direct product, metabelian, supersoluble, monomial
Aliases: C3×C3⋊D12, C33⋊5D4, C32⋊9D12, C6.28S32, (S3×C6)⋊2C6, (S3×C6)⋊2S3, D6⋊2(C3×S3), C3⋊2(C3×D12), C6.4(S3×C6), Dic3⋊(C3×S3), (C3×C6).41D6, C32⋊5(C3×D4), (C3×Dic3)⋊4S3, (C3×Dic3)⋊1C6, C32⋊9(C3⋊D4), (C32×Dic3)⋊2C2, (C32×C6).4C22, (S3×C3×C6)⋊2C2, C2.4(C3×S32), (C6×C3⋊S3)⋊1C2, (C2×C3⋊S3)⋊4C6, C3⋊1(C3×C3⋊D4), (C3×C6).9(C2×C6), SmallGroup(216,122)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×C3⋊D12
G = < a,b,c,d | a3=b3=c12=d2=1, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=c-1 >
Subgroups: 316 in 94 conjugacy classes, 28 normal (all characteristic)
C1, C2, C2, C3, C3, C4, C22, S3, C6, C6, D4, C32, C32, Dic3, C12, D6, D6, C2×C6, C3×S3, C3⋊S3, C3×C6, C3×C6, D12, C3⋊D4, C3×D4, C33, C3×Dic3, C3×Dic3, C3×C12, S3×C6, S3×C6, C2×C3⋊S3, C62, S3×C32, C3×C3⋊S3, C32×C6, C3⋊D12, C3×D12, C3×C3⋊D4, C32×Dic3, S3×C3×C6, C6×C3⋊S3, C3×C3⋊D12
Quotients: C1, C2, C3, C22, S3, C6, D4, D6, C2×C6, C3×S3, D12, C3⋊D4, C3×D4, S32, S3×C6, C3⋊D12, C3×D12, C3×C3⋊D4, C3×S32, C3×C3⋊D12
►On 24 points - transitive group 24T543
(1 9 5)(2 10 6)(3 11 7)(4 12 8)(13 17 21)(14 18 22)(15 19 23)(16 20 24)
(1 9 5)(2 6 10)(3 11 7)(4 8 12)(13 17 21)(14 22 18)(15 19 23)(16 24 20)
(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 18)(2 17)(3 16)(4 15)(5 14)(6 13)(7 24)(8 23)(9 22)(10 21)(11 20)(12 19)
G:=sub<Sym(24)| (1,9,5)(2,10,6)(3,11,7)(4,12,8)(13,17,21)(14,18,22)(15,19,23)(16,20,24), (1,9,5)(2,6,10)(3,11,7)(4,8,12)(13,17,21)(14,22,18)(15,19,23)(16,24,20), (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24), (1,18)(2,17)(3,16)(4,15)(5,14)(6,13)(7,24)(8,23)(9,22)(10,21)(11,20)(12,19)>;
G:=Group( (1,9,5)(2,10,6)(3,11,7)(4,12,8)(13,17,21)(14,18,22)(15,19,23)(16,20,24), (1,9,5)(2,6,10)(3,11,7)(4,8,12)(13,17,21)(14,22,18)(15,19,23)(16,24,20), (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24), (1,18)(2,17)(3,16)(4,15)(5,14)(6,13)(7,24)(8,23)(9,22)(10,21)(11,20)(12,19) );
G=PermutationGroup([[(1,9,5),(2,10,6),(3,11,7),(4,12,8),(13,17,21),(14,18,22),(15,19,23),(16,20,24)], [(1,9,5),(2,6,10),(3,11,7),(4,8,12),(13,17,21),(14,22,18),(15,19,23),(16,24,20)], [(1,2,3,4,5,6,7,8,9,10,11,12),(13,14,15,16,17,18,19,20,21,22,23,24)], [(1,18),(2,17),(3,16),(4,15),(5,14),(6,13),(7,24),(8,23),(9,22),(10,21),(11,20),(12,19)]])
G:=TransitiveGroup(24,543);
C3×C3⋊D12 is a maximal subgroup of
D6⋊S32 (S3×C6)⋊D6 C3⋊S3⋊4D12 D6.S32 D6.4S32 D6.3S32 D6.6S32 C3×S3×D12 C3×S3×C3⋊D4
45 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | ··· | 3H | 3I | 3J | 3K | 4 | 6A | 6B | 6C | ··· | 6H | 6I | 6J | 6K | 6L | ··· | 6S | 6T | 6U | 12A | ··· | 12H |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | ··· | 3 | 3 | 3 | 3 | 4 | 6 | 6 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | ··· | 6 | 6 | 6 | 12 | ··· | 12 |
| size | 1 | 1 | 6 | 18 | 1 | 1 | 2 | ··· | 2 | 4 | 4 | 4 | 6 | 1 | 1 | 2 | ··· | 2 | 4 | 4 | 4 | 6 | ··· | 6 | 18 | 18 | 6 | ··· | 6 |
45 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | |||||||||||||
| image | C1 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | S3 | S3 | D4 | D6 | C3×S3 | C3×S3 | D12 | C3⋊D4 | C3×D4 | S3×C6 | C3×D12 | C3×C3⋊D4 | S32 | C3⋊D12 | C3×S32 | C3×C3⋊D12 |
| kernel | C3×C3⋊D12 | C32×Dic3 | S3×C3×C6 | C6×C3⋊S3 | C3⋊D12 | C3×Dic3 | S3×C6 | C2×C3⋊S3 | C3×Dic3 | S3×C6 | C33 | C3×C6 | Dic3 | D6 | C32 | C32 | C32 | C6 | C3 | C3 | C6 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 1 | 1 | 2 | 2 |
Matrix representation of C3×C3⋊D12 ►in GL4(𝔽7) generated by
| 4 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 4 |
| 2 | 6 | 5 | 6 |
| 4 | 3 | 1 | 3 |
| 1 | 1 | 2 | 5 |
| 1 | 6 | 3 | 5 |
| 0 | 5 | 1 | 5 |
| 4 | 1 | 2 | 6 |
| 5 | 2 | 1 | 0 |
| 5 | 0 | 0 | 5 |
| 6 | 1 | 2 | 4 |
| 1 | 4 | 6 | 5 |
| 4 | 0 | 4 | 3 |
| 3 | 1 | 4 | 0 |
G:=sub<GL(4,GF(7))| [4,0,0,0,0,4,0,0,0,0,4,0,0,0,0,4],[2,4,1,1,6,3,1,6,5,1,2,3,6,3,5,5],[0,4,5,5,5,1,2,0,1,2,1,0,5,6,0,5],[6,1,4,3,1,4,0,1,2,6,4,4,4,5,3,0] >;
C3×C3⋊D12 in GAP, Magma, Sage, TeX
C_3\times C_3\rtimes D_{12} % in TeX
G:=Group("C3xC3:D12"); // GroupNames label
G:=SmallGroup(216,122);
// by ID
G=gap.SmallGroup(216,122);
# by ID
G:=PCGroup([6,-2,-2,-3,-2,-3,-3,169,79,730,5189]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^3=c^12=d^2=1,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^-1>;
// generators/relations