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## G = C4.C42order 64 = 26

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

p-group, metabelian, nilpotent (class 3), monomial

Aliases: C4.3C42, C23.6Q8, M4(2).2C4, (C2×C8).5C4, (C2×C4).112D4, (C22×C8).3C2, C2.3(C8.C4), C4.20(C22⋊C4), C22.16(C4⋊C4), (C2×M4(2)).7C2, C2.6(C2.C42), (C22×C4).103C22, (C2×C4).64(C2×C4), SmallGroup(64,22)

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C4 — C4.C42
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C22×C8 — C4.C42
 Lower central C1 — C2 — C4 — C4.C42
 Upper central C1 — C2×C4 — C22×C4 — C4.C42
 Jennings C1 — C2 — C2 — C22×C4 — C4.C42

Generators and relations for C4.C42
G = < a,b,c | a4=1, b4=c4=a2, bab-1=a-1, ac=ca, cbc-1=a-1b >

Character table of C4.C42

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E 4F 8A 8B 8C 8D 8E 8F 8G 8H 8I 8J 8K 8L 8M 8N 8O 8P 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 -i i -i i i i -i -i linear of order 4 ρ6 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 ρ7 1 -1 -1 1 -1 1 1 1 -1 -1 1 -1 -i -i i i -i i i -i -i -1 i i -i 1 1 -1 linear of order 4 ρ8 1 -1 -1 1 1 -1 -1 -1 1 1 1 -1 -i i -i i -i -i i i -1 -i 1 -1 1 i -i i linear of order 4 ρ9 1 -1 -1 1 1 -1 -1 -1 1 1 1 -1 i -i i -i i i -i -i -1 i 1 -1 1 -i i -i linear of order 4 ρ10 1 -1 -1 1 -1 1 1 1 -1 -1 1 -1 i i -i -i i -i -i i -i 1 i i -i -1 -1 1 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 -i -i i i -i i i -i i 1 -i -i i -1 -1 1 linear of order 4 ρ13 1 -1 -1 1 1 -1 -1 -1 1 1 1 -1 i -i i -i i i -i -i 1 -i -1 1 -1 i -i i linear of order 4 ρ14 1 -1 -1 1 -1 1 1 1 -1 -1 1 -1 i i -i -i i -i -i i i -1 -i -i i 1 1 -1 linear of order 4 ρ15 1 -1 -1 1 1 -1 -1 -1 1 1 1 -1 -i i -i i -i -i i i 1 i -1 1 -1 -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 -2 -2 2 2 -2 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ18 2 2 2 2 -2 -2 2 2 2 2 -2 -2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ19 2 -2 -2 2 2 -2 2 2 -2 -2 -2 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ20 2 2 2 2 2 2 -2 -2 -2 -2 -2 -2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 symplectic lifted from Q8, Schur index 2 ρ21 2 -2 2 -2 0 0 2i -2i -2i 2i 0 0 √2 -√-2 √-2 -√2 -√2 -√-2 √2 √-2 0 0 0 0 0 0 0 0 complex lifted from C8.C4 ρ22 2 2 -2 -2 0 0 2i -2i 2i -2i 0 0 √-2 √2 √2 √-2 -√-2 -√2 -√-2 -√2 0 0 0 0 0 0 0 0 complex lifted from C8.C4 ρ23 2 2 -2 -2 0 0 -2i 2i -2i 2i 0 0 √-2 -√2 -√2 √-2 -√-2 √2 -√-2 √2 0 0 0 0 0 0 0 0 complex lifted from C8.C4 ρ24 2 -2 2 -2 0 0 -2i 2i 2i -2i 0 0 -√2 -√-2 √-2 √2 √2 -√-2 -√2 √-2 0 0 0 0 0 0 0 0 complex lifted from C8.C4 ρ25 2 2 -2 -2 0 0 -2i 2i -2i 2i 0 0 -√-2 √2 √2 -√-2 √-2 -√2 √-2 -√2 0 0 0 0 0 0 0 0 complex lifted from C8.C4 ρ26 2 2 -2 -2 0 0 2i -2i 2i -2i 0 0 -√-2 -√2 -√2 -√-2 √-2 √2 √-2 √2 0 0 0 0 0 0 0 0 complex lifted from C8.C4 ρ27 2 -2 2 -2 0 0 2i -2i -2i 2i 0 0 -√2 √-2 -√-2 √2 √2 √-2 -√2 -√-2 0 0 0 0 0 0 0 0 complex lifted from C8.C4 ρ28 2 -2 2 -2 0 0 -2i 2i 2i -2i 0 0 √2 √-2 -√-2 -√2 -√2 √-2 √2 -√-2 0 0 0 0 0 0 0 0 complex lifted from C8.C4

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

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

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

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

Matrix representation of C4.C42 in GL3(𝔽17) generated by

 1 0 0 0 4 0 0 15 13
,
 4 0 0 0 8 15 0 0 9
,
 13 0 0 0 2 0 0 10 8
`G:=sub<GL(3,GF(17))| [1,0,0,0,4,15,0,0,13],[4,0,0,0,8,0,0,15,9],[13,0,0,0,2,10,0,0,8] >;`

C4.C42 in GAP, Magma, Sage, TeX

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

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

`G:=SmallGroup(64,22);`
`// by ID`

`G=gap.SmallGroup(64,22);`
`# by ID`

`G:=PCGroup([6,-2,2,-2,2,2,-2,48,73,103,650,158,117,1444,88]);`
`// Polycyclic`

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

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