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G = C42.C4order 64 = 26

2nd non-split extension by C42 of C4 acting faithfully

p-group, metabelian, nilpotent (class 4), monomial

Aliases: C42.2C4, (C2×C4).3D4, (C2×D4).3C4, C4.10D45C2, C4.4D4.3C2, (C2×Q8).1C22, C2.10(C23⋊C4), C22.13(C22⋊C4), (C2×C4).3(C2×C4), SmallGroup(64,36)

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C4 — C42.C4
 Chief series C1 — C2 — C22 — C2×C4 — C2×Q8 — C4.4D4 — C42.C4
 Lower central C1 — C2 — C22 — C2×C4 — C42.C4
 Upper central C1 — C2 — C22 — C2×Q8 — C42.C4
 Jennings C1 — C2 — C22 — C2×Q8 — C42.C4

Generators and relations for C42.C4
G = < a,b,c | a4=b4=1, c4=b2, ab=ba, cac-1=a-1b-1, cbc-1=a2b-1 >

Character table of C42.C4

 class 1 2A 2B 2C 4A 4B 4C 4D 4E 8A 8B 8C 8D size 1 1 2 8 4 4 4 4 4 8 8 8 8 ρ1 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 linear of order 2 ρ3 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 linear of order 2 ρ5 1 1 1 -1 1 1 -1 1 -1 -i -i i i linear of order 4 ρ6 1 1 1 1 -1 -1 -1 1 -1 -i i i -i linear of order 4 ρ7 1 1 1 1 -1 -1 -1 1 -1 i -i -i i linear of order 4 ρ8 1 1 1 -1 1 1 -1 1 -1 i i -i -i linear of order 4 ρ9 2 2 2 0 0 0 -2 -2 2 0 0 0 0 orthogonal lifted from D4 ρ10 2 2 2 0 0 0 2 -2 -2 0 0 0 0 orthogonal lifted from D4 ρ11 4 4 -4 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C23⋊C4 ρ12 4 -4 0 0 -2i 2i 0 0 0 0 0 0 0 complex faithful ρ13 4 -4 0 0 2i -2i 0 0 0 0 0 0 0 complex faithful

Permutation representations of C42.C4
On 16 points - transitive group 16T131
Generators in S16
```(2 16 6 12)(3 7)(4 10 8 14)(9 13)
(1 11 5 15)(2 16 6 12)(3 9 7 13)(4 14 8 10)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)```

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

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

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

`G:=TransitiveGroup(16,131);`

On 16 points - transitive group 16T151
Generators in S16
```(1 3 5 7)(2 10)(4 16)(6 14)(8 12)(9 15 13 11)
(1 11 5 15)(2 12 6 16)(3 9 7 13)(4 10 8 14)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)```

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

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

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

`G:=TransitiveGroup(16,151);`

C42.C4 is a maximal subgroup of
(C2×D4).135D4  C41D4.C4  C42.15D4  C42.2F5  (D4×C10).C4
(C2×Q8).D2p: C42.2D4  C8⋊C4⋊C4  (C2×D4).D4  (C4×C8)⋊C4  C4⋊Q8.C4  C42.16D4  C42.17D4  (C2×C4).D12 ...
C42.C4 is a maximal quotient of
(C2×D4)⋊C8  C42⋊C8  (C2×C4).Q16  C42.2F5  (D4×C10).C4
(C2×C4).D4p: C2.C2≀C4  (C2×C4).D12  (C2×C4).D20  (C2×C4).D28 ...
(C2×Q8).D2p: (C2×Q8).Q8  C42.Dic3  C42.Dic5  C42.Dic7 ...

Matrix representation of C42.C4 in GL4(𝔽5) generated by

 0 0 1 0 0 3 0 0 1 0 0 0 0 0 0 3
,
 0 0 2 0 0 0 0 1 2 0 0 0 0 4 0 0
,
 0 0 0 3 1 0 0 0 0 1 0 0 0 0 3 0
`G:=sub<GL(4,GF(5))| [0,0,1,0,0,3,0,0,1,0,0,0,0,0,0,3],[0,0,2,0,0,0,0,4,2,0,0,0,0,1,0,0],[0,1,0,0,0,0,1,0,0,0,0,3,3,0,0,0] >;`

C42.C4 in GAP, Magma, Sage, TeX

`C_4^2.C_4`
`% in TeX`

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

`G:=SmallGroup(64,36);`
`// by ID`

`G=gap.SmallGroup(64,36);`
`# by ID`

`G:=PCGroup([6,-2,2,-2,2,-2,-2,48,73,362,332,158,681,255,117,1444]);`
`// Polycyclic`

`G:=Group<a,b,c|a^4=b^4=1,c^4=b^2,a*b=b*a,c*a*c^-1=a^-1*b^-1,c*b*c^-1=a^2*b^-1>;`
`// generators/relations`

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