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G = C78order 78 = 2·3·13

Cyclic group

direct product, cyclic, abelian, monomial

Aliases: C78, also denoted Z78, SmallGroup(78,6)

Series: Derived Chief Lower central Upper central

C1 — C78
C1C13C39 — C78
C1 — C78
C1 — C78

Generators and relations for C78
 G = < a | a78=1 >


Smallest permutation representation of C78
Regular action on 78 points
Generators in S78
(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)

G:=sub<Sym(78)| (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)>;

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) );

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)])

C78 is a maximal subgroup of   Dic39

78 conjugacy classes

class 1  2 3A3B6A6B13A···13L26A···26L39A···39X78A···78X
order12336613···1326···2639···3978···78
size1111111···11···11···11···1

78 irreducible representations

dim11111111
type++
imageC1C2C3C6C13C26C39C78
kernelC78C39C26C13C6C3C2C1
# reps112212122424

Matrix representation of C78 in GL1(𝔽79) generated by

3
G:=sub<GL(1,GF(79))| [3] >;

C78 in GAP, Magma, Sage, TeX

C_{78}
% in TeX

G:=Group("C78");
// GroupNames label

G:=SmallGroup(78,6);
// by ID

G=gap.SmallGroup(78,6);
# by ID

G:=PCGroup([3,-2,-3,-13]);
// Polycyclic

G:=Group<a|a^78=1>;
// generators/relations

Export

Subgroup lattice of C78 in TeX

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