Copied to
clipboard

## G = Q16.D4order 128 = 27

### 2nd non-split extension by Q16 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 — Q16.D4
 Chief series C1 — C2 — C4 — C2×C4 — C2×C8 — C8○D4 — Q8○D8 — Q16.D4
 Lower central C1 — C2 — C4 — C2×C8 — Q16.D4
 Upper central C1 — C2 — C2×C4 — C8○D4 — Q16.D4
 Jennings C1 — C2 — C2 — C2 — C2 — C4 — C4 — C2×C8 — Q16.D4

Generators and relations for Q16.D4
G = < a,b,c,d | a8=d2=1, b2=a4, c4=a6, bab-1=dad=a-1, cac-1=a5, cbc-1=ab, dbd=a3b, dcd=a2c3 >

Subgroups: 272 in 105 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, SD16, Q16, Q16, Q16, C2×D4, C2×Q8, C4○D4, C4○D4, C4.D4, C8.C4, C2×C16, M5(2), D16, SD32, C8○D4, C2×D8, C2×Q16, C2×Q16, C4○D8, C4○D8, C8⋊C22, C8.C22, 2- 1+4, D4.C8, D8.C4, C8.17D4, D4.4D4, C2×SD32, C16⋊C22, Q8○D8, Q16.D4
Quotients: C1, C2, C22, D4, C23, D8, C2×D4, C22≀C2, C2×D8, C8⋊C22, C22⋊D8, Q16.D4

Character table of Q16.D4

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E 4F 8A 8B 8C 8D 8E 16A 16B 16C 16D 16E 16F size 1 1 2 4 8 16 2 2 4 8 8 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 0 -2 2 0 -2 2 0 -2 -2 2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ10 2 2 -2 0 2 0 -2 2 0 0 0 -2 2 2 -2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ11 2 2 2 -2 0 0 2 2 -2 0 0 0 -2 -2 -2 2 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ12 2 2 -2 0 -2 0 -2 2 0 0 0 2 2 2 -2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ13 2 2 2 2 0 0 2 2 2 0 0 0 -2 -2 -2 -2 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ14 2 2 -2 0 0 0 -2 2 0 2 -2 0 -2 -2 2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ15 2 2 -2 2 0 0 2 -2 -2 0 0 0 0 0 0 0 0 -√2 √2 -√2 √2 -√2 √2 orthogonal lifted from D8 ρ16 2 2 -2 -2 0 0 2 -2 2 0 0 0 0 0 0 0 0 -√2 √2 -√2 √2 √2 -√2 orthogonal lifted from D8 ρ17 2 2 -2 2 0 0 2 -2 -2 0 0 0 0 0 0 0 0 √2 -√2 √2 -√2 √2 -√2 orthogonal lifted from D8 ρ18 2 2 -2 -2 0 0 2 -2 2 0 0 0 0 0 0 0 0 √2 -√2 √2 -√2 -√2 √2 orthogonal lifted from D8 ρ19 4 4 4 0 0 0 -4 -4 0 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 Q16.D4
On 32 points
Generators in S32
```(1 7 13 3 9 15 5 11)(2 16 14 12 10 8 6 4)(17 31 29 27 25 23 21 19)(18 24 30 20 26 32 22 28)
(1 22 9 30)(2 17 10 25)(3 20 11 28)(4 31 12 23)(5 18 13 26)(6 29 14 21)(7 32 15 24)(8 27 16 19)
(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)
(2 16)(3 15)(4 14)(5 13)(6 12)(7 11)(8 10)(17 25)(18 24)(19 23)(20 22)(26 32)(27 31)(28 30)```

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

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

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

Matrix representation of Q16.D4 in GL4(𝔽7) generated by

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

Q16.D4 in GAP, Magma, Sage, TeX

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

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

`G:=SmallGroup(128,925);`
`// by ID`

`G=gap.SmallGroup(128,925);`
`# by ID`

`G:=PCGroup([7,-2,2,2,-2,2,-2,-2,141,422,352,1123,570,360,2804,718,172,4037,2028,124]);`
`// Polycyclic`

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

Export

׿
×
𝔽