p-group, metabelian, nilpotent (class 2), monomial
Aliases: C92⋊4C3, C33.9C32, C32.27C33, C9⋊C9⋊6C3, C32⋊C9.11C3, C3.9(C9○He3), (C3×C9).10C32, SmallGroup(243,44)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C92⋊4C3
G = < a,b,c | a9=b9=c3=1, ab=ba, cac-1=ab6, cbc-1=a6b4 >
Smallest permutation representation of C92⋊4C3
►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 71 44 17 77 50 23 62 35)(2 72 45 18 78 51 24 63 36)(3 64 37 10 79 52 25 55 28)(4 65 38 11 80 53 26 56 29)(5 66 39 12 81 54 27 57 30)(6 67 40 13 73 46 19 58 31)(7 68 41 14 74 47 20 59 32)(8 69 42 15 75 48 21 60 33)(9 70 43 16 76 49 22 61 34)
(2 18 24)(3 25 10)(5 12 27)(6 19 13)(8 15 21)(9 22 16)(28 34 31)(29 44 47)(30 51 42)(32 38 50)(33 54 45)(35 41 53)(36 48 39)(37 43 40)(46 52 49)(55 67 76)(56 74 71)(57 60 63)(58 70 79)(59 77 65)(61 64 73)(62 80 68)(66 69 72)(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), (1,71,44,17,77,50,23,62,35)(2,72,45,18,78,51,24,63,36)(3,64,37,10,79,52,25,55,28)(4,65,38,11,80,53,26,56,29)(5,66,39,12,81,54,27,57,30)(6,67,40,13,73,46,19,58,31)(7,68,41,14,74,47,20,59,32)(8,69,42,15,75,48,21,60,33)(9,70,43,16,76,49,22,61,34), (2,18,24)(3,25,10)(5,12,27)(6,19,13)(8,15,21)(9,22,16)(28,34,31)(29,44,47)(30,51,42)(32,38,50)(33,54,45)(35,41,53)(36,48,39)(37,43,40)(46,52,49)(55,67,76)(56,74,71)(57,60,63)(58,70,79)(59,77,65)(61,64,73)(62,80,68)(66,69,72)(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), (1,71,44,17,77,50,23,62,35)(2,72,45,18,78,51,24,63,36)(3,64,37,10,79,52,25,55,28)(4,65,38,11,80,53,26,56,29)(5,66,39,12,81,54,27,57,30)(6,67,40,13,73,46,19,58,31)(7,68,41,14,74,47,20,59,32)(8,69,42,15,75,48,21,60,33)(9,70,43,16,76,49,22,61,34), (2,18,24)(3,25,10)(5,12,27)(6,19,13)(8,15,21)(9,22,16)(28,34,31)(29,44,47)(30,51,42)(32,38,50)(33,54,45)(35,41,53)(36,48,39)(37,43,40)(46,52,49)(55,67,76)(56,74,71)(57,60,63)(58,70,79)(59,77,65)(61,64,73)(62,80,68)(66,69,72)(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)], [(1,71,44,17,77,50,23,62,35),(2,72,45,18,78,51,24,63,36),(3,64,37,10,79,52,25,55,28),(4,65,38,11,80,53,26,56,29),(5,66,39,12,81,54,27,57,30),(6,67,40,13,73,46,19,58,31),(7,68,41,14,74,47,20,59,32),(8,69,42,15,75,48,21,60,33),(9,70,43,16,76,49,22,61,34)], [(2,18,24),(3,25,10),(5,12,27),(6,19,13),(8,15,21),(9,22,16),(28,34,31),(29,44,47),(30,51,42),(32,38,50),(33,54,45),(35,41,53),(36,48,39),(37,43,40),(46,52,49),(55,67,76),(56,74,71),(57,60,63),(58,70,79),(59,77,65),(61,64,73),(62,80,68),(66,69,72),(75,78,81)]])
C92⋊4C3 is a maximal subgroup of
C92⋊4C6
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⋊4C3 | C92 | C32⋊C9 | C9⋊C9 | C3 |
| # reps | 1 | 2 | 8 | 16 | 24 |
Matrix representation of C92⋊4C3 ►in GL6(𝔽19)
| 7 | 0 | 0 | 0 | 0 | 0 |
| 8 | 12 | 10 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 18 | 18 | 6 |
| 0 | 0 | 0 | 1 | 0 | 1 |
| 5 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 15 | 13 | 14 | 0 | 0 | 0 |
| 0 | 0 | 0 | 9 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 0 | 0 | 9 |
| 7 | 9 | 14 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 11 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 11 | 0 |
| 0 | 0 | 0 | 18 | 7 | 7 |
G:=sub<GL(6,GF(19))| [7,8,0,0,0,0,0,12,16,0,0,0,0,10,0,0,0,0,0,0,0,0,18,1,0,0,0,1,18,0,0,0,0,0,6,1],[5,0,15,0,0,0,0,0,13,0,0,0,0,1,14,0,0,0,0,0,0,9,0,0,0,0,0,0,9,0,0,0,0,0,0,9],[7,0,0,0,0,0,9,1,0,0,0,0,14,0,11,0,0,0,0,0,0,1,0,18,0,0,0,0,11,7,0,0,0,0,0,7] >;
C92⋊4C3 in GAP, Magma, Sage, TeX
C_9^2\rtimes_4C_3
% in TeX
G:=Group("C9^2:4C3"); // GroupNames label
G:=SmallGroup(243,44);
// by ID
G=gap.SmallGroup(243,44);
# by ID
G:=PCGroup([5,-3,3,3,-3,3,301,366,1352,57]);
// 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^4>;
// generators/relations
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