Copied to
clipboard

## G = C82⋊C3order 192 = 26·3

### The semidirect product of C82 and C3 acting faithfully

Aliases: C82⋊C3, C42.2A4, C22.(C42⋊C3), SmallGroup(192,3)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C82 — C82⋊C3
 Chief series C1 — C22 — C42 — C82 — C82⋊C3
 Lower central C82 — C82⋊C3
 Upper central C1

Generators and relations for C82⋊C3
G = < a,b,c | a8=b8=c3=1, ab=ba, cac-1=ab-1, cbc-1=a3b6 >

3C2
64C3
3C4
3C4
3C8
3C8
3C8
3C8
16A4

Character table of C82⋊C3

 class 1 2 3A 3B 4A 4B 4C 4D 8A 8B 8C 8D 8E 8F 8G 8H 8I 8J 8K 8L 8M 8N 8O 8P size 1 3 64 64 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 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 ζ3 ζ32 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 linear of order 3 ρ3 1 1 ζ32 ζ3 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 linear of order 3 ρ4 3 3 0 0 3 3 3 3 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 orthogonal lifted from A4 ρ5 3 3 0 0 -1 -1 -1 -1 -1+2i -1+2i 1 1 1 1 1 1 1 1 -1-2i -1-2i -1-2i -1-2i -1+2i -1+2i complex lifted from C42⋊C3 ρ6 3 3 0 0 -1 -1 -1 -1 -1-2i -1-2i 1 1 1 1 1 1 1 1 -1+2i -1+2i -1+2i -1+2i -1-2i -1-2i complex lifted from C42⋊C3 ρ7 3 3 0 0 -1 -1 -1 -1 1 1 -1+2i -1+2i -1+2i -1+2i -1-2i -1-2i -1-2i -1-2i 1 1 1 1 1 1 complex lifted from C42⋊C3 ρ8 3 3 0 0 -1 -1 -1 -1 1 1 -1-2i -1-2i -1-2i -1-2i -1+2i -1+2i -1+2i -1+2i 1 1 1 1 1 1 complex lifted from C42⋊C3 ρ9 3 -1 0 0 1 -1+2i -1-2i 1 1+√2 -1-√-2 ζ86+2ζ8 i ζ86+2ζ85 i 2ζ83+ζ82 -i 2ζ87+ζ82 -i 1+√2 -1+√-2 1-√2 -1-√-2 1-√2 -1+√-2 complex faithful ρ10 3 -1 0 0 -1-2i 1 1 -1+2i 2ζ83+ζ82 -i 1+√2 -1+√-2 1-√2 -1-√-2 1-√2 -1+√-2 1+√2 -1-√-2 ζ86+2ζ85 i ζ86+2ζ8 i 2ζ87+ζ82 -i complex faithful ρ11 3 -1 0 0 -1+2i 1 1 -1-2i i ζ86+2ζ8 -1+√-2 1-√2 -1-√-2 1+√2 -1+√-2 1+√2 -1-√-2 1-√2 -i 2ζ87+ζ82 -i 2ζ83+ζ82 i ζ86+2ζ85 complex faithful ρ12 3 -1 0 0 -1-2i 1 1 -1+2i -i 2ζ83+ζ82 -1+√-2 1+√2 -1-√-2 1-√2 -1+√-2 1-√2 -1-√-2 1+√2 i ζ86+2ζ85 i ζ86+2ζ8 -i 2ζ87+ζ82 complex faithful ρ13 3 -1 0 0 1 -1-2i -1+2i 1 1-√2 -1-√-2 2ζ83+ζ82 -i 2ζ87+ζ82 -i ζ86+2ζ8 i ζ86+2ζ85 i 1-√2 -1+√-2 1+√2 -1-√-2 1+√2 -1+√-2 complex faithful ρ14 3 -1 0 0 -1+2i 1 1 -1-2i ζ86+2ζ8 i 1-√2 -1+√-2 1+√2 -1-√-2 1+√2 -1+√-2 1-√2 -1-√-2 2ζ87+ζ82 -i 2ζ83+ζ82 -i ζ86+2ζ85 i complex faithful ρ15 3 -1 0 0 -1+2i 1 1 -1-2i ζ86+2ζ85 i 1+√2 -1-√-2 1-√2 -1+√-2 1-√2 -1-√-2 1+√2 -1+√-2 2ζ83+ζ82 -i 2ζ87+ζ82 -i ζ86+2ζ8 i complex faithful ρ16 3 -1 0 0 -1-2i 1 1 -1+2i -i 2ζ87+ζ82 -1-√-2 1-√2 -1+√-2 1+√2 -1-√-2 1+√2 -1+√-2 1-√2 i ζ86+2ζ8 i ζ86+2ζ85 -i 2ζ83+ζ82 complex faithful ρ17 3 -1 0 0 1 -1+2i -1-2i 1 1-√2 -1+√-2 ζ86+2ζ85 i ζ86+2ζ8 i 2ζ87+ζ82 -i 2ζ83+ζ82 -i 1-√2 -1-√-2 1+√2 -1+√-2 1+√2 -1-√-2 complex faithful ρ18 3 -1 0 0 1 -1-2i -1+2i 1 -1-√-2 1-√2 -i 2ζ83+ζ82 -i 2ζ87+ζ82 i ζ86+2ζ8 i ζ86+2ζ85 -1+√-2 1-√2 -1-√-2 1+√2 -1+√-2 1+√2 complex faithful ρ19 3 -1 0 0 1 -1-2i -1+2i 1 -1+√-2 1+√2 -i 2ζ87+ζ82 -i 2ζ83+ζ82 i ζ86+2ζ85 i ζ86+2ζ8 -1-√-2 1+√2 -1+√-2 1-√2 -1-√-2 1-√2 complex faithful ρ20 3 -1 0 0 1 -1-2i -1+2i 1 1+√2 -1+√-2 2ζ87+ζ82 -i 2ζ83+ζ82 -i ζ86+2ζ85 i ζ86+2ζ8 i 1+√2 -1-√-2 1-√2 -1+√-2 1-√2 -1-√-2 complex faithful ρ21 3 -1 0 0 -1-2i 1 1 -1+2i 2ζ87+ζ82 -i 1-√2 -1-√-2 1+√2 -1+√-2 1+√2 -1-√-2 1-√2 -1+√-2 ζ86+2ζ8 i ζ86+2ζ85 i 2ζ83+ζ82 -i complex faithful ρ22 3 -1 0 0 1 -1+2i -1-2i 1 -1-√-2 1+√2 i ζ86+2ζ8 i ζ86+2ζ85 -i 2ζ83+ζ82 -i 2ζ87+ζ82 -1+√-2 1+√2 -1-√-2 1-√2 -1+√-2 1-√2 complex faithful ρ23 3 -1 0 0 1 -1+2i -1-2i 1 -1+√-2 1-√2 i ζ86+2ζ85 i ζ86+2ζ8 -i 2ζ87+ζ82 -i 2ζ83+ζ82 -1-√-2 1-√2 -1+√-2 1+√2 -1-√-2 1+√2 complex faithful ρ24 3 -1 0 0 -1+2i 1 1 -1-2i i ζ86+2ζ85 -1-√-2 1+√2 -1+√-2 1-√2 -1-√-2 1-√2 -1+√-2 1+√2 -i 2ζ83+ζ82 -i 2ζ87+ζ82 i ζ86+2ζ8 complex faithful

Permutation representations of C82⋊C3
On 24 points - transitive group 24T389
Generators in S24
```(1 2 3 4)(5 6 7 8)(9 10 11 12 13 14 15 16)(17 18 19 20 21 22 23 24)
(1 8 2 5 3 6 4 7)(9 16 15 14 13 12 11 10)(17 21)(18 22)(19 23)(20 24)
(1 18 9)(2 20 11)(3 22 13)(4 24 15)(5 21 16)(6 23 10)(7 17 12)(8 19 14)```

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

`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), (1,8,2,5,3,6,4,7)(9,16,15,14,13,12,11,10)(17,21)(18,22)(19,23)(20,24), (1,18,9)(2,20,11)(3,22,13)(4,24,15)(5,21,16)(6,23,10)(7,17,12)(8,19,14) );`

`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)], [(1,8,2,5,3,6,4,7),(9,16,15,14,13,12,11,10),(17,21),(18,22),(19,23),(20,24)], [(1,18,9),(2,20,11),(3,22,13),(4,24,15),(5,21,16),(6,23,10),(7,17,12),(8,19,14)]])`

`G:=TransitiveGroup(24,389);`

Matrix representation of C82⋊C3 in GL3(𝔽73) generated by

 46 0 0 0 63 0 0 0 63
,
 10 0 0 0 22 0 0 0 1
,
 0 1 0 0 0 1 1 0 0
`G:=sub<GL(3,GF(73))| [46,0,0,0,63,0,0,0,63],[10,0,0,0,22,0,0,0,1],[0,0,1,1,0,0,0,1,0] >;`

C82⋊C3 in GAP, Magma, Sage, TeX

`C_8^2\rtimes C_3`
`% in TeX`

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

`G:=SmallGroup(192,3);`
`// by ID`

`G=gap.SmallGroup(192,3);`
`# by ID`

`G:=PCGroup([7,-3,-2,2,-2,2,-2,2,85,176,695,394,4707,360,1264,102,4037,7062]);`
`// Polycyclic`

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

Export

׿
×
𝔽