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## G = C48order 48 = 24·3

### Cyclic group

Aliases: C48, also denoted Z48, SmallGroup(48,2)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C48
 Chief series C1 — C2 — C4 — C8 — C24 — C48
 Lower central C1 — C48
 Upper central C1 — C48

Generators and relations for C48
G = < a | a48=1 >

Smallest permutation representation of C48
Regular action on 48 points
Generators in S48
`(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)`

`G:=sub<Sym(48)| (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)>;`

`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) );`

`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)]])`

C48 is a maximal subgroup of
C3⋊C32  D6.C8  D48  C48⋊C2  Dic24  C16.A4  C7⋊C48  He32C16
C48 is a maximal quotient of
C7⋊C48

48 conjugacy classes

 class 1 2 3A 3B 4A 4B 6A 6B 8A 8B 8C 8D 12A 12B 12C 12D 16A ··· 16H 24A ··· 24H 48A ··· 48P order 1 2 3 3 4 4 6 6 8 8 8 8 12 12 12 12 16 ··· 16 24 ··· 24 48 ··· 48 size 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ··· 1 1 ··· 1 1 ··· 1

48 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 type + + image C1 C2 C3 C4 C6 C8 C12 C16 C24 C48 kernel C48 C24 C16 C12 C8 C6 C4 C3 C2 C1 # reps 1 1 2 2 2 4 4 8 8 16

Matrix representation of C48 in GL2(𝔽7) generated by

 0 2 2 5
`G:=sub<GL(2,GF(7))| [0,2,2,5] >;`

C48 in GAP, Magma, Sage, TeX

`C_{48}`
`% in TeX`

`G:=Group("C48");`
`// GroupNames label`

`G:=SmallGroup(48,2);`
`// by ID`

`G=gap.SmallGroup(48,2);`
`# by ID`

`G:=PCGroup([5,-2,-3,-2,-2,-2,30,42,58]);`
`// Polycyclic`

`G:=Group<a|a^48=1>;`
`// generators/relations`

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