p-group, metabelian, nilpotent (class 2), monomial
Aliases: C92⋊7C3, C33.8C32, C9⋊13- 1+2, C32.26C33, C9⋊C9⋊5C3, (C3×C9).9C32, C32⋊C9.10C3, C3.8(C9○He3), C3.8(C3×3- 1+2), (C3×3- 1+2).5C3, SmallGroup(243,43)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C92⋊7C3
G = < a,b,c | a9=b9=c3=1, ab=ba, cac-1=ab6, cbc-1=a6b >
Subgroups: 108 in 60 conjugacy classes, 39 normal (5 characteristic)
C1, C3, C3, C9, C9, C32, C32, C3×C9, 3- 1+2, C33, C92, C32⋊C9, C9⋊C9, C3×3- 1+2, C92⋊7C3
Quotients: C1, C3, C32, 3- 1+2, C33, C3×3- 1+2, C9○He3, C92⋊7C3
►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 36 80 26 44 47 16 68)(2 57 28 81 27 45 48 17 69)(3 58 29 73 19 37 49 18 70)(4 59 30 74 20 38 50 10 71)(5 60 31 75 21 39 51 11 72)(6 61 32 76 22 40 52 12 64)(7 62 33 77 23 41 53 13 65)(8 63 34 78 24 42 54 14 66)(9 55 35 79 25 43 46 15 67)
(2 81 48)(3 49 73)(5 75 51)(6 52 76)(8 78 54)(9 46 79)(10 13 16)(11 63 27)(12 25 58)(14 57 21)(15 19 61)(17 60 24)(18 22 55)(20 23 26)(28 42 72)(29 67 40)(30 36 33)(31 45 66)(32 70 43)(34 39 69)(35 64 37)(38 44 41)(56 59 62)(65 71 68)
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), (1,56,36,80,26,44,47,16,68)(2,57,28,81,27,45,48,17,69)(3,58,29,73,19,37,49,18,70)(4,59,30,74,20,38,50,10,71)(5,60,31,75,21,39,51,11,72)(6,61,32,76,22,40,52,12,64)(7,62,33,77,23,41,53,13,65)(8,63,34,78,24,42,54,14,66)(9,55,35,79,25,43,46,15,67), (2,81,48)(3,49,73)(5,75,51)(6,52,76)(8,78,54)(9,46,79)(10,13,16)(11,63,27)(12,25,58)(14,57,21)(15,19,61)(17,60,24)(18,22,55)(20,23,26)(28,42,72)(29,67,40)(30,36,33)(31,45,66)(32,70,43)(34,39,69)(35,64,37)(38,44,41)(56,59,62)(65,71,68)>;
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), (1,56,36,80,26,44,47,16,68)(2,57,28,81,27,45,48,17,69)(3,58,29,73,19,37,49,18,70)(4,59,30,74,20,38,50,10,71)(5,60,31,75,21,39,51,11,72)(6,61,32,76,22,40,52,12,64)(7,62,33,77,23,41,53,13,65)(8,63,34,78,24,42,54,14,66)(9,55,35,79,25,43,46,15,67), (2,81,48)(3,49,73)(5,75,51)(6,52,76)(8,78,54)(9,46,79)(10,13,16)(11,63,27)(12,25,58)(14,57,21)(15,19,61)(17,60,24)(18,22,55)(20,23,26)(28,42,72)(29,67,40)(30,36,33)(31,45,66)(32,70,43)(34,39,69)(35,64,37)(38,44,41)(56,59,62)(65,71,68) );
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)], [(1,56,36,80,26,44,47,16,68),(2,57,28,81,27,45,48,17,69),(3,58,29,73,19,37,49,18,70),(4,59,30,74,20,38,50,10,71),(5,60,31,75,21,39,51,11,72),(6,61,32,76,22,40,52,12,64),(7,62,33,77,23,41,53,13,65),(8,63,34,78,24,42,54,14,66),(9,55,35,79,25,43,46,15,67)], [(2,81,48),(3,49,73),(5,75,51),(6,52,76),(8,78,54),(9,46,79),(10,13,16),(11,63,27),(12,25,58),(14,57,21),(15,19,61),(17,60,24),(18,22,55),(20,23,26),(28,42,72),(29,67,40),(30,36,33),(31,45,66),(32,70,43),(34,39,69),(35,64,37),(38,44,41),(56,59,62),(65,71,68)]])
C92⋊7C3 is a maximal subgroup of
C92⋊7C6 C92⋊6S3 C92⋊10C6
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 | 1 | 3 | 3 |
| type | + | ||||||
| image | C1 | C3 | C3 | C3 | C3 | 3- 1+2 | C9○He3 |
| kernel | C92⋊7C3 | C92 | C32⋊C9 | C9⋊C9 | C3×3- 1+2 | C9 | C3 |
| # reps | 1 | 2 | 4 | 16 | 4 | 12 | 12 |
Matrix representation of C92⋊7C3 ►in GL6(𝔽19)
| 9 | 0 | 0 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 9 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 7 | 0 |
| 0 | 0 | 0 | 0 | 0 | 7 |
| 0 | 0 | 0 | 11 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 7 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 11 | 0 | 0 | 0 | 0 |
| 0 | 0 | 7 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 11 | 0 |
| 0 | 0 | 0 | 0 | 0 | 7 |
G:=sub<GL(6,GF(19))| [9,0,0,0,0,0,0,9,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,11,0,0,0,7,0,0,0,0,0,0,7,0],[0,0,1,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,7,0,0,0,1,0,0,0,0,0,0,1,0],[1,0,0,0,0,0,0,11,0,0,0,0,0,0,7,0,0,0,0,0,0,1,0,0,0,0,0,0,11,0,0,0,0,0,0,7] >;
C92⋊7C3 in GAP, Magma, Sage, TeX
C_9^2\rtimes_7C_3
% in TeX
G:=Group("C9^2:7C3"); // GroupNames label
G:=SmallGroup(243,43);
// by ID
G=gap.SmallGroup(243,43);
# by ID
G:=PCGroup([5,-3,3,3,-3,3,301,276,1352,57]);
// Polycyclic
G:=Group<a,b,c|a^9=b^9=c^3=1,a*b=b*a,c*a*c^-1=a*b^6,c*b*c^-1=a^6*b>;
// generators/relations