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## G = C49order 49 = 72

### Cyclic group

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 >

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 ··· 7F 49A ··· 49AP order 1 7 ··· 7 49 ··· 49 size 1 1 ··· 1 1 ··· 1

49 irreducible representations

 dim 1 1 1 type + image C1 C7 C49 kernel C49 C7 C1 # reps 1 6 42

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`

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