p-group, cyclic, abelian, monomial
Aliases: C81, also denoted Z81, SmallGroup(81,1)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
| C1 — C81 |
| C1 — C81 |
| C1 — C81 |
| C1 — C3 — C3 — C3 — C3 — C3 — C3 — C3 — C3 — C3 — C3 — C3 — C3 — C3 — C3 — C3 — C3 — C3 — C3 — C9 — C9 — C9 — C9 — C9 — C9 — C27 — C27 — C81 |
Generators and relations for C81
G = < a | a81=1 >
Smallest permutation representation of C81
►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)
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)>;
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) );
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)]])
C81 is a maximal subgroup of
D81 C243 C81⋊C3 C27.A4
C81 is a maximal quotient of C243 C27.A4
81 conjugacy classes
| class | 1 | 3A | 3B | 9A | ··· | 9F | 27A | ··· | 27R | 81A | ··· | 81BB |
| order | 1 | 3 | 3 | 9 | ··· | 9 | 27 | ··· | 27 | 81 | ··· | 81 |
| size | 1 | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
81 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 |
| type | + | ||||
| image | C1 | C3 | C9 | C27 | C81 |
| kernel | C81 | C27 | C9 | C3 | C1 |
| # reps | 1 | 2 | 6 | 18 | 54 |
Matrix representation of C81 ►in GL1(𝔽163) generated by
| 91 |
G:=sub<GL(1,GF(163))| [91] >;
C81 in GAP, Magma, Sage, TeX
C_{81} % in TeX
G:=Group("C81"); // GroupNames label
G:=SmallGroup(81,1);
// by ID
G=gap.SmallGroup(81,1);
# by ID
G:=PCGroup([4,-3,-3,-3,-3,12,29,46]);
// Polycyclic
G:=Group<a|a^81=1>;
// generators/relations
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