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## G = C4○D4.D4order 128 = 27

### 3rd non-split extension by C4○D4 of D4 acting via D4/C2=C22

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C4 — C4○D4.D4
 Chief series C1 — C2 — C22 — C23 — C22×C4 — C2×C4○D4 — C2×2+ 1+4 — C4○D4.D4
 Lower central C1 — C2 — C2×C4 — C4○D4.D4
 Upper central C1 — C2 — C22×C4 — C4○D4.D4
 Jennings C1 — C2 — C2 — C22×C4 — C4○D4.D4

Generators and relations for C4○D4.D4
G = < a,b,c,d,e | a4=c2=d4=1, b2=a2, e2=dad-1=a-1, ab=ba, ac=ca, ae=ea, cbc=ebe-1=a2b, bd=db, dcd-1=ece-1=a2bc, ede-1=a-1d-1 >

Subgroups: 684 in 294 conjugacy classes, 66 normal (20 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C23, C23, C42, C22⋊C4, C4⋊C4, C2×C8, M4(2), C22×C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×D4, C2×Q8, C4○D4, C4○D4, C24, C23⋊C4, C4.D4, C4.10D4, D4⋊C4, C4≀C2, C42⋊C2, C2×M4(2), C22×D4, C22×D4, C22×D4, C2×C4○D4, C2×C4○D4, C2×C4○D4, 2+ 1+4, C23.C23, M4(2).8C22, C23.37D4, C42⋊C22, C2×2+ 1+4, C4○D4.D4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, C22×C4, C2×D4, C2×C22⋊C4, C22≀C2, C243C4, C4○D4.D4

Character table of C4○D4.D4

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 2J 2K 2L 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 8A 8B 8C 8D size 1 1 2 2 2 4 4 4 4 4 4 4 4 2 2 2 2 4 4 4 4 8 8 8 8 8 8 8 8 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 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 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 -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 -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 -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 -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 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 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 -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 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 -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 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 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 -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 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 -1 i -i -i i -i i -i i linear of order 4 ρ17 2 2 2 -2 -2 -2 0 0 2 2 -2 0 0 -2 2 2 -2 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ18 2 2 -2 -2 2 0 0 -2 0 0 0 0 -2 -2 2 -2 2 2 0 2 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ19 2 2 2 2 2 0 0 2 0 0 0 0 -2 -2 -2 -2 -2 2 0 -2 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ20 2 2 -2 2 -2 0 -2 0 0 0 0 2 0 2 2 -2 -2 0 -2 0 2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ21 2 2 -2 2 -2 -2 0 0 -2 2 2 0 0 -2 -2 2 2 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ22 2 2 2 -2 -2 2 0 0 -2 -2 2 0 0 -2 2 2 -2 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ23 2 2 -2 2 -2 0 2 0 0 0 0 -2 0 2 2 -2 -2 0 2 0 -2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ24 2 2 -2 2 -2 2 0 0 2 -2 -2 0 0 -2 -2 2 2 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ25 2 2 2 -2 -2 0 2 0 0 0 0 -2 0 2 -2 -2 2 0 -2 0 2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ26 2 2 2 -2 -2 0 -2 0 0 0 0 2 0 2 -2 -2 2 0 2 0 -2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ27 2 2 -2 -2 2 0 0 2 0 0 0 0 2 -2 2 -2 2 -2 0 -2 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ28 2 2 2 2 2 0 0 -2 0 0 0 0 2 -2 -2 -2 -2 -2 0 2 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ29 8 -8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal faithful

Permutation representations of C4○D4.D4
On 16 points - transitive group 16T232
Generators in S16
```(1 7 5 3)(2 8 6 4)(9 15 13 11)(10 16 14 12)
(1 7 5 3)(2 4 6 8)(9 11 13 15)(10 16 14 12)
(1 13)(2 16)(3 15)(4 10)(5 9)(6 12)(7 11)(8 14)
(1 8)(2 7)(3 6)(4 5)(9 12 13 16)(10 15 14 11)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)```

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

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

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

`G:=TransitiveGroup(16,232);`

On 16 points - transitive group 16T290
Generators in S16
```(1 7 5 3)(2 8 6 4)(9 15 13 11)(10 16 14 12)
(1 9 5 13)(2 14 6 10)(3 11 7 15)(4 16 8 12)
(1 5)(2 14)(3 7)(4 16)(6 10)(8 12)
(1 14 15 8)(2 7 16 13)(3 12 9 6)(4 5 10 11)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)```

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

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

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

`G:=TransitiveGroup(16,290);`

Matrix representation of C4○D4.D4 in GL8(ℤ)

 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 -1 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 -1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0
,
 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 -1 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 -1 0 0 0 0 0 0 0 0 -1 0 0
,
 -1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 1 0 0 0 0 -1 0 0 0 0 0 0 0 0 1 0 0
,
 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0
,
 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 -1 0 0 0 -1 0 0 0 0 0 0 -1 0 0 0 0 0 0 -1 0 0 0 0 0 0 -1 0 0 0 0 0 0 0

`G:=sub<GL(8,Integers())| [0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0],[0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,0,0],[0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0],[0,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0] >;`

C4○D4.D4 in GAP, Magma, Sage, TeX

`C_4\circ D_4.D_4`
`% in TeX`

`G:=Group("C4oD4.D4");`
`// GroupNames label`

`G:=SmallGroup(128,527);`
`// by ID`

`G=gap.SmallGroup(128,527);`
`# by ID`

`G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,422,2019,1018,248,2804,1027]);`
`// Polycyclic`

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

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