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G = C75order 75 = 3·52

Cyclic group

direct product, cyclic, abelian, monomial

Aliases: C75, also denoted Z75, SmallGroup(75,1)

Series: Derived Chief Lower central Upper central

C1 — C75
C1C5C25 — C75
C1 — C75
C1 — C75

Generators and relations for C75
 G = < a | a75=1 >


Smallest permutation representation of C75
Regular action on 75 points
Generators in S75
(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)

G:=sub<Sym(75)| (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)>;

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) );

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)])

C75 is a maximal subgroup of   D75

75 conjugacy classes

class 1 3A3B5A5B5C5D15A···15H25A···25T75A···75AN
order133555515···1525···2575···75
size11111111···11···11···1

75 irreducible representations

dim111111
type+
imageC1C3C5C15C25C75
kernelC75C25C15C5C3C1
# reps12482040

Matrix representation of C75 in GL1(𝔽151) generated by

47
G:=sub<GL(1,GF(151))| [47] >;

C75 in GAP, Magma, Sage, TeX

C_{75}
% in TeX

G:=Group("C75");
// GroupNames label

G:=SmallGroup(75,1);
// by ID

G=gap.SmallGroup(75,1);
# by ID

G:=PCGroup([3,-3,-5,-5,34]);
// Polycyclic

G:=Group<a|a^75=1>;
// generators/relations

Export

Subgroup lattice of C75 in TeX

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