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## G = Q16.5D4order 128 = 27

### 2nd non-split extension by Q16 of D4 acting via D4/C4=C2

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C8 — Q16.5D4
 Chief series C1 — C2 — C4 — C8 — C2×C8 — C2×Q16 — C4×Q16 — Q16.5D4
 Lower central C1 — C2 — C4 — C2×C8 — Q16.5D4
 Upper central C1 — C22 — C42 — C4×C8 — Q16.5D4
 Jennings C1 — C2 — C2 — C2 — C2 — C4 — C4 — C2×C8 — Q16.5D4

Generators and relations for Q16.5D4
G = < a,b,c,d | a8=c4=1, b2=d2=a4, bab-1=dad-1=a-1, ac=ca, bc=cb, dbd-1=a-1b, dcd-1=a4c-1 >

Subgroups: 204 in 79 conjugacy classes, 32 normal (30 characteristic)
C1, C2 [×3], C2, C4 [×2], C4 [×6], C22, C22 [×3], C8 [×2], C8, C2×C4 [×3], C2×C4 [×3], D4 [×2], Q8 [×5], C23, C16 [×2], C42, C42, C22⋊C4 [×2], C4⋊C4 [×2], C2×C8 [×2], D8 [×2], SD16 [×2], Q16 [×2], Q16 [×3], C2×D4, C2×Q8 [×2], C4×C8, Q8⋊C4, C2.D8, C2×C16 [×2], SD32 [×2], Q32 [×2], C4×Q8, C4.4D4, C2×D8, C2×SD16, C2×Q16 [×2], C2.D16, C2.Q32, C4⋊C16, C4×Q16, C8.12D4, C2×SD32, C2×Q32, Q16.5D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×4], C23, D8 [×2], C2×D4 [×2], C4○D4, C4⋊D4, C2×D8, C8⋊C22, C4⋊D8, C4○D16, Q32⋊C2, Q16.5D4

Character table of Q16.5D4

 class 1 2A 2B 2C 2D 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 8A 8B 8C 8D 8E 8F 16A 16B 16C 16D 16E 16F 16G 16H size 1 1 1 1 16 2 2 2 2 4 8 8 8 8 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 2 0 0 -2 0 2 0 -2 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 2 -2 -2 2 -2 0 0 0 0 0 -2 -2 -2 -2 2 2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ11 2 -2 -2 2 0 2 0 0 -2 0 -2 0 2 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 2 2 2 2 2 0 0 0 0 0 -2 -2 -2 -2 -2 -2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ13 2 2 2 2 0 -2 -2 -2 -2 2 0 0 0 0 0 0 0 0 0 0 0 -√2 √2 -√2 -√2 √2 √2 √2 -√2 orthogonal lifted from D8 ρ14 2 2 2 2 0 -2 2 2 -2 -2 0 0 0 0 0 0 0 0 0 0 0 √2 -√2 √2 -√2 √2 -√2 √2 -√2 orthogonal lifted from D8 ρ15 2 2 2 2 0 -2 2 2 -2 -2 0 0 0 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 0 -2 -2 -2 -2 2 0 0 0 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 0 2 0 0 -2 0 0 -2i 0 2i 0 2 -2 -2 2 0 0 0 0 0 0 0 0 0 0 complex lifted from C4○D4 ρ18 2 -2 -2 2 0 2 0 0 -2 0 0 2i 0 -2i 0 2 -2 -2 2 0 0 0 0 0 0 0 0 0 0 complex lifted from C4○D4 ρ19 2 2 -2 -2 0 0 2i -2i 0 0 0 0 0 0 0 -√2 √2 -√2 √2 -√-2 √-2 ζ167-ζ16 ζ165-ζ163 -ζ167+ζ16 ζ1615+ζ169 ζ1613+ζ1611 -ζ165+ζ163 ζ165+ζ163 ζ167+ζ16 complex lifted from C4○D16 ρ20 2 2 -2 -2 0 0 -2i 2i 0 0 0 0 0 0 0 -√2 √2 -√2 √2 √-2 -√-2 -ζ167+ζ16 -ζ165+ζ163 ζ167-ζ16 ζ1615+ζ169 ζ1613+ζ1611 ζ165-ζ163 ζ165+ζ163 ζ167+ζ16 complex lifted from C4○D16 ρ21 2 2 -2 -2 0 0 -2i 2i 0 0 0 0 0 0 0 -√2 √2 -√2 √2 √-2 -√-2 ζ167-ζ16 ζ165-ζ163 -ζ167+ζ16 ζ167+ζ16 ζ165+ζ163 -ζ165+ζ163 ζ1613+ζ1611 ζ1615+ζ169 complex lifted from C4○D16 ρ22 2 2 -2 -2 0 0 2i -2i 0 0 0 0 0 0 0 -√2 √2 -√2 √2 -√-2 √-2 -ζ167+ζ16 -ζ165+ζ163 ζ167-ζ16 ζ167+ζ16 ζ165+ζ163 ζ165-ζ163 ζ1613+ζ1611 ζ1615+ζ169 complex lifted from C4○D16 ρ23 2 2 -2 -2 0 0 2i -2i 0 0 0 0 0 0 0 √2 -√2 √2 -√2 √-2 -√-2 ζ165-ζ163 -ζ167+ζ16 -ζ165+ζ163 ζ165+ζ163 ζ1615+ζ169 ζ167-ζ16 ζ167+ζ16 ζ1613+ζ1611 complex lifted from C4○D16 ρ24 2 2 -2 -2 0 0 2i -2i 0 0 0 0 0 0 0 √2 -√2 √2 -√2 √-2 -√-2 -ζ165+ζ163 ζ167-ζ16 ζ165-ζ163 ζ1613+ζ1611 ζ167+ζ16 -ζ167+ζ16 ζ1615+ζ169 ζ165+ζ163 complex lifted from C4○D16 ρ25 2 2 -2 -2 0 0 -2i 2i 0 0 0 0 0 0 0 √2 -√2 √2 -√2 -√-2 √-2 ζ165-ζ163 -ζ167+ζ16 -ζ165+ζ163 ζ1613+ζ1611 ζ167+ζ16 ζ167-ζ16 ζ1615+ζ169 ζ165+ζ163 complex lifted from C4○D16 ρ26 2 2 -2 -2 0 0 -2i 2i 0 0 0 0 0 0 0 √2 -√2 √2 -√2 -√-2 √-2 -ζ165+ζ163 ζ167-ζ16 ζ165-ζ163 ζ165+ζ163 ζ1615+ζ169 -ζ167+ζ16 ζ167+ζ16 ζ1613+ζ1611 complex lifted from C4○D16 ρ27 4 -4 -4 4 0 -4 0 0 4 0 0 0 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 symplectic lifted from Q32⋊C2, Schur index 2 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 symplectic lifted from Q32⋊C2, Schur index 2

Smallest permutation representation of Q16.5D4
On 64 points
Generators in S64
```(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 25 5 29)(2 32 6 28)(3 31 7 27)(4 30 8 26)(9 19 13 23)(10 18 14 22)(11 17 15 21)(12 24 16 20)(33 59 37 63)(34 58 38 62)(35 57 39 61)(36 64 40 60)(41 51 45 55)(42 50 46 54)(43 49 47 53)(44 56 48 52)
(1 47 15 35)(2 48 16 36)(3 41 9 37)(4 42 10 38)(5 43 11 39)(6 44 12 40)(7 45 13 33)(8 46 14 34)(17 61 29 49)(18 62 30 50)(19 63 31 51)(20 64 32 52)(21 57 25 53)(22 58 26 54)(23 59 27 55)(24 60 28 56)
(1 39 5 35)(2 38 6 34)(3 37 7 33)(4 36 8 40)(9 41 13 45)(10 48 14 44)(11 47 15 43)(12 46 16 42)(17 54 21 50)(18 53 22 49)(19 52 23 56)(20 51 24 55)(25 62 29 58)(26 61 30 57)(27 60 31 64)(28 59 32 63)```

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

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

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

Matrix representation of Q16.5D4 in GL4(𝔽17) generated by

 14 3 0 0 14 14 0 0 0 0 1 0 0 0 0 1
,
 1 7 0 0 7 16 0 0 0 0 1 0 0 0 0 1
,
 4 0 0 0 0 4 0 0 0 0 1 15 0 0 1 16
,
 4 0 0 0 0 13 0 0 0 0 1 15 0 0 0 16
`G:=sub<GL(4,GF(17))| [14,14,0,0,3,14,0,0,0,0,1,0,0,0,0,1],[1,7,0,0,7,16,0,0,0,0,1,0,0,0,0,1],[4,0,0,0,0,4,0,0,0,0,1,1,0,0,15,16],[4,0,0,0,0,13,0,0,0,0,1,0,0,0,15,16] >;`

Q16.5D4 in GAP, Magma, Sage, TeX

`Q_{16}._5D_4`
`% in TeX`

`G:=Group("Q16.5D4");`
`// GroupNames label`

`G:=SmallGroup(128,943);`
`// by ID`

`G=gap.SmallGroup(128,943);`
`# by ID`

`G:=PCGroup([7,-2,2,2,-2,2,-2,-2,141,512,422,352,1684,438,242,4037,1027,124]);`
`// Polycyclic`

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

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