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## G = C112order 112 = 24·7

### Cyclic group

Aliases: C112, also denoted Z112, SmallGroup(112,2)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C112
 Chief series C1 — C2 — C4 — C8 — C56 — C112
 Lower central C1 — C112
 Upper central C1 — C112

Generators and relations for C112
G = < a | a112=1 >

Smallest permutation representation of C112
Regular action on 112 points
Generators in S112
`(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 99 100 101 102 103 104 105 106 107 108 109 110 111 112)`

`G:=sub<Sym(112)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112)>;`

`G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112) );`

`G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112)]])`

C112 is a maximal subgroup of   C7⋊C32  C16⋊D7  D112  C112⋊C2  Dic56

112 conjugacy classes

 class 1 2 4A 4B 7A ··· 7F 8A 8B 8C 8D 14A ··· 14F 16A ··· 16H 28A ··· 28L 56A ··· 56X 112A ··· 112AV order 1 2 4 4 7 ··· 7 8 8 8 8 14 ··· 14 16 ··· 16 28 ··· 28 56 ··· 56 112 ··· 112 size 1 1 1 1 1 ··· 1 1 1 1 1 1 ··· 1 1 ··· 1 1 ··· 1 1 ··· 1 1 ··· 1

112 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 type + + image C1 C2 C4 C7 C8 C14 C16 C28 C56 C112 kernel C112 C56 C28 C16 C14 C8 C7 C4 C2 C1 # reps 1 1 2 6 4 6 8 12 24 48

Matrix representation of C112 in GL2(𝔽41) generated by

 14 23 39 25
`G:=sub<GL(2,GF(41))| [14,39,23,25] >;`

C112 in GAP, Magma, Sage, TeX

`C_{112}`
`% in TeX`

`G:=Group("C112");`
`// GroupNames label`

`G:=SmallGroup(112,2);`
`// by ID`

`G=gap.SmallGroup(112,2);`
`# by ID`

`G:=PCGroup([5,-2,-7,-2,-2,-2,70,42,58]);`
`// Polycyclic`

`G:=Group<a|a^112=1>;`
`// generators/relations`

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