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G = C81order 81 = 34

Cyclic group

p-group, cyclic, abelian, monomial

Aliases: C81, also denoted Z81, SmallGroup(81,1)

Series: Derived Chief Lower central Upper central Jennings

C1 — C81
C1C3C9C27 — C81
C1 — C81
C1 — C81
C1C3C3C3C3C3C3C3C3C3C3C3C3C3C3C3C3C3C3C9C9C9C9C9C9C27C27 — 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 3A3B9A···9F27A···27R81A···81BB
order1339···927···2781···81
size1111···11···11···1

81 irreducible representations

dim11111
type+
imageC1C3C9C27C81
kernelC81C27C9C3C1
# reps1261854

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

Export

Subgroup lattice of C81 in TeX

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