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## G = D4.4D4order 64 = 26

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

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

Aliases: D4.4D4, C8.19D4, Q8.4D4, M4(2).4C22, C8○D42C2, (C2×D8)⋊8C2, C8⋊C223C2, C4.59(C2×D4), C8.C46C2, C4.D44C2, (C2×C4).9C23, (C2×C8).18C22, C2.24(C4⋊D4), C4○D4.10C22, (C2×D4).20C22, C22.7(C4○D4), SmallGroup(64,153)

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C4 — D4.4D4
 Chief series C1 — C2 — C4 — C2×C4 — C4○D4 — C8○D4 — D4.4D4
 Lower central C1 — C2 — C2×C4 — D4.4D4
 Upper central C1 — C2 — C2×C4 — D4.4D4
 Jennings C1 — C2 — C2 — C2×C4 — D4.4D4

Generators and relations for D4.4D4
G = < a,b,c,d | a4=b2=d2=1, c4=a2, bab=dad=a-1, ac=ca, bc=cb, dbd=ab, dcd=a2c3 >

Character table of D4.4D4

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 8A 8B 8C 8D 8E 8F 8G size 1 1 2 4 8 8 2 2 4 2 2 4 4 4 8 8 ρ1 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 linear of order 2 ρ3 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 linear of order 2 ρ5 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 linear of order 2 ρ7 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 linear of order 2 ρ9 2 2 -2 -2 0 0 2 -2 2 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ10 2 2 -2 0 0 0 -2 2 0 2 2 -2 0 0 0 0 orthogonal lifted from D4 ρ11 2 2 -2 2 0 0 2 -2 -2 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ12 2 2 -2 0 0 0 -2 2 0 -2 -2 2 0 0 0 0 orthogonal lifted from D4 ρ13 2 2 2 0 0 0 -2 -2 0 0 0 0 -2i 2i 0 0 complex lifted from C4○D4 ρ14 2 2 2 0 0 0 -2 -2 0 0 0 0 2i -2i 0 0 complex lifted from C4○D4 ρ15 4 -4 0 0 0 0 0 0 0 -2√2 2√2 0 0 0 0 0 orthogonal faithful ρ16 4 -4 0 0 0 0 0 0 0 2√2 -2√2 0 0 0 0 0 orthogonal faithful

Permutation representations of D4.4D4
On 16 points - transitive group 16T133
Generators in S16
```(1 3 5 7)(2 4 6 8)(9 15 13 11)(10 16 14 12)
(1 13)(2 14)(3 15)(4 16)(5 9)(6 10)(7 11)(8 12)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)
(1 8)(2 7)(3 6)(4 5)(9 10)(11 16)(12 15)(13 14)```

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

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

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

`G:=TransitiveGroup(16,133);`

D4.4D4 is a maximal subgroup of
M4(2).10C23  D811D4  D8○D8  C8.3S4
M4(2).D2p: Q16.10D4  Q16.D4  D4.3D8  D4.5D8  M4(2).37D4  D12.3D4  C24.19D4  Q8.9D12 ...
D4p.D4: D8○SD16  C24.23D4  C40.23D4  C56.23D4 ...
D4.4D4 is a maximal quotient of
M4(2).D2p: M4(2).48D4  M4(2).32D4  M4(2).4D4  M4(2).10D4  M4(2).12D4  D12.3D4  C24.19D4  Q8.9D12 ...
(C2×C8).D2p: C87D8  C813SD16  D4.1Q16  Q8.1Q16  D4.2SD16  Q8.2SD16  C82D8  C82SD16 ...

Matrix representation of D4.4D4 in GL4(𝔽7) generated by

 0 6 5 1 3 0 5 6 3 3 6 1 1 6 3 1
,
 0 4 0 0 2 0 0 0 6 6 4 3 2 5 2 3
,
 5 0 2 6 6 5 5 6 6 1 1 2 2 2 6 2
,
 1 4 1 6 0 2 5 4 0 6 2 1 0 4 2 2
`G:=sub<GL(4,GF(7))| [0,3,3,1,6,0,3,6,5,5,6,3,1,6,1,1],[0,2,6,2,4,0,6,5,0,0,4,2,0,0,3,3],[5,6,6,2,0,5,1,2,2,5,1,6,6,6,2,2],[1,0,0,0,4,2,6,4,1,5,2,2,6,4,1,2] >;`

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

`D_4._4D_4`
`% in TeX`

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

`G:=SmallGroup(64,153);`
`// by ID`

`G=gap.SmallGroup(64,153);`
`# by ID`

`G:=PCGroup([6,-2,2,2,-2,2,-2,121,247,362,963,117,1444,376,88]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^4=b^2=d^2=1,c^4=a^2,b*a*b=d*a*d=a^-1,a*c=c*a,b*c=c*b,d*b*d=a*b,d*c*d=a^2*c^3>;`
`// generators/relations`

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