Copied to
clipboard

## G = D8.13D4order 128 = 27

### 5th non-split extension by D8 of D4 acting via D4/C22=C2

p-group, metabelian, nilpotent (class 3), monomial, rational

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C4 — D8.13D4
 Chief series C1 — C2 — C4 — C2×C4 — C4○D4 — 2+ 1+4 — D4○SD16 — D8.13D4
 Lower central C1 — C2 — C2×C4 — D8.13D4
 Upper central C1 — C2 — C2×C4 — D8.13D4
 Jennings C1 — C2 — C2 — C2×C4 — D8.13D4

Generators and relations for D8.13D4
G = < a,b,c,d | a8=b2=1, c4=d2=a4, bab=dad-1=a-1, cac-1=a3, cbc-1=a2b, dbd-1=a6b, dcd-1=c3 >

Subgroups: 444 in 224 conjugacy classes, 92 normal (18 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), M4(2), D8, D8, SD16, SD16, Q16, Q16, C2×D4, C2×D4, C2×Q8, C2×Q8, C2×Q8, C4○D4, C4○D4, C8⋊C4, C4.D4, C4.10D4, C4.10D4, C4≀C2, C8.C4, C4.4D4, C4⋊Q8, C8○D4, C2×SD16, C2×SD16, C2×Q16, C2×Q16, C4○D8, C4○D8, C8⋊C22, C8⋊C22, C8.C22, C8.C22, 2+ 1+4, 2- 1+4, 2- 1+4, C8.26D4, D4.8D4, D4.9D4, D4.10D4, D4.3D4, D4.5D4, C8.2D4, D4○SD16, Q8○D8, D8.13D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C24, C22×D4, 2+ 1+4, D42, D8.13D4

Character table of D8.13D4

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

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

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

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

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

Matrix representation of D8.13D4 in GL8(𝔽17)

 0 0 0 0 0 1 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 16 0 0 0 0 0 0 1 0 0 0 0 0 0 16 0 0 0 0 0 0 0
,
 0 0 0 0 0 1 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 16 0 0 16 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 1 0 0 0 0 0
,
 12 5 0 0 0 0 0 0 12 12 0 0 0 0 0 0 0 0 5 12 0 0 0 0 0 0 5 5 0 0 0 0 0 0 0 0 0 0 12 5 0 0 0 0 0 0 12 12 0 0 0 0 12 5 0 0 0 0 0 0 12 12 0 0
,
 12 5 0 0 0 0 0 0 5 5 0 0 0 0 0 0 0 0 5 12 0 0 0 0 0 0 12 12 0 0 0 0 0 0 0 0 0 0 12 5 0 0 0 0 0 0 5 5 0 0 0 0 12 5 0 0 0 0 0 0 5 5 0 0

`G:=sub<GL(8,GF(17))| [0,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0],[0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0],[12,12,0,0,0,0,0,0,5,12,0,0,0,0,0,0,0,0,5,5,0,0,0,0,0,0,12,5,0,0,0,0,0,0,0,0,0,0,12,12,0,0,0,0,0,0,5,12,0,0,0,0,12,12,0,0,0,0,0,0,5,12,0,0],[12,5,0,0,0,0,0,0,5,5,0,0,0,0,0,0,0,0,5,12,0,0,0,0,0,0,12,12,0,0,0,0,0,0,0,0,0,0,12,5,0,0,0,0,0,0,5,5,0,0,0,0,12,5,0,0,0,0,0,0,5,5,0,0] >;`

D8.13D4 in GAP, Magma, Sage, TeX

`D_8._{13}D_4`
`% in TeX`

`G:=Group("D8.13D4");`
`// GroupNames label`

`G:=SmallGroup(128,2021);`
`// by ID`

`G=gap.SmallGroup(128,2021);`
`# by ID`

`G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,456,758,346,2804,1411,375,172,4037,1027,124]);`
`// Polycyclic`

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

Export

׿
×
𝔽