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## G = D8⋊3D4order 128 = 27

### 2nd semidirect product of D8 and D4 acting via D4/C4=C2

p-group, metabelian, nilpotent (class 4), monomial

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C8 — D8⋊3D4
 Chief series C1 — C2 — C4 — C8 — C2×C8 — C4○D8 — C8○D8 — D8⋊3D4
 Lower central C1 — C2 — C4 — C2×C8 — D8⋊3D4
 Upper central C1 — C2 — C2×C4 — C4×C8 — D8⋊3D4
 Jennings C1 — C2 — C2 — C2 — C2 — C4 — C4 — C2×C8 — D8⋊3D4

Generators and relations for D83D4
G = < a,b,c,d | a8=b2=c4=d2=1, bab=dad=a-1, ac=ca, cbc-1=a2b, dbd=a-1b, dcd=c-1 >

Subgroups: 268 in 85 conjugacy classes, 30 normal (22 characteristic)
C1, C2, C2 [×4], C4 [×2], C4 [×3], C22, C22 [×7], C8 [×4], C8, C2×C4, C2×C4 [×2], D4 [×10], Q8, C23 [×2], C16 [×2], C42, C2×C8 [×2], C2×C8, M4(2) [×2], D8, D8 [×6], SD16, Q16, C2×D4 [×4], C4○D4, C4×C8, C4≀C2, C8.C4, M5(2) [×2], D16 [×2], SD32 [×2], C41D4, C8○D4, C2×D8 [×2], C2×D8, C4○D8, M5(2)⋊C2 [×2], C8.C8, C8○D8, C84D4, C16⋊C22 [×2], D83D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×4], C23, D8 [×2], C2×D4 [×2], C4○D4, C4⋊D4, C2×D8, C8⋊C22, C4⋊D8, D83D4

Character table of D83D4

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E 8A 8B 8C 8D 8E 8F 8G 8H 16A 16B 16C 16D size 1 1 2 8 16 16 2 2 4 4 8 2 2 2 2 4 4 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 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 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 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 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 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 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 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 linear of order 2 ρ9 2 2 -2 2 0 0 -2 2 0 0 -2 0 2 0 2 -2 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ10 2 2 2 0 0 0 2 2 2 2 0 -2 -2 -2 -2 -2 -2 0 0 0 0 0 0 orthogonal lifted from D4 ρ11 2 2 2 0 0 0 2 2 -2 -2 0 2 -2 2 -2 -2 2 0 0 0 0 0 0 orthogonal lifted from D4 ρ12 2 2 -2 -2 0 0 -2 2 0 0 2 0 2 0 2 -2 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ13 2 2 -2 0 0 0 2 -2 0 0 0 -2 0 -2 0 0 2 0 0 -√2 √2 -√2 √2 orthogonal lifted from D8 ρ14 2 2 -2 0 0 0 2 -2 0 0 0 -2 0 -2 0 0 2 0 0 √2 -√2 √2 -√2 orthogonal lifted from D8 ρ15 2 2 -2 0 0 0 2 -2 0 0 0 2 0 2 0 0 -2 0 0 √2 -√2 -√2 √2 orthogonal lifted from D8 ρ16 2 2 -2 0 0 0 2 -2 0 0 0 2 0 2 0 0 -2 0 0 -√2 √2 √2 -√2 orthogonal lifted from D8 ρ17 2 2 -2 0 0 0 -2 2 0 0 0 0 -2 0 -2 2 0 -2i 2i 0 0 0 0 complex lifted from C4○D4 ρ18 2 2 -2 0 0 0 -2 2 0 0 0 0 -2 0 -2 2 0 2i -2i 0 0 0 0 complex lifted from C4○D4 ρ19 4 4 4 0 0 0 -4 -4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C8⋊C22 ρ20 4 -4 0 0 0 0 0 0 2 -2 0 -2√2 2√2 2√2 -2√2 0 0 0 0 0 0 0 0 orthogonal faithful ρ21 4 -4 0 0 0 0 0 0 2 -2 0 2√2 -2√2 -2√2 2√2 0 0 0 0 0 0 0 0 orthogonal faithful ρ22 4 -4 0 0 0 0 0 0 -2 2 0 2√2 2√2 -2√2 -2√2 0 0 0 0 0 0 0 0 orthogonal faithful ρ23 4 -4 0 0 0 0 0 0 -2 2 0 -2√2 -2√2 2√2 2√2 0 0 0 0 0 0 0 0 orthogonal faithful

Permutation representations of D83D4
On 16 points - transitive group 16T379
Generators in S16
```(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)
(1 12)(2 11)(3 10)(4 9)(5 16)(6 15)(7 14)(8 13)
(1 7 5 3)(2 8 6 4)(9 13)(10 14)(11 15)(12 16)
(1 3)(4 8)(5 7)(9 14)(10 13)(11 12)(15 16)```

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

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

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

`G:=TransitiveGroup(16,379);`

Matrix representation of D83D4 in GL4(𝔽7) generated by

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

D83D4 in GAP, Magma, Sage, TeX

`D_8\rtimes_3D_4`
`% in TeX`

`G:=Group("D8:3D4");`
`// GroupNames label`

`G:=SmallGroup(128,945);`
`// by ID`

`G=gap.SmallGroup(128,945);`
`# by ID`

`G:=PCGroup([7,-2,2,2,-2,2,-2,-2,141,512,422,2019,248,1684,438,242,4037,1027,124]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^8=b^2=c^4=d^2=1,b*a*b=d*a*d=a^-1,a*c=c*a,c*b*c^-1=a^2*b,d*b*d=a^-1*b,d*c*d=c^-1>;`
`// generators/relations`

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