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G = D8.9D4order 128 = 27

1st non-split extension by D8 of D4 acting via D4/C22=C2

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C8 — D8.9D4
 Chief series C1 — C2 — C4 — C2×C4 — C2×C8 — C22×C8 — C22×D8 — D8.9D4
 Lower central C1 — C2 — C4 — C2×C8 — D8.9D4
 Upper central C1 — C22 — C22×C4 — C22×C8 — D8.9D4
 Jennings C1 — C2 — C2 — C2 — C2 — C4 — C4 — C2×C8 — D8.9D4

Generators and relations for D8.9D4
G = < a,b,c,d | a8=b2=1, c4=a2, d2=a4, bab=dad-1=a-1, ac=ca, cbc-1=ab, dbd-1=a5b, dcd-1=a2c3 >

Subgroups: 404 in 128 conjugacy classes, 36 normal (20 characteristic)
C1, C2 [×3], C2 [×6], C4 [×2], C4 [×3], C22, C22 [×2], C22 [×18], C8 [×2], C8, C2×C4 [×2], C2×C4 [×4], D4 [×10], Q8 [×2], C23, C23 [×10], C16 [×2], C22⋊C4, C4⋊C4 [×2], C2×C8 [×2], C2×C8 [×2], D8 [×4], D8 [×6], Q16 [×2], C22×C4, C2×D4 [×9], C2×Q8, C24, Q8⋊C4, C2.D8, C2×C16 [×2], SD32 [×4], C22⋊Q8, C22×C8, C2×D8 [×2], C2×D8 [×5], C2×Q16, C22×D4, C22⋊C16, C2.D16 [×2], C8.18D4, C2×SD32 [×2], C22×D8, D8.9D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], C23, D8 [×2], C2×D4 [×3], SD32 [×2], C22≀C2, C2×D8, C8⋊C22, C22⋊D8, C2×SD32, C16⋊C22, D8.9D4

Character table of D8.9D4

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 4A 4B 4C 4D 4E 8A 8B 8C 8D 8E 8F 16A 16B 16C 16D 16E 16F 16G 16H size 1 1 1 1 2 2 8 8 8 8 2 2 4 16 16 2 2 2 2 4 4 4 4 4 4 4 4 4 4 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 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 -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 -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 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 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 -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 -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 1 1 1 linear of order 2 ρ9 2 -2 -2 2 0 0 0 0 -2 2 2 -2 0 0 0 -2 2 2 -2 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ10 2 -2 -2 2 0 0 0 0 2 -2 2 -2 0 0 0 -2 2 2 -2 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ11 2 -2 -2 2 0 0 -2 2 0 0 2 -2 0 0 0 2 -2 -2 2 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ12 2 -2 -2 2 0 0 2 -2 0 0 2 -2 0 0 0 2 -2 -2 2 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ13 2 2 2 2 -2 -2 0 0 0 0 2 2 -2 0 0 -2 -2 -2 -2 2 2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ14 2 2 2 2 2 2 0 0 0 0 2 2 2 0 0 -2 -2 -2 -2 -2 -2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ15 2 2 2 2 2 2 0 0 0 0 -2 -2 -2 0 0 0 0 0 0 0 0 √2 √2 -√2 -√2 √2 √2 -√2 -√2 orthogonal lifted from D8 ρ16 2 2 2 2 -2 -2 0 0 0 0 -2 -2 2 0 0 0 0 0 0 0 0 √2 -√2 -√2 √2 -√2 √2 -√2 √2 orthogonal lifted from D8 ρ17 2 2 2 2 -2 -2 0 0 0 0 -2 -2 2 0 0 0 0 0 0 0 0 -√2 √2 √2 -√2 √2 -√2 √2 -√2 orthogonal lifted from D8 ρ18 2 2 2 2 2 2 0 0 0 0 -2 -2 -2 0 0 0 0 0 0 0 0 -√2 -√2 √2 √2 -√2 -√2 √2 √2 orthogonal lifted from D8 ρ19 2 2 -2 -2 -2 2 0 0 0 0 0 0 0 0 0 -√2 -√2 √2 √2 √2 -√2 ζ167+ζ16 ζ1615+ζ169 ζ165+ζ163 ζ165+ζ163 ζ167+ζ16 ζ1615+ζ169 ζ1613+ζ1611 ζ1613+ζ1611 complex lifted from SD32 ρ20 2 2 -2 -2 -2 2 0 0 0 0 0 0 0 0 0 -√2 -√2 √2 √2 √2 -√2 ζ1615+ζ169 ζ167+ζ16 ζ1613+ζ1611 ζ1613+ζ1611 ζ1615+ζ169 ζ167+ζ16 ζ165+ζ163 ζ165+ζ163 complex lifted from SD32 ρ21 2 2 -2 -2 2 -2 0 0 0 0 0 0 0 0 0 √2 √2 -√2 -√2 √2 -√2 ζ1613+ζ1611 ζ1613+ζ1611 ζ167+ζ16 ζ1615+ζ169 ζ165+ζ163 ζ165+ζ163 ζ1615+ζ169 ζ167+ζ16 complex lifted from SD32 ρ22 2 2 -2 -2 2 -2 0 0 0 0 0 0 0 0 0 √2 √2 -√2 -√2 √2 -√2 ζ165+ζ163 ζ165+ζ163 ζ1615+ζ169 ζ167+ζ16 ζ1613+ζ1611 ζ1613+ζ1611 ζ167+ζ16 ζ1615+ζ169 complex lifted from SD32 ρ23 2 2 -2 -2 -2 2 0 0 0 0 0 0 0 0 0 √2 √2 -√2 -√2 -√2 √2 ζ1613+ζ1611 ζ165+ζ163 ζ167+ζ16 ζ167+ζ16 ζ1613+ζ1611 ζ165+ζ163 ζ1615+ζ169 ζ1615+ζ169 complex lifted from SD32 ρ24 2 2 -2 -2 -2 2 0 0 0 0 0 0 0 0 0 √2 √2 -√2 -√2 -√2 √2 ζ165+ζ163 ζ1613+ζ1611 ζ1615+ζ169 ζ1615+ζ169 ζ165+ζ163 ζ1613+ζ1611 ζ167+ζ16 ζ167+ζ16 complex lifted from SD32 ρ25 2 2 -2 -2 2 -2 0 0 0 0 0 0 0 0 0 -√2 -√2 √2 √2 -√2 √2 ζ1615+ζ169 ζ1615+ζ169 ζ1613+ζ1611 ζ165+ζ163 ζ167+ζ16 ζ167+ζ16 ζ165+ζ163 ζ1613+ζ1611 complex lifted from SD32 ρ26 2 2 -2 -2 2 -2 0 0 0 0 0 0 0 0 0 -√2 -√2 √2 √2 -√2 √2 ζ167+ζ16 ζ167+ζ16 ζ165+ζ163 ζ1613+ζ1611 ζ1615+ζ169 ζ1615+ζ169 ζ1613+ζ1611 ζ165+ζ163 complex lifted from SD32 ρ27 4 -4 -4 4 0 0 0 0 0 0 -4 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C8⋊C22 ρ28 4 -4 4 -4 0 0 0 0 0 0 0 0 0 0 0 2√2 -2√2 2√2 -2√2 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C16⋊C22 ρ29 4 -4 4 -4 0 0 0 0 0 0 0 0 0 0 0 -2√2 2√2 -2√2 2√2 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C16⋊C22

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

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

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

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

Matrix representation of D8.9D4 in GL4(𝔽17) generated by

 0 6 0 0 14 6 0 0 0 0 16 0 0 0 0 16
,
 16 2 0 0 0 1 0 0 0 0 16 0 0 0 0 1
,
 11 15 0 0 1 9 0 0 0 0 0 16 0 0 16 0
,
 11 15 0 0 10 6 0 0 0 0 0 1 0 0 1 0
`G:=sub<GL(4,GF(17))| [0,14,0,0,6,6,0,0,0,0,16,0,0,0,0,16],[16,0,0,0,2,1,0,0,0,0,16,0,0,0,0,1],[11,1,0,0,15,9,0,0,0,0,0,16,0,0,16,0],[11,10,0,0,15,6,0,0,0,0,0,1,0,0,1,0] >;`

D8.9D4 in GAP, Magma, Sage, TeX

`D_8._9D_4`
`% in TeX`

`G:=Group("D8.9D4");`
`// GroupNames label`

`G:=SmallGroup(128,919);`
`// by ID`

`G=gap.SmallGroup(128,919);`
`# by ID`

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

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

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