Copied to
clipboard

## G = C8.(C32⋊C4)  order 288 = 25·32

### 1st non-split extension by C8 of C32⋊C4 acting via C32⋊C4/C3⋊S3=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C12 — C8.(C32⋊C4)
 Chief series C1 — C32 — C3×C6 — C3⋊Dic3 — C4×C3⋊S3 — C32⋊M4(2) — C8.(C32⋊C4)
 Lower central C32 — C3×C6 — C3×C12 — C8.(C32⋊C4)
 Upper central C1 — C2 — C4 — C8

Generators and relations for C8.(C32⋊C4)
G = < a,b,c,d | a8=b3=c3=1, d4=a4, ab=ba, ac=ca, dad-1=a-1, dcd-1=bc=cb, dbd-1=b-1c >

Character table of C8.(C32⋊C4)

 class 1 2A 2B 3A 3B 4A 4B 4C 6A 6B 8A 8B 8C 8D 8E 8F 8G 8H 12A 12B 12C 12D 24A 24B 24C 24D 24E 24F 24G 24H size 1 1 18 4 4 2 9 9 4 4 2 2 18 18 36 36 36 36 4 4 4 4 4 4 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 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 -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 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 -1 -1 -1 -1 linear of order 2 ρ5 1 1 -1 1 1 1 -1 -1 1 1 1 1 -1 -1 i i -i -i 1 1 1 1 1 1 1 1 1 1 1 1 linear of order 4 ρ6 1 1 -1 1 1 1 -1 -1 1 1 -1 -1 1 1 i -i -i i 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 linear of order 4 ρ7 1 1 -1 1 1 1 -1 -1 1 1 1 1 -1 -1 -i -i i i 1 1 1 1 1 1 1 1 1 1 1 1 linear of order 4 ρ8 1 1 -1 1 1 1 -1 -1 1 1 -1 -1 1 1 -i i i -i 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 linear of order 4 ρ9 2 2 -2 2 2 -2 2 2 2 2 0 0 0 0 0 0 0 0 -2 -2 -2 -2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ10 2 2 2 2 2 -2 -2 -2 2 2 0 0 0 0 0 0 0 0 -2 -2 -2 -2 0 0 0 0 0 0 0 0 symplectic lifted from Q8, Schur index 2 ρ11 2 -2 0 2 2 0 2i -2i -2 -2 -√2 √2 √-2 -√-2 0 0 0 0 0 0 0 0 √2 -√2 -√2 -√2 -√2 √2 √2 √2 complex lifted from C8.C4 ρ12 2 -2 0 2 2 0 -2i 2i -2 -2 √2 -√2 √-2 -√-2 0 0 0 0 0 0 0 0 -√2 √2 √2 √2 √2 -√2 -√2 -√2 complex lifted from C8.C4 ρ13 2 -2 0 2 2 0 2i -2i -2 -2 √2 -√2 -√-2 √-2 0 0 0 0 0 0 0 0 -√2 √2 √2 √2 √2 -√2 -√2 -√2 complex lifted from C8.C4 ρ14 2 -2 0 2 2 0 -2i 2i -2 -2 -√2 √2 -√-2 √-2 0 0 0 0 0 0 0 0 √2 -√2 -√2 -√2 -√2 √2 √2 √2 complex lifted from C8.C4 ρ15 4 4 0 1 -2 4 0 0 -2 1 4 4 0 0 0 0 0 0 -2 -2 1 1 -2 -2 -2 1 1 -2 1 1 orthogonal lifted from C32⋊C4 ρ16 4 4 0 -2 1 4 0 0 1 -2 4 4 0 0 0 0 0 0 1 1 -2 -2 1 1 1 -2 -2 1 -2 -2 orthogonal lifted from C32⋊C4 ρ17 4 4 0 -2 1 4 0 0 1 -2 -4 -4 0 0 0 0 0 0 1 1 -2 -2 -1 -1 -1 2 2 -1 2 2 orthogonal lifted from C2×C32⋊C4 ρ18 4 4 0 1 -2 4 0 0 -2 1 -4 -4 0 0 0 0 0 0 -2 -2 1 1 2 2 2 -1 -1 2 -1 -1 orthogonal lifted from C2×C32⋊C4 ρ19 4 4 0 -2 1 -4 0 0 1 -2 0 0 0 0 0 0 0 0 -1 -1 2 2 3i 3i -3i 0 0 -3i 0 0 complex lifted from C4⋊(C32⋊C4) ρ20 4 4 0 1 -2 -4 0 0 -2 1 0 0 0 0 0 0 0 0 2 2 -1 -1 0 0 0 3i -3i 0 3i -3i complex lifted from C4⋊(C32⋊C4) ρ21 4 4 0 1 -2 -4 0 0 -2 1 0 0 0 0 0 0 0 0 2 2 -1 -1 0 0 0 -3i 3i 0 -3i 3i complex lifted from C4⋊(C32⋊C4) ρ22 4 4 0 -2 1 -4 0 0 1 -2 0 0 0 0 0 0 0 0 -1 -1 2 2 -3i -3i 3i 0 0 3i 0 0 complex lifted from C4⋊(C32⋊C4) ρ23 4 -4 0 1 -2 0 0 0 2 -1 2√2 -2√2 0 0 0 0 0 0 0 0 -3i 3i √2 -√2 -√2 ζ83+2ζ8 2ζ87+ζ85 √2 ζ87+2ζ85 2ζ83+ζ8 complex faithful ρ24 4 -4 0 -2 1 0 0 0 -1 2 -2√2 2√2 0 0 0 0 0 0 3i -3i 0 0 2ζ87+ζ85 2ζ83+ζ8 ζ87+2ζ85 √2 √2 ζ83+2ζ8 -√2 -√2 complex faithful ρ25 4 -4 0 1 -2 0 0 0 2 -1 -2√2 2√2 0 0 0 0 0 0 0 0 3i -3i -√2 √2 √2 2ζ83+ζ8 ζ87+2ζ85 -√2 2ζ87+ζ85 ζ83+2ζ8 complex faithful ρ26 4 -4 0 -2 1 0 0 0 -1 2 -2√2 2√2 0 0 0 0 0 0 -3i 3i 0 0 ζ83+2ζ8 ζ87+2ζ85 2ζ83+ζ8 √2 √2 2ζ87+ζ85 -√2 -√2 complex faithful ρ27 4 -4 0 -2 1 0 0 0 -1 2 2√2 -2√2 0 0 0 0 0 0 3i -3i 0 0 2ζ83+ζ8 2ζ87+ζ85 ζ83+2ζ8 -√2 -√2 ζ87+2ζ85 √2 √2 complex faithful ρ28 4 -4 0 1 -2 0 0 0 2 -1 -2√2 2√2 0 0 0 0 0 0 0 0 -3i 3i -√2 √2 √2 ζ87+2ζ85 2ζ83+ζ8 -√2 ζ83+2ζ8 2ζ87+ζ85 complex faithful ρ29 4 -4 0 -2 1 0 0 0 -1 2 2√2 -2√2 0 0 0 0 0 0 -3i 3i 0 0 ζ87+2ζ85 ζ83+2ζ8 2ζ87+ζ85 -√2 -√2 2ζ83+ζ8 √2 √2 complex faithful ρ30 4 -4 0 1 -2 0 0 0 2 -1 2√2 -2√2 0 0 0 0 0 0 0 0 3i -3i √2 -√2 -√2 2ζ87+ζ85 ζ83+2ζ8 √2 2ζ83+ζ8 ζ87+2ζ85 complex faithful

Smallest permutation representation of C8.(C32⋊C4)
On 48 points
Generators in S48
```(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)
(1 38 25)(2 39 26)(3 40 27)(4 33 28)(5 34 29)(6 35 30)(7 36 31)(8 37 32)(9 46 20)(10 47 21)(11 48 22)(12 41 23)(13 42 24)(14 43 17)(15 44 18)(16 45 19)
(9 20 46)(10 21 47)(11 22 48)(12 23 41)(13 24 42)(14 17 43)(15 18 44)(16 19 45)
(1 11 3 9 5 15 7 13)(2 10 4 16 6 14 8 12)(17 32 41 39 21 28 45 35)(18 31 42 38 22 27 46 34)(19 30 43 37 23 26 47 33)(20 29 44 36 24 25 48 40)```

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

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

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

Matrix representation of C8.(C32⋊C4) in GL4(𝔽73) generated by

 63 0 4 4 0 63 4 4 0 0 51 0 0 0 0 51
,
 72 72 72 0 1 0 0 1 0 0 0 72 0 0 1 72
,
 1 0 1 0 0 1 1 0 0 0 72 1 0 0 72 0
,
 27 0 33 34 27 0 33 33 19 46 46 46 46 27 0 0
`G:=sub<GL(4,GF(73))| [63,0,0,0,0,63,0,0,4,4,51,0,4,4,0,51],[72,1,0,0,72,0,0,0,72,0,0,1,0,1,72,72],[1,0,0,0,0,1,0,0,1,1,72,72,0,0,1,0],[27,27,19,46,0,0,46,27,33,33,46,0,34,33,46,0] >;`

C8.(C32⋊C4) in GAP, Magma, Sage, TeX

`C_8.(C_3^2\rtimes C_4)`
`% in TeX`

`G:=Group("C8.(C3^2:C4)");`
`// GroupNames label`

`G:=SmallGroup(288,419);`
`// by ID`

`G=gap.SmallGroup(288,419);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,3,28,141,176,100,675,80,9413,691,12550,2372]);`
`// Polycyclic`

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

Export

׿
×
𝔽