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## G = C22.D4order 32 = 25

### 3rd non-split extension by C22 of D4 acting via D4/C22=C2

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

Aliases: C22.4D4, C22.13C23, C23.10C22, C4⋊C44C2, C2.7(C2×D4), C22⋊C44C2, (C22×C4)⋊3C2, (C2×D4).4C2, C2.6(C4○D4), (C2×C4).13C22, SmallGroup(32,30)

Series: Derived Chief Lower central Upper central Jennings

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

Generators and relations for C22.D4
G = < a,b,c,d | a2=b2=c4=d2=1, cac-1=dad=ab=ba, bc=cb, bd=db, dcd=bc-1 >

Character table of C22.D4

 class 1 2A 2B 2C 2D 2E 2F 4A 4B 4C 4D 4E 4F 4G size 1 1 1 1 2 2 4 2 2 2 2 4 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 2 -2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ10 2 -2 2 -2 -2 2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ11 2 2 -2 -2 0 0 0 0 2i -2i 0 0 0 0 complex lifted from C4○D4 ρ12 2 2 -2 -2 0 0 0 0 -2i 2i 0 0 0 0 complex lifted from C4○D4 ρ13 2 -2 -2 2 0 0 0 2i 0 0 -2i 0 0 0 complex lifted from C4○D4 ρ14 2 -2 -2 2 0 0 0 -2i 0 0 2i 0 0 0 complex lifted from C4○D4

Permutation representations of C22.D4
On 16 points - transitive group 16T37
Generators in S16
```(1 16)(2 7)(3 14)(4 5)(6 10)(8 12)(9 15)(11 13)
(1 10)(2 11)(3 12)(4 9)(5 15)(6 16)(7 13)(8 14)
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)
(2 9)(4 11)(5 7)(6 16)(8 14)(13 15)```

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

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

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

`G:=TransitiveGroup(16,37);`

On 16 points - transitive group 16T54
Generators in S16
```(2 14)(4 16)(5 12)(7 10)
(1 13)(2 14)(3 15)(4 16)(5 12)(6 9)(7 10)(8 11)
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)
(1 10)(2 6)(3 12)(4 8)(5 15)(7 13)(9 14)(11 16)```

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

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

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

`G:=TransitiveGroup(16,54);`

C22.D4 is a maximal subgroup of
C23.36C23  C233D4  C22.32C24  C22.33C24  C22.34C24  C22.36C24  C22.45C24  C22.46C24  C22.47C24  C22.53C24  C22.54C24  C22.56C24  C22.57C24  C62.9D4
C23.D2p: C23.D4  C23.7D4  C23.9D6  C23.21D6  C23.28D6  C23.23D6  D10.12D4  C22.D20 ...
C2p.(C2×D4): C22.19C24  C23.38C23  D45D4  D46D4  D6.D4  D10.13D4  D14.5D4  D22.5D4 ...
C22.D4 is a maximal quotient of
C62.9D4
C23.D2p: C23.34D4  C23.8Q8  C23.23D4  C23.10D4  C23.11D4  C22.D8  C23.46D4  C23.19D4 ...
(C2×C4).D2p: C23.63C23  C24.C22  C23.81C23  C23.4Q8  C23.83C23  D6.D4  D10.13D4  D14.5D4 ...

Matrix representation of C22.D4 in GL4(𝔽5) generated by

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

C22.D4 in GAP, Magma, Sage, TeX

`C_2^2.D_4`
`% in TeX`

`G:=Group("C2^2.D4");`
`// GroupNames label`

`G:=SmallGroup(32,30);`
`// by ID`

`G=gap.SmallGroup(32,30);`
`# by ID`

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

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

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