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G = D8⋊11D4order 128 = 27

5th semidirect product of D8 and D4 acting via D4/C22=C2

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C4 — D8⋊11D4
 Chief series C1 — C2 — C4 — C2×C4 — C4○D4 — 2+ 1+4 — D4○D8 — D8⋊11D4
 Lower central C1 — C2 — C2×C4 — D8⋊11D4
 Upper central C1 — C2 — C2×C4 — D8⋊11D4
 Jennings C1 — C2 — C2 — C2×C4 — D8⋊11D4

Generators and relations for D811D4
G = < a,b,c,d | a8=b2=c4=d2=1, bab=a-1, cac-1=a5, dad=a3, cbc-1=dbd=a2b, dcd=c-1 >

Subgroups: 524 in 232 conjugacy classes, 92 normal (18 characteristic)
C1, C2, C2 [×8], C4 [×2], C4 [×6], C22, C22 [×13], C8 [×4], C8 [×4], C2×C4, C2×C4 [×11], D4 [×4], D4 [×19], Q8 [×4], Q8 [×3], C23 [×6], C42, C22⋊C4, C2×C8 [×2], C2×C8 [×4], M4(2) [×4], M4(2) [×4], D8 [×2], D8 [×10], SD16 [×4], SD16 [×12], Q16 [×2], Q16 [×2], C2×D4, C2×D4 [×2], C2×D4 [×10], C2×Q8, C2×Q8 [×2], C4○D4 [×4], C4○D4 [×9], C8⋊C4, C4.D4, C4.D4 [×2], C4.10D4, C4≀C2 [×4], C8.C4 [×2], C4.4D4, C41D4, C8○D4 [×4], C2×D8 [×2], C2×D8 [×2], C2×SD16 [×2], C2×SD16 [×2], C4○D8 [×2], C4○D8 [×4], C8⋊C22 [×6], C8⋊C22 [×6], C8.C22 [×2], C8.C22 [×2], 2+ 1+4, 2+ 1+4 [×2], 2- 1+4, C8.26D4 [×2], D44D4 [×2], D4.8D4, D4.9D4, D4.3D4 [×2], D4.4D4 [×2], C83D4, D4○D8 [×2], D4○SD16 [×2], D811D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×8], C23 [×15], C2×D4 [×12], C24, C22×D4 [×2], 2+ 1+4, D42, D811D4

Character table of D811D4

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 4A 4B 4C 4D 4E 4F 4G 4H 8A 8B 8C 8D 8E 8F 8G 8H size 1 1 2 4 4 4 4 8 8 8 2 2 4 4 4 4 8 8 4 4 4 4 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 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 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 -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 -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 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 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 -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 -1 -1 1 linear of order 2 ρ9 1 1 1 -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 ρ10 1 1 1 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 ρ11 1 1 1 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 ρ12 1 1 1 -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 ρ13 1 1 1 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 ρ14 1 1 1 -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 ρ15 1 1 1 -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 ρ16 1 1 1 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 ρ17 2 2 -2 0 0 2 2 0 0 0 -2 2 0 -2 0 -2 0 0 0 -2 2 0 0 0 0 0 orthogonal lifted from D4 ρ18 2 2 -2 0 0 -2 2 0 0 0 -2 2 0 -2 0 2 0 0 0 2 -2 0 0 0 0 0 orthogonal lifted from D4 ρ19 2 2 -2 -2 2 0 0 0 0 0 2 -2 -2 0 2 0 0 0 -2 0 0 2 0 0 0 0 orthogonal lifted from D4 ρ20 2 2 -2 2 -2 0 0 0 0 0 2 -2 2 0 -2 0 0 0 -2 0 0 2 0 0 0 0 orthogonal lifted from D4 ρ21 2 2 -2 -2 -2 0 0 0 0 0 2 -2 2 0 2 0 0 0 2 0 0 -2 0 0 0 0 orthogonal lifted from D4 ρ22 2 2 -2 2 2 0 0 0 0 0 2 -2 -2 0 -2 0 0 0 2 0 0 -2 0 0 0 0 orthogonal lifted from D4 ρ23 2 2 -2 0 0 2 -2 0 0 0 -2 2 0 2 0 -2 0 0 0 2 -2 0 0 0 0 0 orthogonal lifted from D4 ρ24 2 2 -2 0 0 -2 -2 0 0 0 -2 2 0 2 0 2 0 0 0 -2 2 0 0 0 0 0 orthogonal lifted from D4 ρ25 4 4 4 0 0 0 0 0 0 0 -4 -4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from 2+ 1+4 ρ26 8 -8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal faithful

Permutation representations of D811D4
On 16 points - transitive group 16T308
Generators in S16
```(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)
(1 11)(2 10)(3 9)(4 16)(5 15)(6 14)(7 13)(8 12)
(2 6)(4 8)(9 11 13 15)(10 16 14 12)
(2 4)(3 7)(6 8)(9 11)(10 14)(13 15)```

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

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

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

`G:=TransitiveGroup(16,308);`

Matrix representation of D811D4 in GL8(ℤ)

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

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

D811D4 in GAP, Magma, Sage, TeX

`D_8\rtimes_{11}D_4`
`% in TeX`

`G:=Group("D8:11D4");`
`// GroupNames label`

`G:=SmallGroup(128,2020);`
`// by ID`

`G=gap.SmallGroup(128,2020);`
`# by ID`

`G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,758,346,2804,1411,375,172,4037,1027,124]);`
`// Polycyclic`

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

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