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G = C49  order 49 = 72

Cyclic group

p-group, cyclic, abelian, monomial

Aliases: C49, also denoted Z49, SmallGroup(49,1)

Series: Derived ►Chief ►Lower central ►Upper central ►Jennings

Derived series
C1 — C49
Chief series
C1 — C7 — C49
Lower central
C1 — C49
Upper central
C1 — C49
Jennings
C1 — C7 — C7 — C7 — C7 — C7 — C7 — C49

Generators and relations for C49
 G = < a | a49=1 >

C1
C7
C49

Smallest permutation representation of C49
►Regular action on 49 points
Generators in S49
(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)
 
G:=sub<Sym(49)| (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)>;
 
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) );
 
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)]])
 

C49 is a maximal subgroup of   D49  C49⋊C3  C343  7- 1+2  C7.F8
C49 is a maximal quotient of   C343  C7.F8

49 conjugacy classes

class 1 7A···7F49A···49AP
order17···749···49
size11···11···1

49 irreducible representations

dim111
type+
imageC1C7C49
kernelC49C7C1
# reps1642

Matrix representation of C49 ►in GL1(𝔽197) generated by

193
G:=sub<GL(1,GF(197))| [193] >;
 

C49 in GAP, Magma, Sage, TeX 

C_{49}
 
% in TeX
 
G:=Group("C49");
 
// GroupNames label
 
G:=SmallGroup(49,1);
 
// by ID
 
G=gap.SmallGroup(49,1);
 
# by ID
 
G:=PCGroup([2,-7,-7,14]:ExponentLimit:=1);
 
// Polycyclic
 
G:=Group<a|a^49=1>;
 
// generators/relations
 

Export

Subgroup lattice of C49 in TeX

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