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## G = C108order 108 = 22·33

### Cyclic group

Aliases: C108, also denoted Z108, SmallGroup(108,2)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C108
 Chief series C1 — C3 — C9 — C18 — C54 — C108
 Lower central C1 — C108
 Upper central C1 — C108

Generators and relations for C108
G = < a | a108=1 >

Smallest permutation representation of C108
Regular action on 108 points
Generators in S108
`(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 99 100 101 102 103 104 105 106 107 108)`

`G:=sub<Sym(108)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108)>;`

`G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108) );`

`G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108)]])`

C108 is a maximal subgroup of   C27⋊C8  Dic54  D108  Q8.C54

108 conjugacy classes

 class 1 2 3A 3B 4A 4B 6A 6B 9A ··· 9F 12A 12B 12C 12D 18A ··· 18F 27A ··· 27R 36A ··· 36L 54A ··· 54R 108A ··· 108AJ order 1 2 3 3 4 4 6 6 9 ··· 9 12 12 12 12 18 ··· 18 27 ··· 27 36 ··· 36 54 ··· 54 108 ··· 108 size 1 1 1 1 1 1 1 1 1 ··· 1 1 1 1 1 1 ··· 1 1 ··· 1 1 ··· 1 1 ··· 1 1 ··· 1

108 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 type + + image C1 C2 C3 C4 C6 C9 C12 C18 C27 C36 C54 C108 kernel C108 C54 C36 C27 C18 C12 C9 C6 C4 C3 C2 C1 # reps 1 1 2 2 2 6 4 6 18 12 18 36

Matrix representation of C108 in GL2(𝔽109) generated by

 101 0 0 97
`G:=sub<GL(2,GF(109))| [101,0,0,97] >;`

C108 in GAP, Magma, Sage, TeX

`C_{108}`
`% in TeX`

`G:=Group("C108");`
`// GroupNames label`

`G:=SmallGroup(108,2);`
`// by ID`

`G=gap.SmallGroup(108,2);`
`# by ID`

`G:=PCGroup([5,-2,-3,-2,-3,-3,30,66,78]);`
`// Polycyclic`

`G:=Group<a|a^108=1>;`
`// generators/relations`

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