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## G = C42order 16 = 24

### Abelian group of type [4,4]

Aliases: C42, SmallGroup(16,2)

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C42
 Chief series C1 — C2 — C22 — C2×C4 — C42
 Lower central C1 — C42
 Upper central C1 — C42
 Jennings C1 — C22 — C42

Generators and relations for C42
G = < a,b | a4=b4=1, ab=ba >

Character table of C42

 class 1 2A 2B 2C 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 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 -i i i 1 -1 -1 i -i -i i -i 1 linear of order 4 ρ6 1 1 -1 -1 i -i -i 1 -1 -1 -i i i -i i 1 linear of order 4 ρ7 1 -1 -1 1 -1 i -1 -i -i i -i -i i 1 1 i linear of order 4 ρ8 1 -1 -1 1 1 -i 1 -i -i i i i -i -1 -1 i linear of order 4 ρ9 1 -1 1 -1 -i 1 i -i i -i -1 1 -1 -i i i linear of order 4 ρ10 1 1 -1 -1 i i -i -1 1 1 i -i -i -i i -1 linear of order 4 ρ11 1 -1 1 -1 i -1 -i -i i -i 1 -1 1 i -i i linear of order 4 ρ12 1 1 -1 -1 -i -i i -1 1 1 -i i i i -i -1 linear of order 4 ρ13 1 -1 1 -1 i 1 -i i -i i -1 1 -1 i -i -i linear of order 4 ρ14 1 -1 -1 1 1 i 1 i i -i -i -i i -1 -1 -i linear of order 4 ρ15 1 -1 1 -1 -i -1 i i -i i 1 -1 1 -i i -i linear of order 4 ρ16 1 -1 -1 1 -1 -i -1 i i -i i i -i 1 1 -i linear of order 4

Permutation representations of C42
Regular action on 16 points - transitive group 16T4
Generators in S16
```(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)
(1 10 15 7)(2 11 16 8)(3 12 13 5)(4 9 14 6)```

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

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

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

`G:=TransitiveGroup(16,4);`

C42 is a maximal subgroup of
C8⋊C4  C4≀C2  C4⋊C8  C42⋊C2  C4.4D4  C42.C2  C422C2  C41D4  C4⋊Q8  C42⋊C3
C42 is a maximal quotient of
C2.C42  C8⋊C4

Matrix representation of C42 in GL2(𝔽5) generated by

 4 0 0 3
,
 3 0 0 2
`G:=sub<GL(2,GF(5))| [4,0,0,3],[3,0,0,2] >;`

C42 in GAP, Magma, Sage, TeX

`C_4^2`
`% in TeX`

`G:=Group("C4^2");`
`// GroupNames label`

`G:=SmallGroup(16,2);`
`// by ID`

`G=gap.SmallGroup(16,2);`
`# by ID`

`G:=PCGroup([4,-2,2,-2,2,16,37]);`
`// Polycyclic`

`G:=Group<a,b|a^4=b^4=1,a*b=b*a>;`
`// generators/relations`

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