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## G = D4.4D8order 128 = 27

### 4th non-split extension by D4 of D8 acting via D8/C8=C2

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C8 — D4.4D8
 Chief series C1 — C2 — C4 — C2×C4 — C2×C8 — C8○D4 — D4○C16 — D4.4D8
 Lower central C1 — C2 — C4 — C2×C8 — D4.4D8
 Upper central C1 — C2 — C2×C4 — C8○D4 — D4.4D8
 Jennings C1 — C2 — C2 — C2 — C2 — C4 — C4 — C2×C8 — D4.4D8

Generators and relations for D4.4D8
G = < a,b,c,d | a4=b2=1, c8=d2=a2, bab=dad-1=a-1, ac=ca, bc=cb, dbd-1=ab, dcd-1=a2c7 >

Subgroups: 148 in 70 conjugacy classes, 30 normal (22 characteristic)
C1, C2, C2 [×2], C4 [×2], C4 [×3], C22, C22, C8 [×2], C8 [×3], C2×C4, C2×C4 [×3], D4, D4, Q8, Q8 [×4], C16 [×2], C16, C2×C8, C2×C8, M4(2), M4(2) [×3], SD16 [×2], Q16 [×6], C2×Q8 [×2], C4○D4, C4.10D4 [×2], C8.C4 [×2], C2×C16, C2×C16, M5(2), M5(2), Q32 [×2], C8○D4, C2×Q16 [×2], C8.C22 [×2], C8.17D4 [×2], C8.4Q8, D4.5D4 [×2], D4○C16, C2×Q32, D4.4D8
Quotients: C1, C2 [×7], C22 [×7], D4 [×4], C23, D8 [×2], C2×D4 [×2], C4○D4, C4⋊D4, C2×D8, C4○D8, C87D4, D4.4D8

Character table of D4.4D8

 class 1 2A 2B 2C 4A 4B 4C 4D 4E 8A 8B 8C 8D 8E 8F 8G 16A 16B 16C 16D 16E 16F 16G 16H 16I 16J size 1 1 2 4 2 2 4 16 16 2 2 4 4 4 16 16 2 2 2 2 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 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 2 2 2 2 2 2 2 0 0 -2 -2 -2 -2 -2 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ10 2 2 -2 0 2 -2 0 0 0 2 2 -2 0 0 0 0 2 2 2 2 -2 0 0 0 0 -2 orthogonal lifted from D4 ρ11 2 2 -2 0 2 -2 0 0 0 2 2 -2 0 0 0 0 -2 -2 -2 -2 2 0 0 0 0 2 orthogonal lifted from D4 ρ12 2 2 2 -2 2 2 -2 0 0 -2 -2 -2 2 2 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ13 2 2 -2 -2 -2 2 2 0 0 0 0 0 0 0 0 0 √2 -√2 √2 -√2 √2 -√2 √2 -√2 √2 -√2 orthogonal lifted from D8 ρ14 2 2 -2 -2 -2 2 2 0 0 0 0 0 0 0 0 0 -√2 √2 -√2 √2 -√2 √2 -√2 √2 -√2 √2 orthogonal lifted from D8 ρ15 2 2 -2 2 -2 2 -2 0 0 0 0 0 0 0 0 0 √2 -√2 √2 -√2 √2 √2 -√2 √2 -√2 -√2 orthogonal lifted from D8 ρ16 2 2 -2 2 -2 2 -2 0 0 0 0 0 0 0 0 0 -√2 √2 -√2 √2 -√2 -√2 √2 -√2 √2 √2 orthogonal lifted from D8 ρ17 2 2 -2 0 2 -2 0 0 0 -2 -2 2 0 0 0 0 0 0 0 0 0 -2i -2i 2i 2i 0 complex lifted from C4○D4 ρ18 2 2 2 0 -2 -2 0 0 0 0 0 0 -2i 2i 0 0 √2 -√2 √2 -√2 -√2 √-2 -√-2 -√-2 √-2 √2 complex lifted from C4○D8 ρ19 2 2 -2 0 2 -2 0 0 0 -2 -2 2 0 0 0 0 0 0 0 0 0 2i 2i -2i -2i 0 complex lifted from C4○D4 ρ20 2 2 2 0 -2 -2 0 0 0 0 0 0 -2i 2i 0 0 -√2 √2 -√2 √2 √2 -√-2 √-2 √-2 -√-2 -√2 complex lifted from C4○D8 ρ21 2 2 2 0 -2 -2 0 0 0 0 0 0 2i -2i 0 0 √2 -√2 √2 -√2 -√2 -√-2 √-2 √-2 -√-2 √2 complex lifted from C4○D8 ρ22 2 2 2 0 -2 -2 0 0 0 0 0 0 2i -2i 0 0 -√2 √2 -√2 √2 √2 √-2 -√-2 -√-2 √-2 -√2 complex lifted from C4○D8 ρ23 4 -4 0 0 0 0 0 0 0 2√2 -2√2 0 0 0 0 0 -2ζ167+2ζ16 -2ζ165+2ζ163 2ζ167-2ζ16 2ζ165-2ζ163 0 0 0 0 0 0 symplectic faithful, Schur index 2 ρ24 4 -4 0 0 0 0 0 0 0 -2√2 2√2 0 0 0 0 0 2ζ165-2ζ163 -2ζ167+2ζ16 -2ζ165+2ζ163 2ζ167-2ζ16 0 0 0 0 0 0 symplectic faithful, Schur index 2 ρ25 4 -4 0 0 0 0 0 0 0 2√2 -2√2 0 0 0 0 0 2ζ167-2ζ16 2ζ165-2ζ163 -2ζ167+2ζ16 -2ζ165+2ζ163 0 0 0 0 0 0 symplectic faithful, Schur index 2 ρ26 4 -4 0 0 0 0 0 0 0 -2√2 2√2 0 0 0 0 0 -2ζ165+2ζ163 2ζ167-2ζ16 2ζ165-2ζ163 -2ζ167+2ζ16 0 0 0 0 0 0 symplectic faithful, Schur index 2

Smallest permutation representation of D4.4D8
On 64 points
Generators in S64
```(1 53 9 61)(2 54 10 62)(3 55 11 63)(4 56 12 64)(5 57 13 49)(6 58 14 50)(7 59 15 51)(8 60 16 52)(17 45 25 37)(18 46 26 38)(19 47 27 39)(20 48 28 40)(21 33 29 41)(22 34 30 42)(23 35 31 43)(24 36 32 44)
(1 61)(2 62)(3 63)(4 64)(5 49)(6 50)(7 51)(8 52)(9 53)(10 54)(11 55)(12 56)(13 57)(14 58)(15 59)(16 60)(33 41)(34 42)(35 43)(36 44)(37 45)(38 46)(39 47)(40 48)
(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)(33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48)(49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64)
(1 43 9 35)(2 42 10 34)(3 41 11 33)(4 40 12 48)(5 39 13 47)(6 38 14 46)(7 37 15 45)(8 36 16 44)(17 59 25 51)(18 58 26 50)(19 57 27 49)(20 56 28 64)(21 55 29 63)(22 54 30 62)(23 53 31 61)(24 52 32 60)```

`G:=sub<Sym(64)| (1,53,9,61)(2,54,10,62)(3,55,11,63)(4,56,12,64)(5,57,13,49)(6,58,14,50)(7,59,15,51)(8,60,16,52)(17,45,25,37)(18,46,26,38)(19,47,27,39)(20,48,28,40)(21,33,29,41)(22,34,30,42)(23,35,31,43)(24,36,32,44), (1,61)(2,62)(3,63)(4,64)(5,49)(6,50)(7,51)(8,52)(9,53)(10,54)(11,55)(12,56)(13,57)(14,58)(15,59)(16,60)(33,41)(34,42)(35,43)(36,44)(37,45)(38,46)(39,47)(40,48), (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)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64), (1,43,9,35)(2,42,10,34)(3,41,11,33)(4,40,12,48)(5,39,13,47)(6,38,14,46)(7,37,15,45)(8,36,16,44)(17,59,25,51)(18,58,26,50)(19,57,27,49)(20,56,28,64)(21,55,29,63)(22,54,30,62)(23,53,31,61)(24,52,32,60)>;`

`G:=Group( (1,53,9,61)(2,54,10,62)(3,55,11,63)(4,56,12,64)(5,57,13,49)(6,58,14,50)(7,59,15,51)(8,60,16,52)(17,45,25,37)(18,46,26,38)(19,47,27,39)(20,48,28,40)(21,33,29,41)(22,34,30,42)(23,35,31,43)(24,36,32,44), (1,61)(2,62)(3,63)(4,64)(5,49)(6,50)(7,51)(8,52)(9,53)(10,54)(11,55)(12,56)(13,57)(14,58)(15,59)(16,60)(33,41)(34,42)(35,43)(36,44)(37,45)(38,46)(39,47)(40,48), (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)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64), (1,43,9,35)(2,42,10,34)(3,41,11,33)(4,40,12,48)(5,39,13,47)(6,38,14,46)(7,37,15,45)(8,36,16,44)(17,59,25,51)(18,58,26,50)(19,57,27,49)(20,56,28,64)(21,55,29,63)(22,54,30,62)(23,53,31,61)(24,52,32,60) );`

`G=PermutationGroup([(1,53,9,61),(2,54,10,62),(3,55,11,63),(4,56,12,64),(5,57,13,49),(6,58,14,50),(7,59,15,51),(8,60,16,52),(17,45,25,37),(18,46,26,38),(19,47,27,39),(20,48,28,40),(21,33,29,41),(22,34,30,42),(23,35,31,43),(24,36,32,44)], [(1,61),(2,62),(3,63),(4,64),(5,49),(6,50),(7,51),(8,52),(9,53),(10,54),(11,55),(12,56),(13,57),(14,58),(15,59),(16,60),(33,41),(34,42),(35,43),(36,44),(37,45),(38,46),(39,47),(40,48)], [(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),(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48),(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64)], [(1,43,9,35),(2,42,10,34),(3,41,11,33),(4,40,12,48),(5,39,13,47),(6,38,14,46),(7,37,15,45),(8,36,16,44),(17,59,25,51),(18,58,26,50),(19,57,27,49),(20,56,28,64),(21,55,29,63),(22,54,30,62),(23,53,31,61),(24,52,32,60)])`

Matrix representation of D4.4D8 in GL4(𝔽17) generated by

 1 0 15 0 0 0 16 1 1 0 16 0 1 16 16 0
,
 1 0 15 0 0 0 16 1 0 0 16 0 0 1 16 0
,
 2 8 0 0 13 10 0 0 0 4 6 13 13 4 4 6
,
 0 9 6 8 0 10 16 7 7 9 0 0 1 16 16 7
`G:=sub<GL(4,GF(17))| [1,0,1,1,0,0,0,16,15,16,16,16,0,1,0,0],[1,0,0,0,0,0,0,1,15,16,16,16,0,1,0,0],[2,13,0,13,8,10,4,4,0,0,6,4,0,0,13,6],[0,0,7,1,9,10,9,16,6,16,0,16,8,7,0,7] >;`

D4.4D8 in GAP, Magma, Sage, TeX

`D_4._4D_8`
`% in TeX`

`G:=Group("D4.4D8");`
`// GroupNames label`

`G:=SmallGroup(128,954);`
`// by ID`

`G=gap.SmallGroup(128,954);`
`# by ID`

`G:=PCGroup([7,-2,2,2,-2,2,-2,-2,448,141,288,422,1123,360,2804,718,172,4037,124]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^4=b^2=1,c^8=d^2=a^2,b*a*b=d*a*d^-1=a^-1,a*c=c*a,b*c=c*b,d*b*d^-1=a*b,d*c*d^-1=a^2*c^7>;`
`// generators/relations`

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