p-group, metabelian, nilpotent (class 2), monomial
Aliases: C92⋊3C3, C33.24C32, C32.17C33, (C3×C9)⋊5C9, C9⋊C9⋊10C3, C9.3(C3×C9), C3.3(C32×C9), C32⋊C9.14C3, C3.1(C9○He3), (C32×C9).11C3, C32.11(C3×C9), (C3×C9).20C32, SmallGroup(243,34)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C92⋊3C3
G = < a,b,c | a9=b9=c3=1, ab=ba, cac-1=ab3, bc=cb >
Subgroups: 99 in 69 conjugacy classes, 54 normal (7 characteristic)
C1, C3, C3, C3, C9, C9, C32, C32, C32, C3×C9, C3×C9, C33, C92, C32⋊C9, C9⋊C9, C32×C9, C92⋊3C3
Quotients: C1, C3, C9, C32, C3×C9, C33, C32×C9, C9○He3, C92⋊3C3
►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 56 29 80 71 44 17 25 53)(2 57 30 81 72 45 18 26 54)(3 58 31 73 64 37 10 27 46)(4 59 32 74 65 38 11 19 47)(5 60 33 75 66 39 12 20 48)(6 61 34 76 67 40 13 21 49)(7 62 35 77 68 41 14 22 50)(8 63 36 78 69 42 15 23 51)(9 55 28 79 70 43 16 24 52)
(1 7 4)(2 15 75)(3 79 13)(5 18 78)(6 73 16)(8 12 81)(9 76 10)(11 17 14)(19 25 22)(20 72 63)(21 58 70)(23 66 57)(24 61 64)(26 69 60)(27 55 67)(28 40 46)(29 35 32)(30 51 39)(31 43 49)(33 54 42)(34 37 52)(36 48 45)(38 44 41)(47 53 50)(56 62 59)(65 71 68)(74 80 77)
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), (1,56,29,80,71,44,17,25,53)(2,57,30,81,72,45,18,26,54)(3,58,31,73,64,37,10,27,46)(4,59,32,74,65,38,11,19,47)(5,60,33,75,66,39,12,20,48)(6,61,34,76,67,40,13,21,49)(7,62,35,77,68,41,14,22,50)(8,63,36,78,69,42,15,23,51)(9,55,28,79,70,43,16,24,52), (1,7,4)(2,15,75)(3,79,13)(5,18,78)(6,73,16)(8,12,81)(9,76,10)(11,17,14)(19,25,22)(20,72,63)(21,58,70)(23,66,57)(24,61,64)(26,69,60)(27,55,67)(28,40,46)(29,35,32)(30,51,39)(31,43,49)(33,54,42)(34,37,52)(36,48,45)(38,44,41)(47,53,50)(56,62,59)(65,71,68)(74,80,77)>;
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), (1,56,29,80,71,44,17,25,53)(2,57,30,81,72,45,18,26,54)(3,58,31,73,64,37,10,27,46)(4,59,32,74,65,38,11,19,47)(5,60,33,75,66,39,12,20,48)(6,61,34,76,67,40,13,21,49)(7,62,35,77,68,41,14,22,50)(8,63,36,78,69,42,15,23,51)(9,55,28,79,70,43,16,24,52), (1,7,4)(2,15,75)(3,79,13)(5,18,78)(6,73,16)(8,12,81)(9,76,10)(11,17,14)(19,25,22)(20,72,63)(21,58,70)(23,66,57)(24,61,64)(26,69,60)(27,55,67)(28,40,46)(29,35,32)(30,51,39)(31,43,49)(33,54,42)(34,37,52)(36,48,45)(38,44,41)(47,53,50)(56,62,59)(65,71,68)(74,80,77) );
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)], [(1,56,29,80,71,44,17,25,53),(2,57,30,81,72,45,18,26,54),(3,58,31,73,64,37,10,27,46),(4,59,32,74,65,38,11,19,47),(5,60,33,75,66,39,12,20,48),(6,61,34,76,67,40,13,21,49),(7,62,35,77,68,41,14,22,50),(8,63,36,78,69,42,15,23,51),(9,55,28,79,70,43,16,24,52)], [(1,7,4),(2,15,75),(3,79,13),(5,18,78),(6,73,16),(8,12,81),(9,76,10),(11,17,14),(19,25,22),(20,72,63),(21,58,70),(23,66,57),(24,61,64),(26,69,60),(27,55,67),(28,40,46),(29,35,32),(30,51,39),(31,43,49),(33,54,42),(34,37,52),(36,48,45),(38,44,41),(47,53,50),(56,62,59),(65,71,68),(74,80,77)]])
C92⋊3C3 is a maximal subgroup of
C92⋊3S3 C92⋊4S3 C92⋊3C6
99 conjugacy classes
| class | 1 | 3A | ··· | 3H | 3I | ··· | 3N | 9A | ··· | 9R | 9S | ··· | 9CF |
| order | 1 | 3 | ··· | 3 | 3 | ··· | 3 | 9 | ··· | 9 | 9 | ··· | 9 |
| size | 1 | 1 | ··· | 1 | 3 | ··· | 3 | 1 | ··· | 1 | 3 | ··· | 3 |
99 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 3 |
| type | + | ||||||
| image | C1 | C3 | C3 | C3 | C3 | C9 | C9○He3 |
| kernel | C92⋊3C3 | C92 | C32⋊C9 | C9⋊C9 | C32×C9 | C3×C9 | C3 |
| # reps | 1 | 6 | 6 | 12 | 2 | 54 | 18 |
Matrix representation of C92⋊3C3 ►in GL4(𝔽19) generated by
| 9 | 0 | 0 | 0 |
| 0 | 6 | 6 | 0 |
| 0 | 1 | 13 | 1 |
| 0 | 12 | 15 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 |
| 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 9 |
| 7 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 6 | 7 | 0 |
| 0 | 13 | 0 | 11 |
G:=sub<GL(4,GF(19))| [9,0,0,0,0,6,1,12,0,6,13,15,0,0,1,0],[1,0,0,0,0,9,0,0,0,0,9,0,0,0,0,9],[7,0,0,0,0,1,6,13,0,0,7,0,0,0,0,11] >;
C92⋊3C3 in GAP, Magma, Sage, TeX
C_9^2\rtimes_3C_3
% in TeX
G:=Group("C9^2:3C3"); // GroupNames label
G:=SmallGroup(243,34);
// by ID
G=gap.SmallGroup(243,34);
# by ID
G:=PCGroup([5,-3,3,3,-3,3,405,301,57]);
// Polycyclic
G:=Group<a,b,c|a^9=b^9=c^3=1,a*b=b*a,c*a*c^-1=a*b^3,b*c=c*b>;
// generators/relations