direct product, p-group, abelian, monomial
Aliases: C92, SmallGroup(81,2)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
| C1 — C92 |
| C1 — C92 |
| C1 — C92 |
Generators and relations for C92
G = < a,b | a9=b9=1, ab=ba >
Smallest permutation representation of C92
►Regular action 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 39 12)(2 61 34 81 71 54 27 40 13)(3 62 35 73 72 46 19 41 14)(4 63 36 74 64 47 20 42 15)(5 55 28 75 65 48 21 43 16)(6 56 29 76 66 49 22 44 17)(7 57 30 77 67 50 23 45 18)(8 58 31 78 68 51 24 37 10)(9 59 32 79 69 52 25 38 11)
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), (1,60,33,80,70,53,26,39,12)(2,61,34,81,71,54,27,40,13)(3,62,35,73,72,46,19,41,14)(4,63,36,74,64,47,20,42,15)(5,55,28,75,65,48,21,43,16)(6,56,29,76,66,49,22,44,17)(7,57,30,77,67,50,23,45,18)(8,58,31,78,68,51,24,37,10)(9,59,32,79,69,52,25,38,11)>;
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), (1,60,33,80,70,53,26,39,12)(2,61,34,81,71,54,27,40,13)(3,62,35,73,72,46,19,41,14)(4,63,36,74,64,47,20,42,15)(5,55,28,75,65,48,21,43,16)(6,56,29,76,66,49,22,44,17)(7,57,30,77,67,50,23,45,18)(8,58,31,78,68,51,24,37,10)(9,59,32,79,69,52,25,38,11) );
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)], [(1,60,33,80,70,53,26,39,12),(2,61,34,81,71,54,27,40,13),(3,62,35,73,72,46,19,41,14),(4,63,36,74,64,47,20,42,15),(5,55,28,75,65,48,21,43,16),(6,56,29,76,66,49,22,44,17),(7,57,30,77,67,50,23,45,18),(8,58,31,78,68,51,24,37,10),(9,59,32,79,69,52,25,38,11)]])
C92 is a maximal subgroup of
C9⋊D9 C27⋊2C9 C9⋊C27 C92⋊C3 C92⋊2C3 C92.C3 C92⋊3C3 C92⋊7C3 C92⋊4C3 C92⋊5C3 C92⋊8C3 C92⋊9C3
C92 is a maximal quotient of
C3.C92 C27⋊2C9
81 conjugacy classes
| class | 1 | 3A | ··· | 3H | 9A | ··· | 9BT |
| order | 1 | 3 | ··· | 3 | 9 | ··· | 9 |
| size | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
81 irreducible representations
| dim | 1 | 1 | 1 |
| type | + | ||
| image | C1 | C3 | C9 |
| kernel | C92 | C3×C9 | C9 |
| # reps | 1 | 8 | 72 |
Matrix representation of C92 ►in GL2(𝔽19) generated by
| 11 | 0 |
| 0 | 16 |
| 6 | 0 |
| 0 | 1 |
G:=sub<GL(2,GF(19))| [11,0,0,16],[6,0,0,1] >;
C92 in GAP, Magma, Sage, TeX
C_9^2
% in TeX
G:=Group("C9^2"); // GroupNames label
G:=SmallGroup(81,2);
// by ID
G=gap.SmallGroup(81,2);
# by ID
G:=PCGroup([4,-3,3,-3,3,36,77]);
// Polycyclic
G:=Group<a,b|a^9=b^9=1,a*b=b*a>;
// generators/relations
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