metabelian, supersoluble, monomial
Aliases: D6⋊2D9, C9⋊2D12, Dic9⋊S3, C18.7D6, C6.7D18, C6.7S32, (C3×C9)⋊3D4, C2.7(S3×D9), (S3×C18)⋊4C2, (S3×C6).3S3, C3⋊1(C9⋊D4), (C3×C6).28D6, (C3×Dic9)⋊3C2, (C3×C18).7C22, C3.2(C3⋊D12), C32.3(C3⋊D4), (C2×C9⋊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, C2×C6, D9, C18, C18, C3×S3, C3⋊S3, C3×C6, D12, C3⋊D4, C3×C9, Dic9, D18, C2×C18, C3×Dic3, S3×C6, C2×C3⋊S3, S3×C9, C9⋊S3, C3×C18, C9⋊D4, C3⋊D12, C3×Dic9, S3×C18, C2×C9⋊S3, C9⋊D12
Quotients: C1, C2, C22, S3, D4, D6, D9, D12, C3⋊D4, D18, S32, C9⋊D4, C3⋊D12, S3×D9, 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 D9×D12 Dic3.D18 S3×C9⋊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 | S3×D9 | C9⋊D12 |
| kernel | C9⋊D12 | C3×Dic9 | S3×C18 | C2×C9⋊S3 | Dic9 | S3×C6 | C3×C9 | C18 | C3×C6 | 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