 3 | 1 | 1 | 2 | ··· | 2 | 4 | 4 | 4 | 9 | 9 | 9 | 9 | 3 | ··· | 3 | 6 | ··· | 6 |
54 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | |||||||||||
| image | C1 | C2 | C2 | C3 | C4 | C6 | C6 | C12 | S3 | D6 | C3×S3 | C4×S3 | S3×C6 | S3×C12 | S32 | C6.D6 | C3×S32 | C3×C6.D6 |
| kernel | C3×C6.D6 | C32×Dic3 | C6×C3⋊S3 | C6.D6 | C3×C3⋊S3 | C3×Dic3 | C2×C3⋊S3 | C3⋊S3 | C3×Dic3 | C3×C6 | Dic3 | C32 | C6 | C3 | C6 | C3 | C2 | C1 |
| # reps | 1 | 2 | 1 | 2 | 4 | 4 | 2 | 8 | 2 | 2 | 4 | 4 | 4 | 8 | 1 | 1 | 2 | 2 |
Matrix representation of C3×C6.D6 ►in GL4(𝔽7) generated by
| 4 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 4 |
| 1 | 0 | 4 | 6 |
| 2 | 4 | 1 | 0 |
| 6 | 1 | 3 | 2 |
| 4 | 4 | 5 | 1 |
| 3 | 0 | 2 | 2 |
| 3 | 3 | 1 | 6 |
| 2 | 3 | 1 | 1 |
| 3 | 1 | 4 | 0 |
| 2 | 4 | 0 | 2 |
| 4 | 2 | 2 | 3 |
| 1 | 6 | 4 | 5 |
| 1 | 6 | 3 | 6 |
G:=sub<GL(4,GF(7))| [4,0,0,0,0,4,0,0,0,0,4,0,0,0,0,4],[1,2,6,4,0,4,1,4,4,1,3,5,6,0,2,1],[3,3,2,3,0,3,3,1,2,1,1,4,2,6,1,0],[2,4,1,1,4,2,6,6,0,2,4,3,2,3,5,6] >;
C3×C6.D6 in GAP, Magma, Sage, TeX
C_3\times C_6.D_6
% in TeX
G:=Group("C3xC6.D6"); // GroupNames label
G:=SmallGroup(216,120);
// by ID
G=gap.SmallGroup(216,120);
# by ID
G:=PCGroup([6,-2,-2,-3,-2,-3,-3,72,79,730,5189]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^6=d^2=1,c^6=b^3,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d=b^-1,d*c*d=c^5>;
// generators/relations