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G = C89order 89

Cyclic group

p-group, cyclic, elementary abelian, simple, monomial

Aliases: C89, also denoted Z89, SmallGroup(89,1)

Series: Derived Chief Lower central Upper central Jennings

C1 — C89
C1 — C89
C1 — C89
C1 — C89
C1 — C89

Generators and relations for C89
 G = < a | a89=1 >


Smallest permutation representation of C89
Regular action on 89 points
Generators in S89
(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 82 83 84 85 86 87 88 89)

G:=sub<Sym(89)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89)>;

G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89) );

G=PermutationGroup([(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89)])

C89 is a maximal subgroup of   D89

89 conjugacy classes

class 1 89A···89CJ
order189···89
size11···1

89 irreducible representations

dim11
type+
imageC1C89
kernelC89C1
# reps188

Matrix representation of C89 in GL1(𝔽179) generated by

161
G:=sub<GL(1,GF(179))| [161] >;

C89 in GAP, Magma, Sage, TeX

C_{89}
% in TeX

G:=Group("C89");
// GroupNames label

G:=SmallGroup(89,1);
// by ID

G=gap.SmallGroup(89,1);
# by ID

G:=PCGroup([1,-89]:ExponentLimit:=1);
// Polycyclic

G:=Group<a|a^89=1>;
// generators/relations

Export

Subgroup lattice of C89 in TeX

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