metabelian, supersoluble, monomial
Aliases: C92⋊7C6, D9⋊13- 1+2, C9⋊C9⋊2C6, C9⋊C18⋊2C3, C9⋊3(C9⋊C6), (C9×D9)⋊3C3, C92⋊7C3⋊1C2, C33.12(C3×S3), (C3×D9).5C32, C9⋊1(C2×3- 1+2), C32.42(S3×C32), C3.5(S3×3- 1+2), (C3×3- 1+2).1C6, (C3×3- 1+2).11S3, C3.7(C3×C9⋊C6), (C3×C9⋊C6).1C3, (C3×C9).11(C3×S3), (C3×C9).14(C3×C6), SmallGroup(486,109)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C92⋊7C6
G = < a,b,c | a9=b9=c6=1, ab=ba, cac-1=a4, cbc-1=b2 >
Subgroups: 238 in 66 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, C32⋊C9, C9⋊C9, C9⋊C9, C3×3- 1+2, C9×D9, C9⋊C18, C3×C9⋊C6, S3×3- 1+2, C92⋊7C3, C92⋊7C6
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⋊7C6
►On 54 points
(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)(37 38 39 40 41 42 43 44 45)(46 47 48 49 50 51 52 53 54)
(1 44 31 7 41 28 4 38 34)(2 45 32 8 42 29 5 39 35)(3 37 33 9 43 30 6 40 36)(10 23 52 13 26 46 16 20 49)(11 24 53 14 27 47 17 21 50)(12 25 54 15 19 48 18 22 51)
(1 24 33 49 39 18)(2 22 28 50 37 13)(3 20 32 51 44 17)(4 27 36 52 42 12)(5 25 31 53 40 16)(6 23 35 54 38 11)(7 21 30 46 45 15)(8 19 34 47 43 10)(9 26 29 48 41 14)
G:=sub<Sym(54)| (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)(37,38,39,40,41,42,43,44,45)(46,47,48,49,50,51,52,53,54), (1,44,31,7,41,28,4,38,34)(2,45,32,8,42,29,5,39,35)(3,37,33,9,43,30,6,40,36)(10,23,52,13,26,46,16,20,49)(11,24,53,14,27,47,17,21,50)(12,25,54,15,19,48,18,22,51), (1,24,33,49,39,18)(2,22,28,50,37,13)(3,20,32,51,44,17)(4,27,36,52,42,12)(5,25,31,53,40,16)(6,23,35,54,38,11)(7,21,30,46,45,15)(8,19,34,47,43,10)(9,26,29,48,41,14)>;
G:=Group( (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)(37,38,39,40,41,42,43,44,45)(46,47,48,49,50,51,52,53,54), (1,44,31,7,41,28,4,38,34)(2,45,32,8,42,29,5,39,35)(3,37,33,9,43,30,6,40,36)(10,23,52,13,26,46,16,20,49)(11,24,53,14,27,47,17,21,50)(12,25,54,15,19,48,18,22,51), (1,24,33,49,39,18)(2,22,28,50,37,13)(3,20,32,51,44,17)(4,27,36,52,42,12)(5,25,31,53,40,16)(6,23,35,54,38,11)(7,21,30,46,45,15)(8,19,34,47,43,10)(9,26,29,48,41,14) );
G=PermutationGroup([[(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),(37,38,39,40,41,42,43,44,45),(46,47,48,49,50,51,52,53,54)], [(1,44,31,7,41,28,4,38,34),(2,45,32,8,42,29,5,39,35),(3,37,33,9,43,30,6,40,36),(10,23,52,13,26,46,16,20,49),(11,24,53,14,27,47,17,21,50),(12,25,54,15,19,48,18,22,51)], [(1,24,33,49,39,18),(2,22,28,50,37,13),(3,20,32,51,44,17),(4,27,36,52,42,12),(5,25,31,53,40,16),(6,23,35,54,38,11),(7,21,30,46,45,15),(8,19,34,47,43,10),(9,26,29,48,41,14)]])
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⋊7C6 |
| kernel | C92⋊7C6 | C92⋊7C3 | 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⋊7C6 ►in GL6(𝔽19)
| 0 | 5 | 0 | 0 | 0 | 0 |
| 0 | 0 | 5 | 0 | 0 | 0 |
| 17 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 5 | 0 |
| 0 | 0 | 0 | 0 | 0 | 5 |
| 0 | 0 | 0 | 17 | 0 | 0 |
| 0 | 7 | 0 | 0 | 0 | 0 |
| 0 | 0 | 7 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 11 | 0 | 0 |
| 0 | 0 | 0 | 0 | 11 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 |
| 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 17 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 17 | 0 | 0 | 0 | 0 |
G:=sub<GL(6,GF(19))| [0,0,17,0,0,0,5,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,0,17,0,0,0,5,0,0,0,0,0,0,5,0],[0,0,1,0,0,0,7,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,11,0,0,0,1,0,0],[0,0,0,0,16,0,0,0,0,0,0,17,0,0,0,16,0,0,0,16,0,0,0,0,0,0,17,0,0,0,16,0,0,0,0,0] >;
C92⋊7C6 in GAP, Magma, Sage, TeX
C_9^2\rtimes_7C_6
% in TeX
G:=Group("C9^2:7C6"); // GroupNames label
G:=SmallGroup(486,109);
// by ID
G=gap.SmallGroup(486,109);
# by ID
G:=PCGroup([6,-2,-3,-3,-3,-3,-3,224,68,8104,3250,208,11669]);
// Polycyclic
G:=Group<a,b,c|a^9=b^9=c^6=1,a*b=b*a,c*a*c^-1=a^4,c*b*c^-1=b^2>;
// generators/relations