metabelian, supersoluble, monomial
Aliases: C92⋊9C6, 3- 1+2⋊2D9, C9⋊(C3×D9), C9⋊D9⋊6C3, C9⋊5(C9⋊C6), C32⋊C9.22S3, C32.5(C9⋊S3), (C32×C9).14S3, C33.28(C3⋊S3), (C9×3- 1+2)⋊2C2, C3.3(He3.4S3), C3.2(C33.S3), (C3×3- 1+2).13S3, C3.5(C3×C9⋊S3), (C3×C9).59(C3×S3), C32.34(C3×C3⋊S3), SmallGroup(486,144)
Series: Derived ►Chief ►Lower central ►Upper central
| C92 — C92⋊9C6 |
Generators and relations for C92⋊9C6
G = < a,b,c | a9=b9=c6=1, ab=ba, cac-1=a-1, cbc-1=b2 >
Subgroups: 692 in 91 conjugacy classes, 29 normal (14 characteristic)
C1, C2, C3, C3, C3, S3, C6, C9, C9, C32, C32, D9, C3×S3, C3⋊S3, C3×C9, C3×C9, C3×C9, 3- 1+2, 3- 1+2, C33, C3×D9, C9⋊C6, C9⋊S3, C3×C3⋊S3, C92, C92, C32⋊C9, C9⋊C9, C32×C9, C3×3- 1+2, C32⋊D9, C9⋊D9, C3×C9⋊S3, C33.S3, C9×3- 1+2, C92⋊9C6
Quotients: C1, C2, C3, S3, C6, D9, C3×S3, C3⋊S3, C3×D9, C9⋊C6, C9⋊S3, C3×C3⋊S3, C3×C9⋊S3, C33.S3, He3.4S3, C92⋊9C6
►On 81 points
Generators in S81
(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)(55 56 57 58 59 60 61 62 63)(64 65 66 67 68 69 70 71 72)(73 74 75 76 77 78 79 80 81)
(1 42 36 20 67 11 74 57 50)(2 43 28 21 68 12 75 58 51)(3 44 29 22 69 13 76 59 52)(4 45 30 23 70 14 77 60 53)(5 37 31 24 71 15 78 61 54)(6 38 32 25 72 16 79 62 46)(7 39 33 26 64 17 80 63 47)(8 40 34 27 65 18 81 55 48)(9 41 35 19 66 10 73 56 49)
(2 9)(3 8)(4 7)(5 6)(10 58 49 68 35 43)(11 57 50 67 36 42)(12 56 51 66 28 41)(13 55 52 65 29 40)(14 63 53 64 30 39)(15 62 54 72 31 38)(16 61 46 71 32 37)(17 60 47 70 33 45)(18 59 48 69 34 44)(19 75)(20 74)(21 73)(22 81)(23 80)(24 79)(25 78)(26 77)(27 76)
G:=sub<Sym(81)| (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)(55,56,57,58,59,60,61,62,63)(64,65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80,81), (1,42,36,20,67,11,74,57,50)(2,43,28,21,68,12,75,58,51)(3,44,29,22,69,13,76,59,52)(4,45,30,23,70,14,77,60,53)(5,37,31,24,71,15,78,61,54)(6,38,32,25,72,16,79,62,46)(7,39,33,26,64,17,80,63,47)(8,40,34,27,65,18,81,55,48)(9,41,35,19,66,10,73,56,49), (2,9)(3,8)(4,7)(5,6)(10,58,49,68,35,43)(11,57,50,67,36,42)(12,56,51,66,28,41)(13,55,52,65,29,40)(14,63,53,64,30,39)(15,62,54,72,31,38)(16,61,46,71,32,37)(17,60,47,70,33,45)(18,59,48,69,34,44)(19,75)(20,74)(21,73)(22,81)(23,80)(24,79)(25,78)(26,77)(27,76)>;
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)(55,56,57,58,59,60,61,62,63)(64,65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80,81), (1,42,36,20,67,11,74,57,50)(2,43,28,21,68,12,75,58,51)(3,44,29,22,69,13,76,59,52)(4,45,30,23,70,14,77,60,53)(5,37,31,24,71,15,78,61,54)(6,38,32,25,72,16,79,62,46)(7,39,33,26,64,17,80,63,47)(8,40,34,27,65,18,81,55,48)(9,41,35,19,66,10,73,56,49), (2,9)(3,8)(4,7)(5,6)(10,58,49,68,35,43)(11,57,50,67,36,42)(12,56,51,66,28,41)(13,55,52,65,29,40)(14,63,53,64,30,39)(15,62,54,72,31,38)(16,61,46,71,32,37)(17,60,47,70,33,45)(18,59,48,69,34,44)(19,75)(20,74)(21,73)(22,81)(23,80)(24,79)(25,78)(26,77)(27,76) );
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),(55,56,57,58,59,60,61,62,63),(64,65,66,67,68,69,70,71,72),(73,74,75,76,77,78,79,80,81)], [(1,42,36,20,67,11,74,57,50),(2,43,28,21,68,12,75,58,51),(3,44,29,22,69,13,76,59,52),(4,45,30,23,70,14,77,60,53),(5,37,31,24,71,15,78,61,54),(6,38,32,25,72,16,79,62,46),(7,39,33,26,64,17,80,63,47),(8,40,34,27,65,18,81,55,48),(9,41,35,19,66,10,73,56,49)], [(2,9),(3,8),(4,7),(5,6),(10,58,49,68,35,43),(11,57,50,67,36,42),(12,56,51,66,28,41),(13,55,52,65,29,40),(14,63,53,64,30,39),(15,62,54,72,31,38),(16,61,46,71,32,37),(17,60,47,70,33,45),(18,59,48,69,34,44),(19,75),(20,74),(21,73),(22,81),(23,80),(24,79),(25,78),(26,77),(27,76)]])
54 conjugacy classes
| class | 1 | 2 | 3A | 3B | 3C | 3D | 3E | 3F | 3G | 3H | 6A | 6B | 9A | ··· | 9I | 9J | ··· | 9AP |
| order | 1 | 2 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 6 | 6 | 9 | ··· | 9 | 9 | ··· | 9 |
| size | 1 | 81 | 2 | 2 | 2 | 2 | 3 | 3 | 6 | 6 | 81 | 81 | 2 | ··· | 2 | 6 | ··· | 6 |
54 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 6 | 6 |
| type | + | + | + | + | + | + | + | + | ||||
| image | C1 | C2 | C3 | C6 | S3 | S3 | S3 | C3×S3 | D9 | C3×D9 | C9⋊C6 | He3.4S3 |
| kernel | C92⋊9C6 | C9×3- 1+2 | C9⋊D9 | C92 | C32⋊C9 | C32×C9 | C3×3- 1+2 | C3×C9 | 3- 1+2 | C9 | C9 | C3 |
| # reps | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 8 | 9 | 18 | 3 | 6 |
Matrix representation of C92⋊9C6 ►in GL8(𝔽19)
| 14 | 17 | 0 | 0 | 0 | 0 | 0 | 0 |
| 2 | 12 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 2 | 5 | 0 | 0 | 0 | 0 |
| 0 | 0 | 14 | 7 | 0 | 0 | 0 | 0 |
| 0 | 0 | 5 | 14 | 7 | 5 | 0 | 0 |
| 0 | 0 | 14 | 0 | 14 | 2 | 0 | 0 |
| 0 | 0 | 0 | 5 | 0 | 0 | 7 | 5 |
| 0 | 0 | 5 | 14 | 0 | 0 | 14 | 2 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 17 | 1 | 18 | 17 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 18 | 0 | 0 |
| 0 | 0 | 1 | 0 | 1 | 1 | 0 | 1 |
| 0 | 0 | 2 | 17 | 1 | 1 | 18 | 18 |
| 0 | 0 | 18 | 0 | 18 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 18 | 0 | 0 | 0 |
| 0 | 7 | 0 | 0 | 0 | 0 | 0 | 0 |
| 7 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 18 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 18 | 0 | 0 | 0 | 18 |
| 0 | 0 | 0 | 18 | 0 | 0 | 18 | 0 |
| 0 | 0 | 18 | 1 | 18 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 1 | 1 | 0 | 0 |
G:=sub<GL(8,GF(19))| [14,2,0,0,0,0,0,0,17,12,0,0,0,0,0,0,0,0,2,14,5,14,0,5,0,0,5,7,14,0,5,14,0,0,0,0,7,14,0,0,0,0,0,0,5,2,0,0,0,0,0,0,0,0,7,14,0,0,0,0,0,0,5,2],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,17,0,1,2,18,0,0,0,1,0,0,17,0,0,0,0,18,1,1,1,18,18,0,0,17,18,1,1,0,0,0,0,0,0,0,18,0,0,0,0,0,0,1,18,0,0],[0,7,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,1,1,0,0,18,1,0,0,0,18,18,18,1,0,0,0,0,0,0,0,18,1,0,0,0,0,0,0,0,1,0,0,0,0,0,18,0,0,0,0,0,0,18,0,0,0] >;
C92⋊9C6 in GAP, Magma, Sage, TeX
C_9^2\rtimes_9C_6
% in TeX
G:=Group("C9^2:9C6"); // GroupNames label
G:=SmallGroup(486,144);
// by ID
G=gap.SmallGroup(486,144);
# by ID
G:=PCGroup([6,-2,-3,-3,-3,-3,-3,1190,338,4755,2169,453,3244,11669]);
// Polycyclic
G:=Group<a,b,c|a^9=b^9=c^6=1,a*b=b*a,c*a*c^-1=a^-1,c*b*c^-1=b^2>;
// generators/relations