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G = C65order 65 = 5·13

Cyclic group

direct product, cyclic, abelian, monomial

Aliases: C65, also denoted Z65, SmallGroup(65,1)

Series: Derived Chief Lower central Upper central

C1 — C65
C1C13 — C65
C1 — C65
C1 — C65

Generators and relations for C65
 G = < a | a65=1 >


Smallest permutation representation of C65
Regular action on 65 points
Generators in S65
(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)

G:=sub<Sym(65)| (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)>;

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) );

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)])

C65 is a maximal subgroup of   D65

65 conjugacy classes

class 1 5A5B5C5D13A···13L65A···65AV
order1555513···1365···65
size111111···11···1

65 irreducible representations

dim1111
type+
imageC1C5C13C65
kernelC65C13C5C1
# reps141248

Matrix representation of C65 in GL1(𝔽131) generated by

15
G:=sub<GL(1,GF(131))| [15] >;

C65 in GAP, Magma, Sage, TeX

C_{65}
% in TeX

G:=Group("C65");
// GroupNames label

G:=SmallGroup(65,1);
// by ID

G=gap.SmallGroup(65,1);
# by ID

G:=PCGroup([2,-5,-13]);
// Polycyclic

G:=Group<a|a^65=1>;
// generators/relations

Export

Subgroup lattice of C65 in TeX

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