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## G = C42.6C4order 64 = 26

### 3rd non-split extension by C42 of C4 acting via C4/C2=C2

p-group, metabelian, nilpotent (class 2), monomial

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C22 — C42.6C4
 Chief series C1 — C2 — C4 — C2×C4 — C42 — C2×C42 — C42.6C4
 Lower central C1 — C22 — C42.6C4
 Upper central C1 — C2×C4 — C42.6C4
 Jennings C1 — C2 — C2 — C2×C4 — C42.6C4

Generators and relations for C42.6C4
G = < a,b,c | a4=b4=1, c4=a2, ab=ba, cac-1=a-1b2, cbc-1=a2b >

Character table of C42.6C4

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 4M 4N 8A 8B 8C 8D 8E 8F 8G 8H size 1 1 1 1 2 2 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 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 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 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 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 -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 -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 -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 -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 1 -1 1 1 -i -i i i i i -i -i linear of order 4 ρ10 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 1 1 1 -1 1 -1 -1 -1 i i i i -i -i -i -i linear of order 4 ρ11 1 1 1 1 -1 -1 -1 -1 -1 -1 1 1 -1 1 -1 -1 1 -1 1 1 i i -i -i -i -i i i linear of order 4 ρ12 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 1 1 1 -1 1 -1 -1 -1 -i -i -i -i i i i i linear of order 4 ρ13 1 1 1 1 1 1 -1 -1 -1 -1 1 1 -1 -1 -1 1 -1 1 -1 -1 -i i -i i i -i i -i linear of order 4 ρ14 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 1 -1 1 1 -1 1 1 1 i -i -i i -i i i -i linear of order 4 ρ15 1 1 1 1 1 1 -1 -1 -1 -1 1 1 -1 -1 -1 1 -1 1 -1 -1 i -i i -i -i i -i i linear of order 4 ρ16 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 1 -1 1 1 -1 1 1 1 -i i i -i i -i -i i linear of order 4 ρ17 2 -2 2 -2 2 -2 2i -2i 2i -2i 0 0 0 0 0 0 0 0 -2i 2i 0 0 0 0 0 0 0 0 complex lifted from M4(2) ρ18 2 -2 -2 2 0 0 -2i -2i 2i 2i 2 -2 2i 0 -2i 0 0 0 0 0 0 0 0 0 0 0 0 0 complex lifted from M4(2) ρ19 2 -2 -2 2 0 0 2i 2i -2i -2i -2 2 2i 0 -2i 0 0 0 0 0 0 0 0 0 0 0 0 0 complex lifted from M4(2) ρ20 2 -2 2 -2 2 -2 -2i 2i -2i 2i 0 0 0 0 0 0 0 0 2i -2i 0 0 0 0 0 0 0 0 complex lifted from M4(2) ρ21 2 2 -2 -2 0 0 2 -2 -2 2 0 0 0 -2i 0 2i 2i -2i 0 0 0 0 0 0 0 0 0 0 complex lifted from C4○D4 ρ22 2 -2 -2 2 0 0 -2i -2i 2i 2i -2 2 -2i 0 2i 0 0 0 0 0 0 0 0 0 0 0 0 0 complex lifted from M4(2) ρ23 2 2 -2 -2 0 0 -2 2 2 -2 0 0 0 -2i 0 -2i 2i 2i 0 0 0 0 0 0 0 0 0 0 complex lifted from C4○D4 ρ24 2 -2 2 -2 -2 2 -2i 2i -2i 2i 0 0 0 0 0 0 0 0 -2i 2i 0 0 0 0 0 0 0 0 complex lifted from M4(2) ρ25 2 -2 2 -2 -2 2 2i -2i 2i -2i 0 0 0 0 0 0 0 0 2i -2i 0 0 0 0 0 0 0 0 complex lifted from M4(2) ρ26 2 2 -2 -2 0 0 -2 2 2 -2 0 0 0 2i 0 2i -2i -2i 0 0 0 0 0 0 0 0 0 0 complex lifted from C4○D4 ρ27 2 -2 -2 2 0 0 2i 2i -2i -2i 2 -2 -2i 0 2i 0 0 0 0 0 0 0 0 0 0 0 0 0 complex lifted from M4(2) ρ28 2 2 -2 -2 0 0 2 -2 -2 2 0 0 0 2i 0 -2i -2i 2i 0 0 0 0 0 0 0 0 0 0 complex lifted from C4○D4

Smallest permutation representation of C42.6C4
On 32 points
Generators in S32
```(1 3 5 7)(2 26 6 30)(4 28 8 32)(9 22 13 18)(10 12 14 16)(11 24 15 20)(17 19 21 23)(25 27 29 31)
(1 21 27 14)(2 18 28 11)(3 23 29 16)(4 20 30 13)(5 17 31 10)(6 22 32 15)(7 19 25 12)(8 24 26 9)
(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)```

`G:=sub<Sym(32)| (1,3,5,7)(2,26,6,30)(4,28,8,32)(9,22,13,18)(10,12,14,16)(11,24,15,20)(17,19,21,23)(25,27,29,31), (1,21,27,14)(2,18,28,11)(3,23,29,16)(4,20,30,13)(5,17,31,10)(6,22,32,15)(7,19,25,12)(8,24,26,9), (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)>;`

`G:=Group( (1,3,5,7)(2,26,6,30)(4,28,8,32)(9,22,13,18)(10,12,14,16)(11,24,15,20)(17,19,21,23)(25,27,29,31), (1,21,27,14)(2,18,28,11)(3,23,29,16)(4,20,30,13)(5,17,31,10)(6,22,32,15)(7,19,25,12)(8,24,26,9), (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) );`

`G=PermutationGroup([(1,3,5,7),(2,26,6,30),(4,28,8,32),(9,22,13,18),(10,12,14,16),(11,24,15,20),(17,19,21,23),(25,27,29,31)], [(1,21,27,14),(2,18,28,11),(3,23,29,16),(4,20,30,13),(5,17,31,10),(6,22,32,15),(7,19,25,12),(8,24,26,9)], [(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)])`

Matrix representation of C42.6C4 in GL4(𝔽17) generated by

 4 0 0 0 0 4 0 0 0 0 4 0 0 0 0 13
,
 13 0 0 0 0 4 0 0 0 0 1 0 0 0 0 16
,
 0 1 0 0 4 0 0 0 0 0 0 1 0 0 4 0
`G:=sub<GL(4,GF(17))| [4,0,0,0,0,4,0,0,0,0,4,0,0,0,0,13],[13,0,0,0,0,4,0,0,0,0,1,0,0,0,0,16],[0,4,0,0,1,0,0,0,0,0,0,4,0,0,1,0] >;`

C42.6C4 in GAP, Magma, Sage, TeX

`C_4^2._6C_4`
`% in TeX`

`G:=Group("C4^2.6C4");`
`// GroupNames label`

`G:=SmallGroup(64,113);`
`// by ID`

`G=gap.SmallGroup(64,113);`
`# by ID`

`G:=PCGroup([6,-2,2,2,-2,2,-2,96,121,650,50,88]);`
`// Polycyclic`

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

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