Copied to
clipboard

G = C8.32D8order 128 = 27

9th non-split extension by C8 of D8 acting via D8/D4=C2

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C8 — C8.32D8
 Chief series C1 — C2 — C4 — C2×C4 — C2×C8 — C4×C8 — C8○D8 — C8.32D8
 Lower central C1 — C2 — C4 — C8 — C8.32D8
 Upper central C1 — C4 — C2×C8 — C4×C8 — C8.32D8
 Jennings C1 — C2 — C2 — C2 — C2 — C2×C4 — C2×C4 — C4×C8 — C8.32D8

Generators and relations for C8.32D8
G = < a,b,c | a8=b8=1, c2=a, bab-1=a5, ac=ca, cbc-1=ab-1 >

Permutation representations of C8.32D8
On 16 points - transitive group 16T260
Generators in S16
```(1 3 5 7 9 11 13 15)(2 4 6 8 10 12 14 16)
(1 13 9 5)(2 16 6 4 10 8 14 12)(3 7 11 15)
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16)```

`G:=sub<Sym(16)| (1,3,5,7,9,11,13,15)(2,4,6,8,10,12,14,16), (1,13,9,5)(2,16,6,4,10,8,14,12)(3,7,11,15), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16)>;`

`G:=Group( (1,3,5,7,9,11,13,15)(2,4,6,8,10,12,14,16), (1,13,9,5)(2,16,6,4,10,8,14,12)(3,7,11,15), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16) );`

`G=PermutationGroup([[(1,3,5,7,9,11,13,15),(2,4,6,8,10,12,14,16)], [(1,13,9,5),(2,16,6,4,10,8,14,12),(3,7,11,15)], [(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16)]])`

`G:=TransitiveGroup(16,260);`

32 conjugacy classes

 class 1 2A 2B 2C 4A 4B 4C ··· 4G 4H 8A 8B 8C 8D 8E ··· 8N 8O 8P 16A 16B 16C 16D order 1 2 2 2 4 4 4 ··· 4 4 8 8 8 8 8 ··· 8 8 8 16 16 16 16 size 1 1 2 8 1 1 2 ··· 2 8 2 2 2 2 4 ··· 4 8 8 8 8 8 8

32 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 4 type + + + + + + + image C1 C2 C2 C2 C4 C4 C8 C8 D4 D4 M4(2) D8 SD16 C4≀C2 C8.32D8 kernel C8.32D8 C8⋊C8 C8.C8 C8○D8 C8.C4 C4○D8 D8 Q16 C42 C2×C8 C8 C8 C8 C22 C1 # reps 1 1 1 1 2 2 4 4 1 1 2 2 2 4 4

Matrix representation of C8.32D8 in GL4(𝔽5) generated by

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

C8.32D8 in GAP, Magma, Sage, TeX

`C_8._{32}D_8`
`% in TeX`

`G:=Group("C8.32D8");`
`// GroupNames label`

`G:=SmallGroup(128,68);`
`// by ID`

`G=gap.SmallGroup(128,68);`
`# by ID`

`G:=PCGroup([7,-2,2,-2,2,-2,2,-2,56,85,422,891,100,1018,136,2804,1411,172,124]);`
`// Polycyclic`

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

Export

׿
×
𝔽