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G = C80order 80 = 24·5

Cyclic group

direct product, cyclic, abelian, monomial

Aliases: C80, also denoted Z80, SmallGroup(80,2)

Series: Derived Chief Lower central Upper central

C1 — C80
C1C2C4C8C40 — C80
C1 — C80
C1 — C80

Generators and relations for C80
 G = < a | a80=1 >


Smallest permutation representation of C80
Regular action on 80 points
Generators in S80
(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)

G:=sub<Sym(80)| (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)>;

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) );

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)])

C80 is a maximal subgroup of   C52C32  C80⋊C2  D80  C16⋊D5  Dic40

80 conjugacy classes

class 1  2 4A4B5A5B5C5D8A8B8C8D10A10B10C10D16A···16H20A···20H40A···40P80A···80AF
order1244555588881010101016···1620···2040···4080···80
size11111111111111111···11···11···11···1

80 irreducible representations

dim1111111111
type++
imageC1C2C4C5C8C10C16C20C40C80
kernelC80C40C20C16C10C8C5C4C2C1
# reps112444881632

Matrix representation of C80 in GL1(𝔽241) generated by

140
G:=sub<GL(1,GF(241))| [140] >;

C80 in GAP, Magma, Sage, TeX

C_{80}
% in TeX

G:=Group("C80");
// GroupNames label

G:=SmallGroup(80,2);
// by ID

G=gap.SmallGroup(80,2);
# by ID

G:=PCGroup([5,-2,-5,-2,-2,-2,50,42,58]);
// Polycyclic

G:=Group<a|a^80=1>;
// generators/relations

Export

Subgroup lattice of C80 in TeX

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