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## G = C74order 74 = 2·37

### Cyclic group

Aliases: C74, also denoted Z74, SmallGroup(74,2)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C74
 Chief series C1 — C37 — C74
 Lower central C1 — C74
 Upper central C1 — C74

Generators and relations for C74
G = < a | a74=1 >

Smallest permutation representation of C74
Regular action on 74 points
Generators in S74
`(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)`

`G:=sub<Sym(74)| (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)>;`

`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) );`

`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)]])`

C74 is a maximal subgroup of   Dic37

74 conjugacy classes

 class 1 2 37A ··· 37AJ 74A ··· 74AJ order 1 2 37 ··· 37 74 ··· 74 size 1 1 1 ··· 1 1 ··· 1

74 irreducible representations

 dim 1 1 1 1 type + + image C1 C2 C37 C74 kernel C74 C37 C2 C1 # reps 1 1 36 36

Matrix representation of C74 in GL1(𝔽149) generated by

 69
`G:=sub<GL(1,GF(149))| [69] >;`

C74 in GAP, Magma, Sage, TeX

`C_{74}`
`% in TeX`

`G:=Group("C74");`
`// GroupNames label`

`G:=SmallGroup(74,2);`
`// by ID`

`G=gap.SmallGroup(74,2);`
`# by ID`

`G:=PCGroup([2,-2,-37]);`
`// Polycyclic`

`G:=Group<a|a^74=1>;`
`// generators/relations`

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