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## G = C24order 16 = 24

### Elementary abelian group of type [2,2,2,2]

Aliases: C24, SmallGroup(16,14)

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C24
 Chief series C1 — C2 — C22 — C23 — C24
 Lower central C1 — C24
 Upper central C1 — C24
 Jennings C1 — C24

Generators and relations for C24
G = < a,b,c,d | a2=b2=c2=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, cd=dc >

Subgroups: 67, all normal (2 characteristic)
C1, C2 [×15], C22 [×35], C23 [×15], C24
Quotients: C1, C2 [×15], C22 [×35], C23 [×15], C24

Character table of C24

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 2J 2K 2L 2M 2N 2O size 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 -1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 1 linear of order 2 ρ3 1 -1 1 1 -1 -1 -1 -1 1 1 1 1 -1 -1 -1 1 linear of order 2 ρ4 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 1 1 1 1 linear of order 2 ρ5 1 -1 -1 -1 1 1 -1 -1 1 1 -1 -1 1 1 -1 1 linear of order 2 ρ6 1 1 -1 -1 1 1 -1 -1 -1 -1 1 1 -1 -1 1 1 linear of order 2 ρ7 1 1 -1 -1 -1 -1 1 1 1 1 -1 -1 -1 -1 1 1 linear of order 2 ρ8 1 -1 -1 -1 -1 -1 1 1 -1 -1 1 1 1 1 -1 1 linear of order 2 ρ9 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 linear of order 2 ρ10 1 1 1 -1 1 -1 1 -1 -1 1 -1 1 -1 1 -1 -1 linear of order 2 ρ11 1 1 1 -1 -1 1 -1 1 1 -1 1 -1 -1 1 -1 -1 linear of order 2 ρ12 1 -1 1 -1 -1 1 -1 1 -1 1 -1 1 1 -1 1 -1 linear of order 2 ρ13 1 1 -1 1 1 -1 -1 1 1 -1 -1 1 1 -1 -1 -1 linear of order 2 ρ14 1 -1 -1 1 1 -1 -1 1 -1 1 1 -1 -1 1 1 -1 linear of order 2 ρ15 1 -1 -1 1 -1 1 1 -1 1 -1 -1 1 -1 1 1 -1 linear of order 2 ρ16 1 1 -1 1 -1 1 1 -1 -1 1 1 -1 1 -1 -1 -1 linear of order 2

Permutation representations of C24
Regular action on 16 points - transitive group 16T3
Generators in S16
```(1 2)(3 4)(5 6)(7 8)(9 10)(11 12)(13 14)(15 16)
(1 5)(2 6)(3 11)(4 12)(7 10)(8 9)(13 15)(14 16)
(1 11)(2 12)(3 5)(4 6)(7 13)(8 14)(9 16)(10 15)
(1 13)(2 14)(3 10)(4 9)(5 15)(6 16)(7 11)(8 12)```

`G:=sub<Sym(16)| (1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16), (1,5)(2,6)(3,11)(4,12)(7,10)(8,9)(13,15)(14,16), (1,11)(2,12)(3,5)(4,6)(7,13)(8,14)(9,16)(10,15), (1,13)(2,14)(3,10)(4,9)(5,15)(6,16)(7,11)(8,12)>;`

`G:=Group( (1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16), (1,5)(2,6)(3,11)(4,12)(7,10)(8,9)(13,15)(14,16), (1,11)(2,12)(3,5)(4,6)(7,13)(8,14)(9,16)(10,15), (1,13)(2,14)(3,10)(4,9)(5,15)(6,16)(7,11)(8,12) );`

`G=PermutationGroup([(1,2),(3,4),(5,6),(7,8),(9,10),(11,12),(13,14),(15,16)], [(1,5),(2,6),(3,11),(4,12),(7,10),(8,9),(13,15),(14,16)], [(1,11),(2,12),(3,5),(4,6),(7,13),(8,14),(9,16),(10,15)], [(1,13),(2,14),(3,10),(4,9),(5,15),(6,16),(7,11),(8,12)])`

`G:=TransitiveGroup(16,3);`

C24 is a maximal subgroup of   C22≀C2  C22⋊A4  C24⋊C5
C24 is a maximal quotient of   2+ 1+4  2- 1+4

Matrix representation of C24 in GL4(ℤ) generated by

 1 0 0 0 0 -1 0 0 0 0 1 0 0 0 0 -1
,
 -1 0 0 0 0 1 0 0 0 0 -1 0 0 0 0 1
,
 1 0 0 0 0 1 0 0 0 0 -1 0 0 0 0 1
,
 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 -1
`G:=sub<GL(4,Integers())| [1,0,0,0,0,-1,0,0,0,0,1,0,0,0,0,-1],[-1,0,0,0,0,1,0,0,0,0,-1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,-1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,1,0,0,0,0,-1] >;`

C24 in GAP, Magma, Sage, TeX

`C_2^4`
`% in TeX`

`G:=Group("C2^4");`
`// GroupNames label`

`G:=SmallGroup(16,14);`
`// by ID`

`G=gap.SmallGroup(16,14);`
`# by ID`

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

`G:=Group<a,b,c,d|a^2=b^2=c^2=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,c*d=d*c>;`
`// generators/relations`

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