Copied to
clipboard

## G = C42.7C22order 64 = 26

### 7th non-split extension by C42 of C22 acting faithfully

p-group, metabelian, nilpotent (class 2), monomial

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C22 — C42.7C22
 Chief series C1 — C2 — C4 — C2×C4 — C42 — C42⋊C2 — C42.7C22
 Lower central C1 — C22 — C42.7C22
 Upper central C1 — C2×C4 — C42.7C22
 Jennings C1 — C2 — C2 — C2×C4 — C42.7C22

Generators and relations for C42.7C22
G = < a,b,c,d | a4=b4=d2=1, c2=b, ab=ba, cac-1=a-1b2, dad=ab2, bc=cb, bd=db, dcd=a2b2c >

Character table of C42.7C22

 class 1 2A 2B 2C 2D 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 8A 8B 8C 8D 8E 8F 8G 8H 8I 8J 8K 8L size 1 1 1 1 4 1 1 1 1 2 2 2 2 4 4 4 2 2 2 2 2 2 2 2 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 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 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 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 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 -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 -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 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 1 -1 linear of order 2 ρ9 1 1 1 1 -1 -1 -1 -1 -1 -1 1 -1 1 1 1 -1 -i i i i -i -i -i i -i i i -i linear of order 4 ρ10 1 1 1 1 -1 -1 -1 -1 -1 -1 1 -1 1 1 1 -1 i -i -i -i i i i -i i -i -i i linear of order 4 ρ11 1 1 1 1 1 -1 -1 -1 -1 -1 1 -1 1 -1 -1 1 -i i i i -i -i -i i i -i -i i linear of order 4 ρ12 1 1 1 1 1 -1 -1 -1 -1 -1 1 -1 1 -1 -1 1 i -i -i -i i i i -i -i i i -i linear of order 4 ρ13 1 1 1 1 1 -1 -1 -1 -1 1 -1 1 -1 -1 1 -1 -i -i i i i i -i -i i i -i -i linear of order 4 ρ14 1 1 1 1 1 -1 -1 -1 -1 1 -1 1 -1 -1 1 -1 i i -i -i -i -i i i -i -i i i linear of order 4 ρ15 1 1 1 1 -1 -1 -1 -1 -1 1 -1 1 -1 1 -1 1 -i -i i i i i -i -i -i -i i i linear of order 4 ρ16 1 1 1 1 -1 -1 -1 -1 -1 1 -1 1 -1 1 -1 1 i i -i -i -i -i i i i i -i -i linear of order 4 ρ17 2 -2 -2 2 0 -2 -2 2 2 -2i -2i 2i 2i 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 complex lifted from C4○D4 ρ18 2 -2 -2 2 0 2 2 -2 -2 -2i 2i 2i -2i 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 complex lifted from C4○D4 ρ19 2 -2 -2 2 0 -2 -2 2 2 2i 2i -2i -2i 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 complex lifted from C4○D4 ρ20 2 -2 -2 2 0 2 2 -2 -2 2i -2i -2i 2i 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 complex lifted from C4○D4 ρ21 2 -2 2 -2 0 2i -2i -2i 2i 0 0 0 0 0 0 0 0 2ζ8 0 0 2ζ83 2ζ87 0 2ζ85 0 0 0 0 complex lifted from C8○D4 ρ22 2 2 -2 -2 0 2i -2i 2i -2i 0 0 0 0 0 0 0 2ζ85 0 2ζ87 2ζ83 0 0 2ζ8 0 0 0 0 0 complex lifted from C8○D4 ρ23 2 2 -2 -2 0 -2i 2i -2i 2i 0 0 0 0 0 0 0 2ζ83 0 2ζ8 2ζ85 0 0 2ζ87 0 0 0 0 0 complex lifted from C8○D4 ρ24 2 -2 2 -2 0 -2i 2i 2i -2i 0 0 0 0 0 0 0 0 2ζ83 0 0 2ζ8 2ζ85 0 2ζ87 0 0 0 0 complex lifted from C8○D4 ρ25 2 2 -2 -2 0 2i -2i 2i -2i 0 0 0 0 0 0 0 2ζ8 0 2ζ83 2ζ87 0 0 2ζ85 0 0 0 0 0 complex lifted from C8○D4 ρ26 2 -2 2 -2 0 -2i 2i 2i -2i 0 0 0 0 0 0 0 0 2ζ87 0 0 2ζ85 2ζ8 0 2ζ83 0 0 0 0 complex lifted from C8○D4 ρ27 2 2 -2 -2 0 -2i 2i -2i 2i 0 0 0 0 0 0 0 2ζ87 0 2ζ85 2ζ8 0 0 2ζ83 0 0 0 0 0 complex lifted from C8○D4 ρ28 2 -2 2 -2 0 2i -2i -2i 2i 0 0 0 0 0 0 0 0 2ζ85 0 0 2ζ87 2ζ83 0 2ζ8 0 0 0 0 complex lifted from C8○D4

Smallest permutation representation of C42.7C22
On 32 points
Generators in S32
```(1 19 27 12)(2 9 28 24)(3 21 29 14)(4 11 30 18)(5 23 31 16)(6 13 32 20)(7 17 25 10)(8 15 26 22)
(1 3 5 7)(2 4 6 8)(9 11 13 15)(10 12 14 16)(17 19 21 23)(18 20 22 24)(25 27 29 31)(26 28 30 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)
(2 32)(4 26)(6 28)(8 30)(9 24)(10 14)(11 18)(12 16)(13 20)(15 22)(17 21)(19 23)```

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

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

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

Matrix representation of C42.7C22 in GL4(𝔽17) generated by

 0 16 0 0 1 0 0 0 0 0 0 16 0 0 16 0
,
 4 0 0 0 0 4 0 0 0 0 4 0 0 0 0 4
,
 15 0 0 0 0 15 0 0 0 0 0 8 0 0 9 0
,
 1 0 0 0 0 16 0 0 0 0 1 0 0 0 0 16
`G:=sub<GL(4,GF(17))| [0,1,0,0,16,0,0,0,0,0,0,16,0,0,16,0],[4,0,0,0,0,4,0,0,0,0,4,0,0,0,0,4],[15,0,0,0,0,15,0,0,0,0,0,9,0,0,8,0],[1,0,0,0,0,16,0,0,0,0,1,0,0,0,0,16] >;`

C42.7C22 in GAP, Magma, Sage, TeX

`C_4^2._7C_2^2`
`% in TeX`

`G:=Group("C4^2.7C2^2");`
`// GroupNames label`

`G:=SmallGroup(64,114);`
`// by ID`

`G=gap.SmallGroup(64,114);`
`# by ID`

`G:=PCGroup([6,-2,2,2,-2,2,-2,96,121,332,50,88]);`
`// Polycyclic`

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

Export

׿
×
𝔽