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## G = D8⋊D4order 128 = 27

### The semidirect product of D8 and D4 acting via D4/C2=C22

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C8 — D8⋊D4
 Chief series C1 — C2 — C4 — C2×C4 — C2×C8 — C2×M4(2) — D8⋊C22 — D8⋊D4
 Lower central C1 — C2 — C4 — C2×C8 — D8⋊D4
 Upper central C1 — C2 — C2×C4 — C2×M4(2) — D8⋊D4
 Jennings C1 — C2 — C2 — C2 — C2 — C4 — C4 — C2×C8 — D8⋊D4

Generators and relations for D8⋊D4
G = < a,b,c,d | a8=b2=c4=d2=1, bab=dad=a-1, cac-1=a3, cbc-1=ab, dbd=a5b, dcd=c-1 >

Subgroups: 324 in 113 conjugacy classes, 32 normal (18 characteristic)
C1, C2, C2 [×5], C4 [×2], C4 [×4], C22, C22 [×9], C8 [×2], C8, C2×C4 [×2], C2×C4 [×8], D4 [×11], Q8 [×3], C23, C23 [×2], C16 [×2], C22⋊C4, C4⋊C4, C2×C8 [×2], M4(2) [×2], D8 [×2], D8 [×3], SD16 [×4], Q16 [×2], Q16, C22×C4, C22×C4, C2×D4 [×4], C2×Q8, C4○D4 [×6], D4⋊C4, C4.Q8, M5(2) [×2], D16 [×2], SD32 [×2], C4⋊D4, C2×M4(2), C2×D8, C4○D8 [×2], C4○D8, C8⋊C22 [×2], C8.C22 [×2], C2×C4○D4, C23.C8, D82C4 [×2], C82D4, C16⋊C22 [×2], D8⋊C22, D8⋊D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], C23, D8 [×2], C2×D4 [×3], C22≀C2, C2×D8, C8⋊C22, C22⋊D8, D8⋊D4

Character table of D8⋊D4

 class 1 2A 2B 2C 2D 2E 2F 4A 4B 4C 4D 4E 4F 8A 8B 8C 16A 16B 16C 16D size 1 1 2 4 8 8 16 2 2 4 8 8 16 4 4 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 trivial ρ2 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 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 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 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 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 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 linear of order 2 ρ9 2 2 -2 0 2 0 0 -2 2 0 -2 0 0 2 -2 0 0 0 0 0 orthogonal lifted from D4 ρ10 2 2 2 2 0 0 0 2 2 2 0 0 0 -2 -2 -2 0 0 0 0 orthogonal lifted from D4 ρ11 2 2 -2 0 0 2 0 -2 2 0 0 -2 0 -2 2 0 0 0 0 0 orthogonal lifted from D4 ρ12 2 2 2 -2 0 0 0 2 2 -2 0 0 0 -2 -2 2 0 0 0 0 orthogonal lifted from D4 ρ13 2 2 -2 0 -2 0 0 -2 2 0 2 0 0 2 -2 0 0 0 0 0 orthogonal lifted from D4 ρ14 2 2 -2 0 0 -2 0 -2 2 0 0 2 0 -2 2 0 0 0 0 0 orthogonal lifted from D4 ρ15 2 2 2 2 0 0 0 -2 -2 -2 0 0 0 0 0 0 -√2 √2 -√2 √2 orthogonal lifted from D8 ρ16 2 2 2 -2 0 0 0 -2 -2 2 0 0 0 0 0 0 √2 √2 -√2 -√2 orthogonal lifted from D8 ρ17 2 2 2 -2 0 0 0 -2 -2 2 0 0 0 0 0 0 -√2 -√2 √2 √2 orthogonal lifted from D8 ρ18 2 2 2 2 0 0 0 -2 -2 -2 0 0 0 0 0 0 √2 -√2 √2 -√2 orthogonal lifted from D8 ρ19 4 4 -4 0 0 0 0 4 -4 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C8⋊C22 ρ20 8 -8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal faithful

Permutation representations of D8⋊D4
On 16 points - transitive group 16T385
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 5)(2 8)(4 6)(9 10 13 14)(11 16 15 12)
(2 8)(3 7)(4 6)(9 10)(11 16)(12 15)(13 14)```

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

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

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

`G:=TransitiveGroup(16,385);`

Matrix representation of D8⋊D4 in GL8(ℤ)

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

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

D8⋊D4 in GAP, Magma, Sage, TeX

`D_8\rtimes D_4`
`% in TeX`

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

`G:=SmallGroup(128,922);`
`// by ID`

`G=gap.SmallGroup(128,922);`
`# by ID`

`G:=PCGroup([7,-2,2,2,-2,2,-2,-2,141,422,1123,570,521,360,1411,4037,2028,124]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^8=b^2=c^4=d^2=1,b*a*b=d*a*d=a^-1,c*a*c^-1=a^3,c*b*c^-1=a*b,d*b*d=a^5*b,d*c*d=c^-1>;`
`// generators/relations`

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