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## G = D8.D4order 128 = 27

### 1st non-split extension by D8 of 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=1, d2=a4, bab=dad-1=a-1, cac-1=a3, cbc-1=ab, dbd-1=a5b, dcd-1=a4c-1 >

Subgroups: 276 in 108 conjugacy classes, 32 normal (18 characteristic)
C1, C2, C2 [×4], C4 [×2], C4 [×5], C22, C22 [×6], C8 [×2], C8, C2×C4 [×2], C2×C4 [×9], D4 [×7], Q8 [×5], C23, C23, C16 [×2], C22⋊C4, C4⋊C4 [×2], C2×C8 [×2], M4(2) [×2], D8 [×2], D8, SD16 [×4], Q16 [×2], Q16 [×3], C22×C4, C22×C4, C2×D4 [×2], C2×Q8 [×2], C4○D4 [×6], Q8⋊C4, C4.Q8, M5(2) [×2], SD32 [×2], Q32 [×2], C22⋊Q8, C2×M4(2), C2×Q16, C4○D8 [×2], C4○D8, C8⋊C22 [×2], C8.C22 [×2], C2×C4○D4, C23.C8, D82C4 [×2], C8.D4, Q32⋊C2 [×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 4A 4B 4C 4D 4E 4F 4G 8A 8B 8C 16A 16B 16C 16D size 1 1 2 4 8 8 2 2 4 8 8 16 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 0 -2 -2 2 0 0 2 0 0 -2 2 0 0 0 0 0 orthogonal lifted from D4 ρ10 2 2 -2 0 -2 0 -2 2 0 2 0 0 0 2 -2 0 0 0 0 0 orthogonal lifted from D4 ρ11 2 2 -2 0 2 0 -2 2 0 -2 0 0 0 2 -2 0 0 0 0 0 orthogonal lifted from D4 ρ12 2 2 2 -2 0 0 2 2 -2 0 0 0 0 -2 -2 2 0 0 0 0 orthogonal lifted from D4 ρ13 2 2 2 2 0 0 2 2 2 0 0 0 0 -2 -2 -2 0 0 0 0 orthogonal lifted from D4 ρ14 2 2 -2 0 0 2 -2 2 0 0 -2 0 0 -2 2 0 0 0 0 0 orthogonal lifted from D4 ρ15 2 2 2 -2 0 0 -2 -2 2 0 0 0 0 0 0 0 -√2 -√2 √2 √2 orthogonal lifted from D8 ρ16 2 2 2 -2 0 0 -2 -2 2 0 0 0 0 0 0 0 √2 √2 -√2 -√2 orthogonal lifted from D8 ρ17 2 2 2 2 0 0 -2 -2 -2 0 0 0 0 0 0 0 -√2 √2 -√2 √2 orthogonal lifted from D8 ρ18 2 2 2 2 0 0 -2 -2 -2 0 0 0 0 0 0 0 √2 -√2 √2 -√2 orthogonal lifted from D8 ρ19 4 4 -4 0 0 0 4 -4 0 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 symplectic faithful, Schur index 2

Smallest permutation representation of D8.D4
On 32 points
Generators in S32
```(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)(17 18 19 20 21 22 23 24)(25 26 27 28 29 30 31 32)
(1 8)(2 7)(3 6)(4 5)(10 16)(11 15)(12 14)(17 20)(18 19)(21 24)(22 23)(26 32)(27 31)(28 30)
(1 12 21 26)(2 15 22 29)(3 10 23 32)(4 13 24 27)(5 16 17 30)(6 11 18 25)(7 14 19 28)(8 9 20 31)
(1 30 5 26)(2 29 6 25)(3 28 7 32)(4 27 8 31)(9 24 13 20)(10 23 14 19)(11 22 15 18)(12 21 16 17)```

`G:=sub<Sym(32)| (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32), (1,8)(2,7)(3,6)(4,5)(10,16)(11,15)(12,14)(17,20)(18,19)(21,24)(22,23)(26,32)(27,31)(28,30), (1,12,21,26)(2,15,22,29)(3,10,23,32)(4,13,24,27)(5,16,17,30)(6,11,18,25)(7,14,19,28)(8,9,20,31), (1,30,5,26)(2,29,6,25)(3,28,7,32)(4,27,8,31)(9,24,13,20)(10,23,14,19)(11,22,15,18)(12,21,16,17)>;`

`G:=Group( (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32), (1,8)(2,7)(3,6)(4,5)(10,16)(11,15)(12,14)(17,20)(18,19)(21,24)(22,23)(26,32)(27,31)(28,30), (1,12,21,26)(2,15,22,29)(3,10,23,32)(4,13,24,27)(5,16,17,30)(6,11,18,25)(7,14,19,28)(8,9,20,31), (1,30,5,26)(2,29,6,25)(3,28,7,32)(4,27,8,31)(9,24,13,20)(10,23,14,19)(11,22,15,18)(12,21,16,17) );`

`G=PermutationGroup([(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16),(17,18,19,20,21,22,23,24),(25,26,27,28,29,30,31,32)], [(1,8),(2,7),(3,6),(4,5),(10,16),(11,15),(12,14),(17,20),(18,19),(21,24),(22,23),(26,32),(27,31),(28,30)], [(1,12,21,26),(2,15,22,29),(3,10,23,32),(4,13,24,27),(5,16,17,30),(6,11,18,25),(7,14,19,28),(8,9,20,31)], [(1,30,5,26),(2,29,6,25),(3,28,7,32),(4,27,8,31),(9,24,13,20),(10,23,14,19),(11,22,15,18),(12,21,16,17)])`

Matrix representation of D8.D4 in GL8(𝔽17)

 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 16 0 1 0 1 0 15 0 0 0 0 0 0 0 16 1 0 16 0 0 0 0 0 0 1 0 0 0 0 0 0 0 16 0 1 16 0 0 16 0 0 0 0 0 0 0 16 0
,
 0 0 0 0 1 0 0 0 0 0 0 0 0 16 0 0 1 0 16 0 16 0 2 0 1 0 16 0 16 16 1 1 1 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 1 0 0 0 16 0 1 0 1 16 16 1 16 0 1 0
,
 10 16 0 1 0 0 16 10 10 7 6 10 0 0 10 1 1 16 9 1 0 0 1 7 1 0 16 0 0 0 11 8 1 0 16 7 16 7 1 10 16 0 1 16 7 1 10 16 8 0 9 7 0 0 1 7 7 0 10 16 0 0 1 7
,
 10 16 0 1 0 0 16 10 10 7 6 10 0 0 10 1 1 16 9 1 0 0 1 7 1 0 16 0 0 0 11 8 16 0 1 10 1 10 16 7 1 0 16 1 10 16 7 1 0 0 0 10 0 0 1 7 1 0 16 1 0 0 1 7

`G:=sub<GL(8,GF(17))| [0,0,16,0,0,1,16,0,0,0,0,0,16,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,0,16,0,1,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,15,16,0,0,16,16,0,0,0,1,0,0,0,0],[0,0,1,1,1,0,1,1,0,0,0,0,0,16,0,16,0,0,16,16,0,0,0,16,0,0,0,0,0,0,0,1,1,0,16,16,0,0,16,16,0,16,0,16,0,0,0,0,0,0,2,1,0,0,1,1,0,0,0,1,0,0,0,0],[10,10,1,1,1,16,8,7,16,7,16,0,0,0,0,0,0,6,9,16,16,1,9,10,1,10,1,0,7,16,7,16,0,0,0,0,16,7,0,0,0,0,0,0,7,1,0,0,16,10,1,11,1,10,1,1,10,1,7,8,10,16,7,7],[10,10,1,1,16,1,0,1,16,7,16,0,0,0,0,0,0,6,9,16,1,16,0,16,1,10,1,0,10,1,10,1,0,0,0,0,1,10,0,0,0,0,0,0,10,16,0,0,16,10,1,11,16,7,1,1,10,1,7,8,7,1,7,7] >;`

D8.D4 in GAP, Magma, Sage, TeX

`D_8.D_4`
`% in TeX`

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

`G:=SmallGroup(128,923);`
`// by ID`

`G=gap.SmallGroup(128,923);`
`# by ID`

`G:=PCGroup([7,-2,2,2,-2,2,-2,-2,448,141,422,1123,570,521,360,1411,4037,2028,124]);`
`// Polycyclic`

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

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