metabelian, supersoluble, monomial
Aliases: D6:2D9, C9:2D12, Dic9:S3, C18.7D6, C6.7D18, C6.7S32, (C3xC9):3D4, C2.7(S3xD9), (S3xC18):4C2, (S3xC6).3S3, C3:1(C9:D4), (C3xC6).28D6, (C3xDic9):3C2, (C3xC18).7C22, C3.2(C3:D12), C32.3(C3:D4), (C2xC9:S3):2C2, SmallGroup(216,32)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C9:D12
G = < a,b,c | a9=b12=c2=1, bab-1=cac=a-1, cbc=b-1 >
Subgroups: 362 in 58 conjugacy classes, 19 normal (all characteristic)
C1, C2, C2, C3, C3, C4, C22, S3, C6, C6, D4, C9, C9, C32, Dic3, C12, D6, D6, C2xC6, D9, C18, C18, C3xS3, C3:S3, C3xC6, D12, C3:D4, C3xC9, Dic9, D18, C2xC18, C3xDic3, S3xC6, C2xC3:S3, S3xC9, C9:S3, C3xC18, C9:D4, C3:D12, C3xDic9, S3xC18, C2xC9:S3, C9:D12
Quotients: C1, C2, C22, S3, D4, D6, D9, D12, C3:D4, D18, S32, C9:D4, C3:D12, S3xD9, C9:D12
►On 36 points
(1 26 24 9 34 20 5 30 16)(2 17 31 6 21 35 10 13 27)(3 28 14 11 36 22 7 32 18)(4 19 33 8 23 25 12 15 29)
(1 2 3 4 5 6 7 8 9 10 11 12)(13 14 15 16 17 18 19 20 21 22 23 24)(25 26 27 28 29 30 31 32 33 34 35 36)
(1 3)(4 12)(5 11)(6 10)(7 9)(13 31)(14 30)(15 29)(16 28)(17 27)(18 26)(19 25)(20 36)(21 35)(22 34)(23 33)(24 32)
G:=sub<Sym(36)| (1,26,24,9,34,20,5,30,16)(2,17,31,6,21,35,10,13,27)(3,28,14,11,36,22,7,32,18)(4,19,33,8,23,25,12,15,29), (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32,33,34,35,36), (1,3)(4,12)(5,11)(6,10)(7,9)(13,31)(14,30)(15,29)(16,28)(17,27)(18,26)(19,25)(20,36)(21,35)(22,34)(23,33)(24,32)>;
G:=Group( (1,26,24,9,34,20,5,30,16)(2,17,31,6,21,35,10,13,27)(3,28,14,11,36,22,7,32,18)(4,19,33,8,23,25,12,15,29), (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32,33,34,35,36), (1,3)(4,12)(5,11)(6,10)(7,9)(13,31)(14,30)(15,29)(16,28)(17,27)(18,26)(19,25)(20,36)(21,35)(22,34)(23,33)(24,32) );
G=PermutationGroup([[(1,26,24,9,34,20,5,30,16),(2,17,31,6,21,35,10,13,27),(3,28,14,11,36,22,7,32,18),(4,19,33,8,23,25,12,15,29)], [(1,2,3,4,5,6,7,8,9,10,11,12),(13,14,15,16,17,18,19,20,21,22,23,24),(25,26,27,28,29,30,31,32,33,34,35,36)], [(1,3),(4,12),(5,11),(6,10),(7,9),(13,31),(14,30),(15,29),(16,28),(17,27),(18,26),(19,25),(20,36),(21,35),(22,34),(23,33),(24,32)]])
C9:D12 is a maximal subgroup of
D12:D9 D6.D18 Dic9.D6 D9xD12 Dic3.D18 S3xC9:D4 D18:D6
C9:D12 is a maximal quotient of C9:D24 C36.D6 C18.D12 C9:Dic12 Dic9:Dic3 C6.18D36 D6:Dic9
33 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 4 | 6A | 6B | 6C | 6D | 6E | 9A | 9B | 9C | 9D | 9E | 9F | 12A | 12B | 18A | 18B | 18C | 18D | 18E | 18F | 18G | ··· | 18L |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 4 | 6 | 6 | 6 | 6 | 6 | 9 | 9 | 9 | 9 | 9 | 9 | 12 | 12 | 18 | 18 | 18 | 18 | 18 | 18 | 18 | ··· | 18 |
| size | 1 | 1 | 6 | 54 | 2 | 2 | 4 | 18 | 2 | 2 | 4 | 6 | 6 | 2 | 2 | 2 | 4 | 4 | 4 | 18 | 18 | 2 | 2 | 2 | 4 | 4 | 4 | 6 | ··· | 6 |
33 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | S3 | S3 | D4 | D6 | D6 | D9 | D12 | C3:D4 | D18 | C9:D4 | S32 | C3:D12 | S3xD9 | C9:D12 |
| kernel | C9:D12 | C3xDic9 | S3xC18 | C2xC9:S3 | Dic9 | S3xC6 | C3xC9 | C18 | C3xC6 | D6 | C9 | C32 | C6 | C3 | C6 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 2 | 2 | 3 | 6 | 1 | 1 | 3 | 3 |
Matrix representation of C9:D12 ►in GL4(F37) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 17 | 26 |
| 0 | 0 | 11 | 6 |
| 5 | 5 | 0 | 0 |
| 32 | 10 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
| 1 | 0 | 0 | 0 |
| 1 | 36 | 0 | 0 |
| 0 | 0 | 0 | 36 |
| 0 | 0 | 36 | 0 |
G:=sub<GL(4,GF(37))| [1,0,0,0,0,1,0,0,0,0,17,11,0,0,26,6],[5,32,0,0,5,10,0,0,0,0,0,1,0,0,1,0],[1,1,0,0,0,36,0,0,0,0,0,36,0,0,36,0] >;
C9:D12 in GAP, Magma, Sage, TeX
C_9\rtimes D_{12} % in TeX
G:=Group("C9:D12"); // GroupNames label
G:=SmallGroup(216,32);
// by ID
G=gap.SmallGroup(216,32);
# by ID
G:=PCGroup([6,-2,-2,-2,-3,-3,-3,73,31,1065,453,1444,2603]);
// Polycyclic
G:=Group<a,b,c|a^9=b^12=c^2=1,b*a*b^-1=c*a*c=a^-1,c*b*c=b^-1>;
// generators/relations