Copied to
clipboard

G = C6.D8order 96 = 25·3

2nd non-split extension by C6 of D8 acting via D8/D4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — C6.D8
 Chief series C1 — C3 — C6 — C2×C6 — C2×C12 — C2×D12 — C6.D8
 Lower central C3 — C6 — C12 — C6.D8
 Upper central C1 — C22 — C2×C4 — C4⋊C4

Generators and relations for C6.D8
G = < a,b,c | a6=b8=c2=1, bab-1=cac=a-1, cbc=a3b-1 >

Character table of C6.D8

 class 1 2A 2B 2C 2D 2E 3 4A 4B 4C 4D 6A 6B 6C 8A 8B 8C 8D 12A 12B 12C 12D 12E 12F size 1 1 1 1 12 12 2 2 2 4 4 2 2 2 6 6 6 6 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 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 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 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 linear of order 2 ρ5 1 1 -1 -1 1 -1 1 -1 1 -i i 1 -1 -1 -i i i -i i -i 1 i -1 -i linear of order 4 ρ6 1 1 -1 -1 -1 1 1 -1 1 -i i 1 -1 -1 i -i -i i i -i 1 i -1 -i linear of order 4 ρ7 1 1 -1 -1 -1 1 1 -1 1 i -i 1 -1 -1 -i i i -i -i i 1 -i -1 i linear of order 4 ρ8 1 1 -1 -1 1 -1 1 -1 1 i -i 1 -1 -1 i -i -i i -i i 1 -i -1 i linear of order 4 ρ9 2 2 2 2 0 0 2 -2 -2 0 0 2 2 2 0 0 0 0 0 0 -2 0 -2 0 orthogonal lifted from D4 ρ10 2 2 2 2 0 0 -1 2 2 -2 -2 -1 -1 -1 0 0 0 0 1 1 -1 1 -1 1 orthogonal lifted from D6 ρ11 2 2 -2 -2 0 0 2 2 -2 0 0 2 -2 -2 0 0 0 0 0 0 -2 0 2 0 orthogonal lifted from D4 ρ12 2 2 2 2 0 0 -1 2 2 2 2 -1 -1 -1 0 0 0 0 -1 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ13 2 -2 2 -2 0 0 2 0 0 0 0 -2 2 -2 √2 -√2 √2 -√2 0 0 0 0 0 0 orthogonal lifted from D8 ρ14 2 2 -2 -2 0 0 -1 2 -2 0 0 -1 1 1 0 0 0 0 √3 -√3 1 -√3 -1 √3 orthogonal lifted from D12 ρ15 2 -2 2 -2 0 0 2 0 0 0 0 -2 2 -2 -√2 √2 -√2 √2 0 0 0 0 0 0 orthogonal lifted from D8 ρ16 2 2 -2 -2 0 0 -1 2 -2 0 0 -1 1 1 0 0 0 0 -√3 √3 1 √3 -1 -√3 orthogonal lifted from D12 ρ17 2 2 -2 -2 0 0 -1 -2 2 2i -2i -1 1 1 0 0 0 0 i -i -1 i 1 -i complex lifted from C4×S3 ρ18 2 2 2 2 0 0 -1 -2 -2 0 0 -1 -1 -1 0 0 0 0 -√-3 -√-3 1 √-3 1 √-3 complex lifted from C3⋊D4 ρ19 2 -2 -2 2 0 0 2 0 0 0 0 -2 -2 2 √-2 √-2 -√-2 -√-2 0 0 0 0 0 0 complex lifted from SD16 ρ20 2 2 -2 -2 0 0 -1 -2 2 -2i 2i -1 1 1 0 0 0 0 -i i -1 -i 1 i complex lifted from C4×S3 ρ21 2 2 2 2 0 0 -1 -2 -2 0 0 -1 -1 -1 0 0 0 0 √-3 √-3 1 -√-3 1 -√-3 complex lifted from C3⋊D4 ρ22 2 -2 -2 2 0 0 2 0 0 0 0 -2 -2 2 -√-2 -√-2 √-2 √-2 0 0 0 0 0 0 complex lifted from SD16 ρ23 4 -4 4 -4 0 0 -2 0 0 0 0 2 -2 2 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4⋊S3, Schur index 2 ρ24 4 -4 -4 4 0 0 -2 0 0 0 0 2 2 -2 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from Q8⋊2S3

Smallest permutation representation of C6.D8
On 48 points
Generators in S48
```(1 34 20 15 26 41)(2 42 27 16 21 35)(3 36 22 9 28 43)(4 44 29 10 23 37)(5 38 24 11 30 45)(6 46 31 12 17 39)(7 40 18 13 32 47)(8 48 25 14 19 33)
(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)
(2 14)(3 7)(4 12)(6 10)(8 16)(9 13)(17 37)(18 28)(19 35)(20 26)(21 33)(22 32)(23 39)(24 30)(25 42)(27 48)(29 46)(31 44)(34 41)(36 47)(38 45)(40 43)```

`G:=sub<Sym(48)| (1,34,20,15,26,41)(2,42,27,16,21,35)(3,36,22,9,28,43)(4,44,29,10,23,37)(5,38,24,11,30,45)(6,46,31,12,17,39)(7,40,18,13,32,47)(8,48,25,14,19,33), (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), (2,14)(3,7)(4,12)(6,10)(8,16)(9,13)(17,37)(18,28)(19,35)(20,26)(21,33)(22,32)(23,39)(24,30)(25,42)(27,48)(29,46)(31,44)(34,41)(36,47)(38,45)(40,43)>;`

`G:=Group( (1,34,20,15,26,41)(2,42,27,16,21,35)(3,36,22,9,28,43)(4,44,29,10,23,37)(5,38,24,11,30,45)(6,46,31,12,17,39)(7,40,18,13,32,47)(8,48,25,14,19,33), (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), (2,14)(3,7)(4,12)(6,10)(8,16)(9,13)(17,37)(18,28)(19,35)(20,26)(21,33)(22,32)(23,39)(24,30)(25,42)(27,48)(29,46)(31,44)(34,41)(36,47)(38,45)(40,43) );`

`G=PermutationGroup([[(1,34,20,15,26,41),(2,42,27,16,21,35),(3,36,22,9,28,43),(4,44,29,10,23,37),(5,38,24,11,30,45),(6,46,31,12,17,39),(7,40,18,13,32,47),(8,48,25,14,19,33)], [(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)], [(2,14),(3,7),(4,12),(6,10),(8,16),(9,13),(17,37),(18,28),(19,35),(20,26),(21,33),(22,32),(23,39),(24,30),(25,42),(27,48),(29,46),(31,44),(34,41),(36,47),(38,45),(40,43)]])`

Matrix representation of C6.D8 in GL4(𝔽73) generated by

 1 0 0 0 0 1 0 0 0 0 0 1 0 0 72 1
,
 57 16 0 0 57 57 0 0 0 0 0 46 0 0 46 0
,
 1 0 0 0 0 72 0 0 0 0 0 1 0 0 1 0
`G:=sub<GL(4,GF(73))| [1,0,0,0,0,1,0,0,0,0,0,72,0,0,1,1],[57,57,0,0,16,57,0,0,0,0,0,46,0,0,46,0],[1,0,0,0,0,72,0,0,0,0,0,1,0,0,1,0] >;`

C6.D8 in GAP, Magma, Sage, TeX

`C_6.D_8`
`% in TeX`

`G:=Group("C6.D8");`
`// GroupNames label`

`G:=SmallGroup(96,16);`
`// by ID`

`G=gap.SmallGroup(96,16);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-3,121,31,579,297,69,2309]);`
`// Polycyclic`

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

Export

׿
×
𝔽