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## G = C4.4D4order 32 = 25

### 4th non-split extension by C4 of D4 acting via D4/C4=C2

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

Aliases: C4.4D4, C425C2, C23.3C22, C22.14C23, (C2×Q8)⋊2C2, C2.8(C2×D4), C22⋊C45C2, (C2×D4).5C2, C2.7(C4○D4), (C2×C4).22C22, SmallGroup(32,31)

Series: Derived Chief Lower central Upper central Jennings

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

Generators and relations for C4.4D4
G = < a,b,c | a4=b4=1, c2=a2, ab=ba, cac-1=a-1, cbc-1=a2b-1 >

Character table of C4.4D4

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E 4F 4G 4H size 1 1 1 1 4 4 2 2 2 2 2 2 4 4 ρ1 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 linear of order 2 ρ3 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 linear of order 2 ρ5 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 linear of order 2 ρ7 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 linear of order 2 ρ9 2 2 -2 -2 0 0 -2 0 2 0 0 0 0 0 orthogonal lifted from D4 ρ10 2 2 -2 -2 0 0 2 0 -2 0 0 0 0 0 orthogonal lifted from D4 ρ11 2 -2 2 -2 0 0 0 0 0 -2i 2i 0 0 0 complex lifted from C4○D4 ρ12 2 -2 2 -2 0 0 0 0 0 2i -2i 0 0 0 complex lifted from C4○D4 ρ13 2 -2 -2 2 0 0 0 2i 0 0 0 -2i 0 0 complex lifted from C4○D4 ρ14 2 -2 -2 2 0 0 0 -2i 0 0 0 2i 0 0 complex lifted from C4○D4

Permutation representations of C4.4D4
On 16 points - transitive group 16T30
Generators in S16
```(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)
(1 10 15 5)(2 11 16 6)(3 12 13 7)(4 9 14 8)
(1 6 3 8)(2 5 4 7)(9 15 11 13)(10 14 12 16)```

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

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

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

`G:=TransitiveGroup(16,30);`

Matrix representation of C4.4D4 in GL4(𝔽5) generated by

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

C4.4D4 in GAP, Magma, Sage, TeX

`C_4._4D_4`
`% in TeX`

`G:=Group("C4.4D4");`
`// GroupNames label`

`G:=SmallGroup(32,31);`
`// by ID`

`G=gap.SmallGroup(32,31);`
`# by ID`

`G:=PCGroup([5,-2,2,2,-2,2,101,86,302,42]);`
`// Polycyclic`

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

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