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G = C76order 76 = 22·19

Cyclic group

direct product, cyclic, abelian, monomial

Aliases: C76, also denoted Z76, SmallGroup(76,2)

Series: Derived Chief Lower central Upper central

C1 — C76
C1C2C38 — C76
C1 — C76
C1 — C76

Generators and relations for C76
 G = < a | a76=1 >


Smallest permutation representation of C76
Regular action on 76 points
Generators in S76
(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)

G:=sub<Sym(76)| (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)>;

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) );

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)])

C76 is a maximal subgroup of   C19⋊C8  Dic38  D76

76 conjugacy classes

class 1  2 4A4B19A···19R38A···38R76A···76AJ
order124419···1938···3876···76
size11111···11···11···1

76 irreducible representations

dim111111
type++
imageC1C2C4C19C38C76
kernelC76C38C19C4C2C1
# reps112181836

Matrix representation of C76 in GL1(𝔽229) generated by

52
G:=sub<GL(1,GF(229))| [52] >;

C76 in GAP, Magma, Sage, TeX

C_{76}
% in TeX

G:=Group("C76");
// GroupNames label

G:=SmallGroup(76,2);
// by ID

G=gap.SmallGroup(76,2);
# by ID

G:=PCGroup([3,-2,-19,-2,114]);
// Polycyclic

G:=Group<a|a^76=1>;
// generators/relations

Export

Subgroup lattice of C76 in TeX

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