p-group, metabelian, nilpotent (class 2), monomial
Aliases: C92⋊5C3, C33.10C32, C32.28C33, C9⋊C9⋊7C3, C32⋊C9.12C3, (C3×C9).11C32, C3.10(C9○He3), SmallGroup(243,45)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C92⋊5C3
G = < a,b,c | a9=b9=c3=1, ab=ba, cac-1=ab3, cbc-1=a6b >
Smallest permutation representation of C92⋊5C3
►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 46 68 41 56 29 80 15 23)(2 47 69 42 57 30 81 16 24)(3 48 70 43 58 31 73 17 25)(4 49 71 44 59 32 74 18 26)(5 50 72 45 60 33 75 10 27)(6 51 64 37 61 34 76 11 19)(7 52 65 38 62 35 77 12 20)(8 53 66 39 63 36 78 13 21)(9 54 67 40 55 28 79 14 22)
(2 81 42)(3 43 73)(5 75 45)(6 37 76)(8 78 39)(9 40 79)(10 63 47)(11 54 58)(12 15 18)(13 57 50)(14 48 61)(16 60 53)(17 51 55)(19 70 28)(20 26 23)(21 33 69)(22 64 31)(24 36 72)(25 67 34)(27 30 66)(29 35 32)(46 49 52)(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), (1,46,68,41,56,29,80,15,23)(2,47,69,42,57,30,81,16,24)(3,48,70,43,58,31,73,17,25)(4,49,71,44,59,32,74,18,26)(5,50,72,45,60,33,75,10,27)(6,51,64,37,61,34,76,11,19)(7,52,65,38,62,35,77,12,20)(8,53,66,39,63,36,78,13,21)(9,54,67,40,55,28,79,14,22), (2,81,42)(3,43,73)(5,75,45)(6,37,76)(8,78,39)(9,40,79)(10,63,47)(11,54,58)(12,15,18)(13,57,50)(14,48,61)(16,60,53)(17,51,55)(19,70,28)(20,26,23)(21,33,69)(22,64,31)(24,36,72)(25,67,34)(27,30,66)(29,35,32)(46,49,52)(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), (1,46,68,41,56,29,80,15,23)(2,47,69,42,57,30,81,16,24)(3,48,70,43,58,31,73,17,25)(4,49,71,44,59,32,74,18,26)(5,50,72,45,60,33,75,10,27)(6,51,64,37,61,34,76,11,19)(7,52,65,38,62,35,77,12,20)(8,53,66,39,63,36,78,13,21)(9,54,67,40,55,28,79,14,22), (2,81,42)(3,43,73)(5,75,45)(6,37,76)(8,78,39)(9,40,79)(10,63,47)(11,54,58)(12,15,18)(13,57,50)(14,48,61)(16,60,53)(17,51,55)(19,70,28)(20,26,23)(21,33,69)(22,64,31)(24,36,72)(25,67,34)(27,30,66)(29,35,32)(46,49,52)(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)], [(1,46,68,41,56,29,80,15,23),(2,47,69,42,57,30,81,16,24),(3,48,70,43,58,31,73,17,25),(4,49,71,44,59,32,74,18,26),(5,50,72,45,60,33,75,10,27),(6,51,64,37,61,34,76,11,19),(7,52,65,38,62,35,77,12,20),(8,53,66,39,63,36,78,13,21),(9,54,67,40,55,28,79,14,22)], [(2,81,42),(3,43,73),(5,75,45),(6,37,76),(8,78,39),(9,40,79),(10,63,47),(11,54,58),(12,15,18),(13,57,50),(14,48,61),(16,60,53),(17,51,55),(19,70,28),(20,26,23),(21,33,69),(22,64,31),(24,36,72),(25,67,34),(27,30,66),(29,35,32),(46,49,52),(56,59,62),(65,71,68)]])
C92⋊5C3 is a maximal subgroup of
C92⋊5S3 C92⋊5C6
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 | C9○He3 |
| kernel | C92⋊5C3 | C92 | C32⋊C9 | C9⋊C9 | C3 |
| # reps | 1 | 2 | 8 | 16 | 24 |
Matrix representation of C92⋊5C3 ►in GL6(𝔽19)
| 0 | 0 | 17 | 0 | 0 | 0 |
| 5 | 0 | 0 | 0 | 0 | 0 |
| 0 | 5 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 17 | 2 | 0 |
| 0 | 0 | 0 | 0 | 2 | 4 |
| 0 | 0 | 0 | 0 | 18 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 7 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 9 | 10 | 0 |
| 0 | 0 | 0 | 0 | 10 | 1 |
| 0 | 0 | 0 | 0 | 14 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 11 | 0 | 0 | 0 | 0 |
| 0 | 0 | 7 | 0 | 0 | 0 |
| 0 | 0 | 0 | 7 | 0 | 0 |
| 0 | 0 | 0 | 15 | 11 | 0 |
| 0 | 0 | 0 | 16 | 0 | 1 |
G:=sub<GL(6,GF(19))| [0,5,0,0,0,0,0,0,5,0,0,0,17,0,0,0,0,0,0,0,0,17,0,0,0,0,0,2,2,18,0,0,0,0,4,0],[0,0,7,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,9,0,0,0,0,0,10,10,14,0,0,0,0,1,0],[1,0,0,0,0,0,0,11,0,0,0,0,0,0,7,0,0,0,0,0,0,7,15,16,0,0,0,0,11,0,0,0,0,0,0,1] >;
C92⋊5C3 in GAP, Magma, Sage, TeX
C_9^2\rtimes_5C_3
% in TeX
G:=Group("C9^2:5C3"); // GroupNames label
G:=SmallGroup(243,45);
// by ID
G=gap.SmallGroup(243,45);
# by ID
G:=PCGroup([5,-3,3,3,-3,3,301,276,1352,102]);
// Polycyclic
G:=Group<a,b,c|a^9=b^9=c^3=1,a*b=b*a,c*a*c^-1=a*b^3,c*b*c^-1=a^6*b>;
// generators/relations
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