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## G = C104order 104 = 23·13

### Cyclic group

Aliases: C104, also denoted Z104, SmallGroup(104,2)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C104
 Chief series C1 — C2 — C4 — C52 — C104
 Lower central C1 — C104
 Upper central C1 — C104

Generators and relations for C104
G = < a | a104=1 >

Smallest permutation representation of C104
Regular action on 104 points
Generators in S104
`(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 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104)`

`G:=sub<Sym(104)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104)>;`

`G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104) );`

`G=PermutationGroup([(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104)])`

C104 is a maximal subgroup of   C132C16  C8⋊D13  C104⋊C2  D104  Dic52

104 conjugacy classes

 class 1 2 4A 4B 8A 8B 8C 8D 13A ··· 13L 26A ··· 26L 52A ··· 52X 104A ··· 104AV order 1 2 4 4 8 8 8 8 13 ··· 13 26 ··· 26 52 ··· 52 104 ··· 104 size 1 1 1 1 1 1 1 1 1 ··· 1 1 ··· 1 1 ··· 1 1 ··· 1

104 irreducible representations

 dim 1 1 1 1 1 1 1 1 type + + image C1 C2 C4 C8 C13 C26 C52 C104 kernel C104 C52 C26 C13 C8 C4 C2 C1 # reps 1 1 2 4 12 12 24 48

Matrix representation of C104 in GL1(𝔽313) generated by

 245
`G:=sub<GL(1,GF(313))| [245] >;`

C104 in GAP, Magma, Sage, TeX

`C_{104}`
`% in TeX`

`G:=Group("C104");`
`// GroupNames label`

`G:=SmallGroup(104,2);`
`// by ID`

`G=gap.SmallGroup(104,2);`
`# by ID`

`G:=PCGroup([4,-2,-13,-2,-2,104,34]);`
`// Polycyclic`

`G:=Group<a|a^104=1>;`
`// generators/relations`

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