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G = C68order 68 = 22·17

Cyclic group

direct product, cyclic, abelian, monomial

Aliases: C68, also denoted Z68, SmallGroup(68,2)

Series: Derived Chief Lower central Upper central

C1 — C68
C1C2C34 — C68
C1 — C68
C1 — C68

Generators and relations for C68
 G = < a | a68=1 >


Smallest permutation representation of C68
Regular action on 68 points
Generators in S68
(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)

G:=sub<Sym(68)| (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)>;

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) );

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)])

C68 is a maximal subgroup of   C173C8  Dic34  D68

68 conjugacy classes

class 1  2 4A4B17A···17P34A···34P68A···68AF
order124417···1734···3468···68
size11111···11···11···1

68 irreducible representations

dim111111
type++
imageC1C2C4C17C34C68
kernelC68C34C17C4C2C1
# reps112161632

Matrix representation of C68 in GL1(𝔽137) generated by

118
G:=sub<GL(1,GF(137))| [118] >;

C68 in GAP, Magma, Sage, TeX

C_{68}
% in TeX

G:=Group("C68");
// GroupNames label

G:=SmallGroup(68,2);
// by ID

G=gap.SmallGroup(68,2);
# by ID

G:=PCGroup([3,-2,-17,-2,102]);
// Polycyclic

G:=Group<a|a^68=1>;
// generators/relations

Export

Subgroup lattice of C68 in TeX

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