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## G = D4.3D4order 64 = 26

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

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C4 — D4.3D4
 Chief series C1 — C2 — C4 — C2×C4 — C4○D4 — C8○D4 — D4.3D4
 Lower central C1 — C2 — C2×C4 — D4.3D4
 Upper central C1 — C2 — C2×C4 — D4.3D4
 Jennings C1 — C2 — C2 — C2×C4 — D4.3D4

Generators and relations for D4.3D4
G = < a,b,c,d | a4=b2=1, c4=d2=a2, bab=dad-1=a-1, ac=ca, bc=cb, dbd-1=ab, dcd-1=c3 >

Character table of D4.3D4

 class 1 2A 2B 2C 2D 4A 4B 4C 4D 8A 8B 8C 8D 8E 8F 8G size 1 1 2 4 8 2 2 4 8 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 0 0 -2 2 0 0 -2 -2 0 2 0 0 0 orthogonal lifted from D4 ρ10 2 2 -2 2 0 2 -2 -2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ11 2 2 -2 0 0 -2 2 0 0 2 2 0 -2 0 0 0 orthogonal lifted from D4 ρ12 2 2 -2 -2 0 2 -2 2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ13 2 2 2 0 0 -2 -2 0 0 0 0 2i 0 -2i 0 0 complex lifted from C4○D4 ρ14 2 2 2 0 0 -2 -2 0 0 0 0 -2i 0 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 complex faithful ρ16 4 -4 0 0 0 0 0 0 0 2√-2 -2√-2 0 0 0 0 0 complex faithful

Permutation representations of D4.3D4
On 16 points - transitive group 16T132
Generators in S16
```(1 7 5 3)(2 8 6 4)(9 11 13 15)(10 12 14 16)
(1 12)(2 13)(3 14)(4 15)(5 16)(6 9)(7 10)(8 11)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)
(1 4 5 8)(2 7 6 3)(9 12 13 16)(10 15 14 11)```

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

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

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

`G:=TransitiveGroup(16,132);`

D4.3D4 is a maximal subgroup of
M4(2).10C23  D811D4  D86D4  C8.4S4
D4p.D4: D8.13D4  D8○SD16  D12.2D4  D12.6D4  C24.42D4  C24.44D4  D20.2D4  D20.6D4 ...
M4(2).D2p: M4(2).37D4  M4(2).38D4  Q8.8D12  M4(2).13D6  M4(2).15D6  D4.3D20  M4(2).13D10  M4(2).15D10 ...
D4.3D4 is a maximal quotient of
M4(2).D2p: M4(2).48D4  M4(2).49D4  M4(2).31D4  M4(2).5D4  M4(2).10D4  M4(2).11D4  M4(2).12D4  M4(2).13D4 ...
(C2×C8).D2p: C88D8  C814SD16  C811SD16  C88Q16  D4.2D8  Q8.2D8  D4.Q16  Q8.2Q16 ...

Matrix representation of D4.3D4 in GL4(𝔽3) generated by

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

D4.3D4 in GAP, Magma, Sage, TeX

`D_4._3D_4`
`% in TeX`

`G:=Group("D4.3D4");`
`// GroupNames label`

`G:=SmallGroup(64,152);`
`// by ID`

`G=gap.SmallGroup(64,152);`
`# by ID`

`G:=PCGroup([6,-2,2,2,-2,2,-2,121,55,362,963,117,1444,376,88]);`
`// Polycyclic`

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

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