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G = C61order 61

Cyclic group

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

Aliases: C61, also denoted Z61, SmallGroup(61,1)

Series: Derived Chief Lower central Upper central Jennings

C1 — C61
C1 — C61
C1 — C61
C1 — C61
C1 — C61

Generators and relations for C61
 G = < a | a61=1 >


Smallest permutation representation of C61
Regular action on 61 points
Generators in S61
(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)

G:=sub<Sym(61)| (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)>;

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) );

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)])

C61 is a maximal subgroup of   D61  C61⋊C3  C61⋊C5

61 conjugacy classes

class 1 61A···61BH
order161···61
size11···1

61 irreducible representations

dim11
type+
imageC1C61
kernelC61C1
# reps160

Matrix representation of C61 in GL1(𝔽367) generated by

137
G:=sub<GL(1,GF(367))| [137] >;

C61 in GAP, Magma, Sage, TeX

C_{61}
% in TeX

G:=Group("C61");
// GroupNames label

G:=SmallGroup(61,1);
// by ID

G=gap.SmallGroup(61,1);
# by ID

G:=PCGroup([1,-61]:ExponentLimit:=1);
// Polycyclic

G:=Group<a|a^61=1>;
// generators/relations

Export

Subgroup lattice of C61 in TeX

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