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G = C70order 70 = 2·5·7

Cyclic group

direct product, cyclic, abelian, monomial

Aliases: C70, also denoted Z70, SmallGroup(70,4)

Series: Derived Chief Lower central Upper central

C1 — C70
C1C7C35 — C70
C1 — C70
C1 — C70

Generators and relations for C70
 G = < a | a70=1 >


Smallest permutation representation of C70
Regular action on 70 points
Generators in S70
(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)

G:=sub<Sym(70)| (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)>;

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) );

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)])

C70 is a maximal subgroup of   Dic35

70 conjugacy classes

class 1  2 5A5B5C5D7A···7F10A10B10C10D14A···14F35A···35X70A···70X
order1255557···71010101014···1435···3570···70
size1111111···111111···11···11···1

70 irreducible representations

dim11111111
type++
imageC1C2C5C7C10C14C35C70
kernelC70C35C14C10C7C5C2C1
# reps1146462424

Matrix representation of C70 in GL1(𝔽71) generated by

62
G:=sub<GL(1,GF(71))| [62] >;

C70 in GAP, Magma, Sage, TeX

C_{70}
% in TeX

G:=Group("C70");
// GroupNames label

G:=SmallGroup(70,4);
// by ID

G=gap.SmallGroup(70,4);
# by ID

G:=PCGroup([3,-2,-5,-7]);
// Polycyclic

G:=Group<a|a^70=1>;
// generators/relations

Export

Subgroup lattice of C70 in TeX

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