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## G = D8.3D4order 128 = 27

### 3rd 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 — D8.3D4
 Chief series C1 — C2 — C4 — C2×C4 — C2×C8 — C8○D4 — D4○D8 — D8.3D4
 Lower central C1 — C2 — C4 — C2×C8 — D8.3D4
 Upper central C1 — C2 — C2×C4 — C8○D4 — D8.3D4
 Jennings C1 — C2 — C2 — C2 — C2 — C4 — C4 — C2×C8 — D8.3D4

Generators and relations for D8.3D4
G = < a,b,c,d | a8=b2=1, c4=a6, d2=a4, bab=dad-1=a-1, cac-1=a5, cbc-1=a5b, dbd-1=ab, dcd-1=a6c3 >

Subgroups: 304 in 108 conjugacy classes, 32 normal (30 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C16, C2×C8, C2×C8, M4(2), M4(2), D8, D8, D8, SD16, Q16, Q16, C2×D4, C2×Q8, C4○D4, C4○D4, C4.10D4, C8.C4, C2×C16, M5(2), SD32, Q32, C8○D4, C2×D8, C2×D8, C2×Q16, C4○D8, C4○D8, C8⋊C22, C8.C22, 2+ 1+4, D4.C8, D8.C4, M5(2)⋊C2, D4.5D4, C2×SD32, Q32⋊C2, D4○D8, D8.3D4
Quotients: C1, C2, C22, D4, C23, D8, C2×D4, C22≀C2, C2×D8, C8⋊C22, C22⋊D8, D8.3D4

Character table of D8.3D4

 class 1 2A 2B 2C 2D 2E 2F 4A 4B 4C 4D 4E 8A 8B 8C 8D 8E 16A 16B 16C 16D 16E 16F size 1 1 2 4 8 8 8 2 2 4 8 16 2 2 4 8 16 4 4 4 4 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 0 0 -2 2 -2 2 0 0 0 -2 -2 2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ10 2 2 2 -2 0 0 0 2 2 -2 0 0 -2 -2 -2 2 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ11 2 2 -2 0 2 0 0 -2 2 0 -2 0 2 2 -2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ12 2 2 -2 0 0 2 -2 -2 2 0 0 0 -2 -2 2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ13 2 2 -2 0 -2 0 0 -2 2 0 2 0 2 2 -2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ14 2 2 2 2 0 0 0 2 2 2 0 0 -2 -2 -2 -2 0 0 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 0 -√2 √2 -√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 0 -√2 √2 -√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 0 √2 -√2 √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 0 √2 -√2 √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 0 0 0 orthogonal lifted from C8⋊C22 ρ20 4 -4 0 0 0 0 0 0 0 0 0 0 2√2 -2√2 0 0 0 ζ167+ζ165+ζ163+ζ16 ζ1613+ζ1611+ζ167+ζ16 ζ1615+ζ1613+ζ1611+ζ169 ζ1615+ζ169+ζ165+ζ163 0 0 complex faithful ρ21 4 -4 0 0 0 0 0 0 0 0 0 0 -2√2 2√2 0 0 0 ζ1615+ζ169+ζ165+ζ163 ζ167+ζ165+ζ163+ζ16 ζ1613+ζ1611+ζ167+ζ16 ζ1615+ζ1613+ζ1611+ζ169 0 0 complex faithful ρ22 4 -4 0 0 0 0 0 0 0 0 0 0 -2√2 2√2 0 0 0 ζ1613+ζ1611+ζ167+ζ16 ζ1615+ζ1613+ζ1611+ζ169 ζ1615+ζ169+ζ165+ζ163 ζ167+ζ165+ζ163+ζ16 0 0 complex faithful ρ23 4 -4 0 0 0 0 0 0 0 0 0 0 2√2 -2√2 0 0 0 ζ1615+ζ1613+ζ1611+ζ169 ζ1615+ζ169+ζ165+ζ163 ζ167+ζ165+ζ163+ζ16 ζ1613+ζ1611+ζ167+ζ16 0 0 complex faithful

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

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

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

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

Matrix representation of D8.3D4 in GL4(𝔽7) generated by

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

D8.3D4 in GAP, Magma, Sage, TeX

`D_8._3D_4`
`% in TeX`

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

`G:=SmallGroup(128,926);`
`// by ID`

`G=gap.SmallGroup(128,926);`
`# by ID`

`G:=PCGroup([7,-2,2,2,-2,2,-2,-2,448,141,422,1123,570,360,2804,718,172,4037,2028,124]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^8=b^2=1,c^4=a^6,d^2=a^4,b*a*b=d*a*d^-1=a^-1,c*a*c^-1=a^5,c*b*c^-1=a^5*b,d*b*d^-1=a*b,d*c*d^-1=a^6*c^3>;`
`// generators/relations`

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