p-group, metabelian, nilpotent (class 2), monomial
Aliases: C92⋊9C3, C9⋊23- 1+2, C32.30C33, C33.12C32, C9⋊C9⋊9C3, (C3×C9).13C32, (C3×3- 1+2).7C3, C3.10(C3×3- 1+2), 3-Sylow(AGammaL(2,64)), SmallGroup(243,47)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C92⋊9C3
G = < a,b,c | a9=b9=c3=1, ab=ba, cac-1=a7, cbc-1=b7 >
Subgroups: 126 in 70 conjugacy classes, 45 normal (5 characteristic)
C1, C3, C3, C9, C9, C32, C32, C3×C9, 3- 1+2, C33, C92, C9⋊C9, C3×3- 1+2, C92⋊9C3
Quotients: C1, C3, C32, 3- 1+2, C33, C3×3- 1+2, C92⋊9C3
►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 60 33 80 70 53 26 14 40)(2 61 34 81 71 54 27 15 41)(3 62 35 73 72 46 19 16 42)(4 63 36 74 64 47 20 17 43)(5 55 28 75 65 48 21 18 44)(6 56 29 76 66 49 22 10 45)(7 57 30 77 67 50 23 11 37)(8 58 31 78 68 51 24 12 38)(9 59 32 79 69 52 25 13 39)
(2 5 8)(3 9 6)(10 62 69)(11 57 67)(12 61 65)(13 56 72)(14 60 70)(15 55 68)(16 59 66)(17 63 64)(18 58 71)(19 25 22)(21 24 27)(28 38 54)(29 42 52)(30 37 50)(31 41 48)(32 45 46)(33 40 53)(34 44 51)(35 39 49)(36 43 47)(73 79 76)(75 78 81)
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), (1,60,33,80,70,53,26,14,40)(2,61,34,81,71,54,27,15,41)(3,62,35,73,72,46,19,16,42)(4,63,36,74,64,47,20,17,43)(5,55,28,75,65,48,21,18,44)(6,56,29,76,66,49,22,10,45)(7,57,30,77,67,50,23,11,37)(8,58,31,78,68,51,24,12,38)(9,59,32,79,69,52,25,13,39), (2,5,8)(3,9,6)(10,62,69)(11,57,67)(12,61,65)(13,56,72)(14,60,70)(15,55,68)(16,59,66)(17,63,64)(18,58,71)(19,25,22)(21,24,27)(28,38,54)(29,42,52)(30,37,50)(31,41,48)(32,45,46)(33,40,53)(34,44,51)(35,39,49)(36,43,47)(73,79,76)(75,78,81)>;
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), (1,60,33,80,70,53,26,14,40)(2,61,34,81,71,54,27,15,41)(3,62,35,73,72,46,19,16,42)(4,63,36,74,64,47,20,17,43)(5,55,28,75,65,48,21,18,44)(6,56,29,76,66,49,22,10,45)(7,57,30,77,67,50,23,11,37)(8,58,31,78,68,51,24,12,38)(9,59,32,79,69,52,25,13,39), (2,5,8)(3,9,6)(10,62,69)(11,57,67)(12,61,65)(13,56,72)(14,60,70)(15,55,68)(16,59,66)(17,63,64)(18,58,71)(19,25,22)(21,24,27)(28,38,54)(29,42,52)(30,37,50)(31,41,48)(32,45,46)(33,40,53)(34,44,51)(35,39,49)(36,43,47)(73,79,76)(75,78,81) );
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)], [(1,60,33,80,70,53,26,14,40),(2,61,34,81,71,54,27,15,41),(3,62,35,73,72,46,19,16,42),(4,63,36,74,64,47,20,17,43),(5,55,28,75,65,48,21,18,44),(6,56,29,76,66,49,22,10,45),(7,57,30,77,67,50,23,11,37),(8,58,31,78,68,51,24,12,38),(9,59,32,79,69,52,25,13,39)], [(2,5,8),(3,9,6),(10,62,69),(11,57,67),(12,61,65),(13,56,72),(14,60,70),(15,55,68),(16,59,66),(17,63,64),(18,58,71),(19,25,22),(21,24,27),(28,38,54),(29,42,52),(30,37,50),(31,41,48),(32,45,46),(33,40,53),(34,44,51),(35,39,49),(36,43,47),(73,79,76),(75,78,81)]])
C92⋊9C3 is a maximal subgroup of
C92⋊8C6 C92⋊12C6
51 conjugacy classes
| class | 1 | 3A | ··· | 3H | 3I | 3J | 9A | ··· | 9X | 9Y | ··· | 9AN |
| order | 1 | 3 | ··· | 3 | 3 | 3 | 9 | ··· | 9 | 9 | ··· | 9 |
| size | 1 | 1 | ··· | 1 | 9 | 9 | 3 | ··· | 3 | 9 | ··· | 9 |
51 irreducible representations
| dim | 1 | 1 | 1 | 1 | 3 |
| type | + | ||||
| image | C1 | C3 | C3 | C3 | 3- 1+2 |
| kernel | C92⋊9C3 | C92 | C9⋊C9 | C3×3- 1+2 | C9 |
| # reps | 1 | 2 | 16 | 8 | 24 |
Matrix representation of C92⋊9C3 ►in GL6(𝔽19)
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 11 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 7 | 0 | 0 |
| 0 | 0 | 0 | 0 | 7 | 0 |
| 0 | 0 | 0 | 0 | 0 | 7 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 11 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 7 | 0 | 0 | 0 | 0 |
| 0 | 0 | 11 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 7 | 0 |
| 0 | 0 | 0 | 0 | 0 | 11 |
G:=sub<GL(6,GF(19))| [0,0,11,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,7,0,0,0,0,0,0,7,0,0,0,0,0,0,7],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,11,0,0,0,1,0,0,0,0,0,0,1,0],[1,0,0,0,0,0,0,7,0,0,0,0,0,0,11,0,0,0,0,0,0,1,0,0,0,0,0,0,7,0,0,0,0,0,0,11] >;
C92⋊9C3 in GAP, Magma, Sage, TeX
C_9^2\rtimes_9C_3
% in TeX
G:=Group("C9^2:9C3"); // GroupNames label
G:=SmallGroup(243,47);
// by ID
G=gap.SmallGroup(243,47);
# by ID
G:=PCGroup([5,-3,3,3,-3,3,301,96,1352,147]);
// Polycyclic
G:=Group<a,b,c|a^9=b^9=c^3=1,a*b=b*a,c*a*c^-1=a^7,c*b*c^-1=b^7>;
// generators/relations