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G = C90order 90 = 2·32·5

Cyclic group

direct product, cyclic, abelian, monomial

Aliases: C90, also denoted Z90, SmallGroup(90,4)

Series: Derived Chief Lower central Upper central

C1 — C90
C1C3C15C45 — C90
C1 — C90
C1 — C90

Generators and relations for C90
 G = < a | a90=1 >


Smallest permutation representation of C90
Regular action on 90 points
Generators in S90
(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 90)

G:=sub<Sym(90)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90)>;

G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90) );

G=PermutationGroup([(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90)])

C90 is a maximal subgroup of   Dic45

90 conjugacy classes

class 1  2 3A3B5A5B5C5D6A6B9A···9F10A10B10C10D15A···15H18A···18F30A···30H45A···45X90A···90X
order12335555669···91010101015···1518···1830···3045···4590···90
size11111111111···111111···11···11···11···11···1

90 irreducible representations

dim111111111111
type++
imageC1C2C3C5C6C9C10C15C18C30C45C90
kernelC90C45C30C18C15C10C9C6C5C3C2C1
# reps11242648682424

Matrix representation of C90 in GL2(𝔽19) generated by

010
97
G:=sub<GL(2,GF(19))| [0,9,10,7] >;

C90 in GAP, Magma, Sage, TeX

C_{90}
% in TeX

G:=Group("C90");
// GroupNames label

G:=SmallGroup(90,4);
// by ID

G=gap.SmallGroup(90,4);
# by ID

G:=PCGroup([4,-2,-3,-5,-3,125]);
// Polycyclic

G:=Group<a|a^90=1>;
// generators/relations

Export

Subgroup lattice of C90 in TeX

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