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G = C85order 85 = 5·17

Cyclic group

direct product, cyclic, abelian, monomial

Aliases: C85, also denoted Z85, SmallGroup(85,1)

Series: Derived Chief Lower central Upper central

C1 — C85
C1C17 — C85
C1 — C85
C1 — C85

Generators and relations for C85
 G = < a | a85=1 >


Smallest permutation representation of C85
Regular action on 85 points
Generators in S85
(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)

G:=sub<Sym(85)| (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)>;

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) );

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)])

C85 is a maximal subgroup of   D85

85 conjugacy classes

class 1 5A5B5C5D17A···17P85A···85BL
order1555517···1785···85
size111111···11···1

85 irreducible representations

dim1111
type+
imageC1C5C17C85
kernelC85C17C5C1
# reps141664

Matrix representation of C85 in GL1(𝔽1021) generated by

709
G:=sub<GL(1,GF(1021))| [709] >;

C85 in GAP, Magma, Sage, TeX

C_{85}
% in TeX

G:=Group("C85");
// GroupNames label

G:=SmallGroup(85,1);
// by ID

G=gap.SmallGroup(85,1);
# by ID

G:=PCGroup([2,-5,-17]);
// Polycyclic

G:=Group<a|a^85=1>;
// generators/relations

Export

Subgroup lattice of C85 in TeX

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