Copied to
clipboard

## G = Q16.10D4order 128 = 27

### 3rd non-split extension by Q16 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 — Q16.10D4
 Chief series C1 — C2 — C4 — C2×C4 — C2×C8 — C8○D4 — D4○D8 — Q16.10D4
 Lower central C1 — C2 — C4 — C2×C8 — Q16.10D4
 Upper central C1 — C2 — C2×C4 — C8○D4 — Q16.10D4
 Jennings C1 — C2 — C2 — C2 — C2 — C4 — C4 — C2×C8 — Q16.10D4

Generators and relations for Q16.10D4
G = < a,b,c,d | a8=d2=1, b2=c4=a4, bab-1=cac-1=dad=a-1, cbc-1=a-1b, dbd=a3b, dcd=a4c3 >

Subgroups: 336 in 110 conjugacy classes, 32 normal (30 characteristic)
C1, C2, C2 [×6], C4 [×2], C4 [×2], C22, C22 [×9], C8 [×2], C8 [×2], C2×C4, C2×C4 [×4], D4, D4 [×12], Q8, Q8, C23 [×4], C16 [×2], C2×C8, C2×C8, M4(2), M4(2) [×2], D8, D8 [×2], D8 [×6], SD16 [×4], Q16, C2×D4 [×7], C4○D4, C4○D4 [×4], C4.D4, C8.C4, C2×C16, M5(2), D16 [×3], SD32, C8○D4, C2×D8 [×2], C2×D8, C4○D8, C4○D8, C8⋊C22 [×4], 2+ 1+4, D4.C8, D8.C4, M5(2)⋊C2, D4.4D4, C2×D16, C16⋊C22, D4○D8, Q16.10D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], C23, D8 [×2], C2×D4 [×3], C22≀C2, C2×D8, C8⋊C22, C22⋊D8, Q16.10D4

Character table of Q16.10D4

 class 1 2A 2B 2C 2D 2E 2F 2G 4A 4B 4C 4D 8A 8B 8C 8D 8E 16A 16B 16C 16D 16E 16F size 1 1 2 4 8 8 8 16 2 2 4 8 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 0 2 -2 0 0 -2 -2 2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ10 2 2 -2 0 2 0 0 0 2 -2 0 -2 2 2 -2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ11 2 2 -2 0 0 2 -2 0 2 -2 0 0 -2 -2 2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ12 2 2 -2 0 -2 0 0 0 2 -2 0 2 2 2 -2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ13 2 2 2 -2 0 0 0 0 2 2 -2 0 -2 -2 -2 2 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ14 2 2 2 2 0 0 0 0 2 2 2 0 -2 -2 -2 -2 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ15 2 2 -2 2 0 0 0 0 -2 2 -2 0 0 0 0 0 0 √2 -√2 √2 -√2 -√2 √2 orthogonal lifted from D8 ρ16 2 2 -2 -2 0 0 0 0 -2 2 2 0 0 0 0 0 0 -√2 √2 -√2 √2 -√2 √2 orthogonal lifted from D8 ρ17 2 2 -2 2 0 0 0 0 -2 2 -2 0 0 0 0 0 0 -√2 √2 -√2 √2 √2 -√2 orthogonal lifted from D8 ρ18 2 2 -2 -2 0 0 0 0 -2 2 2 0 0 0 0 0 0 √2 -√2 √2 -√2 √2 -√2 orthogonal lifted from D8 ρ19 4 4 4 0 0 0 0 0 -4 -4 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 ζ1615-ζ169-ζ165+ζ163 ζ167-ζ165+ζ163-ζ16 -ζ1615+ζ169+ζ165-ζ163 0 0 orthogonal faithful ρ21 4 -4 0 0 0 0 0 0 0 0 0 0 -2√2 2√2 0 0 0 ζ167-ζ165+ζ163-ζ16 -ζ1615+ζ169+ζ165-ζ163 -ζ167+ζ165-ζ163+ζ16 ζ1615-ζ169-ζ165+ζ163 0 0 orthogonal faithful ρ22 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 ζ1615-ζ169-ζ165+ζ163 ζ167-ζ165+ζ163-ζ16 0 0 orthogonal faithful ρ23 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 -ζ1615+ζ169+ζ165-ζ163 -ζ167+ζ165-ζ163+ζ16 0 0 orthogonal faithful

Smallest permutation representation of Q16.10D4
On 32 points
Generators in S32
```(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 21 5 17)(2 20 6 24)(3 19 7 23)(4 18 8 22)(9 26 13 30)(10 25 14 29)(11 32 15 28)(12 31 16 27)
(1 12 17 28 5 16 21 32)(2 11 18 27 6 15 22 31)(3 10 19 26 7 14 23 30)(4 9 20 25 8 13 24 29)
(1 32)(2 31)(3 30)(4 29)(5 28)(6 27)(7 26)(8 25)(9 24)(10 23)(11 22)(12 21)(13 20)(14 19)(15 18)(16 17)```

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

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

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

Matrix representation of Q16.10D4 in GL4(𝔽17) generated by

 14 3 0 0 14 14 0 0 14 3 0 11 14 0 3 11
,
 0 0 1 0 1 16 16 2 16 0 0 0 0 16 16 1
,
 13 0 4 2 6 0 11 10 11 2 13 15 4 13 7 8
,
 13 0 4 2 6 0 11 10 6 15 4 2 6 4 0 0
`G:=sub<GL(4,GF(17))| [14,14,14,14,3,14,3,0,0,0,0,3,0,0,11,11],[0,1,16,0,0,16,0,16,1,16,0,16,0,2,0,1],[13,6,11,4,0,0,2,13,4,11,13,7,2,10,15,8],[13,6,6,6,0,0,15,4,4,11,4,0,2,10,2,0] >;`

Q16.10D4 in GAP, Magma, Sage, TeX

`Q_{16}._{10}D_4`
`% in TeX`

`G:=Group("Q16.10D4");`
`// GroupNames label`

`G:=SmallGroup(128,924);`
`// by ID`

`G=gap.SmallGroup(128,924);`
`# by ID`

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

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

Export

׿
×
𝔽