metabelian, supersoluble, monomial
Aliases: C92⋊8C6, D9⋊23- 1+2, C9⋊C9⋊3C6, C9⋊C18⋊3C3, C9⋊4(C9⋊C6), (C9×D9)⋊4C3, C92⋊9C3⋊1C2, C33.13(C3×S3), (C3×D9).6C32, C9⋊2(C2×3- 1+2), C32.43(S3×C32), C3.6(S3×3- 1+2), (C3×3- 1+2).2C6, (C3×3- 1+2).12S3, C3.8(C3×C9⋊C6), (C3×C9⋊C6).2C3, (C3×C9).12(C3×S3), (C3×C9).15(C3×C6), SmallGroup(486,110)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C92⋊8C6
G = < a,b,c | a9=b9=c6=1, ab=ba, cac-1=a7, cbc-1=b2 >
Subgroups: 256 in 71 conjugacy classes, 24 normal (18 characteristic)
C1, C2, C3, C3, S3, C6, C9, C9, C32, C32, D9, C18, C3×S3, C3×C6, C3×C9, C3×C9, C3×C9, 3- 1+2, C33, C3×D9, S3×C9, C9⋊C6, C2×3- 1+2, S3×C32, C92, C9⋊C9, C9⋊C9, C3×3- 1+2, C3×3- 1+2, C9×D9, C9⋊C18, C3×C9⋊C6, S3×3- 1+2, C92⋊9C3, C92⋊8C6
Quotients: C1, C2, C3, S3, C6, C32, C3×S3, C3×C6, 3- 1+2, C9⋊C6, C2×3- 1+2, S3×C32, C3×C9⋊C6, S3×3- 1+2, C92⋊8C6
►On 18 points - transitive group 18T158
(1 2 3 4 5 6 7 8 9)(10 11 12 13 14 15 16 17 18)
(1 9 8 7 6 5 4 3 2)(10 11 12 13 14 15 16 17 18)
(1 16)(2 11 8 17 5 14)(3 15 6 18 9 12)(4 10)(7 13)
G:=sub<Sym(18)| (1,2,3,4,5,6,7,8,9)(10,11,12,13,14,15,16,17,18), (1,9,8,7,6,5,4,3,2)(10,11,12,13,14,15,16,17,18), (1,16)(2,11,8,17,5,14)(3,15,6,18,9,12)(4,10)(7,13)>;
G:=Group( (1,2,3,4,5,6,7,8,9)(10,11,12,13,14,15,16,17,18), (1,9,8,7,6,5,4,3,2)(10,11,12,13,14,15,16,17,18), (1,16)(2,11,8,17,5,14)(3,15,6,18,9,12)(4,10)(7,13) );
G=PermutationGroup([[(1,2,3,4,5,6,7,8,9),(10,11,12,13,14,15,16,17,18)], [(1,9,8,7,6,5,4,3,2),(10,11,12,13,14,15,16,17,18)], [(1,16),(2,11,8,17,5,14),(3,15,6,18,9,12),(4,10),(7,13)]])
G:=TransitiveGroup(18,158);
42 conjugacy classes
| class | 1 | 2 | 3A | 3B | 3C | 3D | 3E | 3F | 3G | 6A | 6B | 6C | 6D | 9A | 9B | 9C | ··· | 9M | 9N | 9O | 9P | 9Q | 9R | ··· | 9W | 18A | ··· | 18F |
| order | 1 | 2 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 6 | 6 | 6 | 6 | 9 | 9 | 9 | ··· | 9 | 9 | 9 | 9 | 9 | 9 | ··· | 9 | 18 | ··· | 18 |
| size | 1 | 9 | 1 | 1 | 2 | 2 | 2 | 9 | 9 | 9 | 9 | 27 | 27 | 3 | 3 | 6 | ··· | 6 | 9 | 9 | 9 | 9 | 18 | ··· | 18 | 27 | ··· | 27 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 3 | 3 | 6 | 6 | 6 | 6 |
| type | + | + | + | + | |||||||||||||
| image | C1 | C2 | C3 | C3 | C3 | C6 | C6 | C6 | S3 | C3×S3 | C3×S3 | 3- 1+2 | C2×3- 1+2 | C9⋊C6 | C3×C9⋊C6 | S3×3- 1+2 | C92⋊8C6 |
| kernel | C92⋊8C6 | C92⋊9C3 | C9×D9 | C9⋊C18 | C3×C9⋊C6 | C92 | C9⋊C9 | C3×3- 1+2 | C3×3- 1+2 | C3×C9 | C33 | D9 | C9 | C9 | C3 | C3 | C1 |
| # reps | 1 | 1 | 2 | 4 | 2 | 2 | 4 | 2 | 1 | 6 | 2 | 2 | 2 | 1 | 2 | 2 | 6 |
Matrix representation of C92⋊8C6 ►in GL6(𝔽19)
| 1 | 0 | 6 | 0 | 0 | 0 |
| 8 | 0 | 18 | 0 | 0 | 0 |
| 8 | 11 | 18 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 1 |
| 11 | 0 | 1 | 11 | 0 | 0 |
| 11 | 0 | 1 | 0 | 11 | 0 |
| 7 | 4 | 0 | 0 | 0 | 0 |
| 0 | 12 | 7 | 0 | 0 | 0 |
| 18 | 12 | 0 | 0 | 0 | 0 |
| 1 | 7 | 0 | 0 | 0 | 1 |
| 12 | 7 | 0 | 11 | 0 | 0 |
| 12 | 7 | 0 | 0 | 11 | 0 |
| 18 | 0 | 0 | 6 | 0 | 0 |
| 18 | 0 | 0 | 18 | 7 | 0 |
| 11 | 0 | 0 | 18 | 0 | 11 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 1 | 7 | 0 | 1 | 0 | 0 |
| 8 | 0 | 11 | 1 | 0 | 0 |
G:=sub<GL(6,GF(19))| [1,8,8,0,11,11,0,0,11,0,0,0,6,18,18,1,1,1,0,0,0,0,11,0,0,0,0,0,0,11,0,0,0,1,0,0],[7,0,18,1,12,12,4,12,12,7,7,7,0,7,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,11,0,0,0,1,0,0],[18,18,11,0,1,8,0,0,0,0,7,0,0,0,0,0,0,11,6,18,18,1,1,1,0,7,0,0,0,0,0,0,11,0,0,0] >;
C92⋊8C6 in GAP, Magma, Sage, TeX
C_9^2\rtimes_8C_6
% in TeX
G:=Group("C9^2:8C6"); // GroupNames label
G:=SmallGroup(486,110);
// by ID
G=gap.SmallGroup(486,110);
# by ID
G:=PCGroup([6,-2,-3,-3,-3,-3,-3,224,122,8104,3250,208,11669]);
// Polycyclic
G:=Group<a,b,c|a^9=b^9=c^6=1,a*b=b*a,c*a*c^-1=a^7,c*b*c^-1=b^2>;
// generators/relations