Copied to
clipboard

G = C52⋊C8order 200 = 23·52

The semidirect product of C52 and C8 acting faithfully

Aliases: C52⋊C8, C5⋊D5.C4, C5⋊F5.C2, SmallGroup(200,40)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C52 — C52⋊C8
 Chief series C1 — C52 — C5⋊D5 — C5⋊F5 — C52⋊C8
 Lower central C52 — C52⋊C8
 Upper central C1

Generators and relations for C52⋊C8
G = < a,b,c | a5=b5=c8=1, ab=ba, cac-1=ab2, cbc-1=ab-1 >

25C2
2C5
2C5
2C5
25C4
10D5
10D5
10D5
25C8
10F5
10F5
10F5

Character table of C52⋊C8

 class 1 2 4A 4B 5A 5B 5C 8A 8B 8C 8D size 1 25 25 25 8 8 8 25 25 25 25 ρ1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 1 1 1 1 1 -1 -1 -1 -1 linear of order 2 ρ3 1 1 -1 -1 1 1 1 i -i i -i linear of order 4 ρ4 1 1 -1 -1 1 1 1 -i i -i i linear of order 4 ρ5 1 -1 i -i 1 1 1 ζ83 ζ8 ζ87 ζ85 linear of order 8 ρ6 1 -1 -i i 1 1 1 ζ8 ζ83 ζ85 ζ87 linear of order 8 ρ7 1 -1 -i i 1 1 1 ζ85 ζ87 ζ8 ζ83 linear of order 8 ρ8 1 -1 i -i 1 1 1 ζ87 ζ85 ζ83 ζ8 linear of order 8 ρ9 8 0 0 0 -2 -2 3 0 0 0 0 orthogonal faithful ρ10 8 0 0 0 -2 3 -2 0 0 0 0 orthogonal faithful ρ11 8 0 0 0 3 -2 -2 0 0 0 0 orthogonal faithful

Permutation representations of C52⋊C8
On 10 points - transitive group 10T18
Generators in S10
```(1 6 8 4 10)(2 5 7 3 9)
(1 6 8 4 10)
(1 2)(3 4 5 6 7 8 9 10)```

`G:=sub<Sym(10)| (1,6,8,4,10)(2,5,7,3,9), (1,6,8,4,10), (1,2)(3,4,5,6,7,8,9,10)>;`

`G:=Group( (1,6,8,4,10)(2,5,7,3,9), (1,6,8,4,10), (1,2)(3,4,5,6,7,8,9,10) );`

`G=PermutationGroup([[(1,6,8,4,10),(2,5,7,3,9)], [(1,6,8,4,10)], [(1,2),(3,4,5,6,7,8,9,10)]])`

`G:=TransitiveGroup(10,18);`

On 20 points - transitive group 20T56
Generators in S20
```(1 17 6 10 13)(2 11 18 14 7)(3 12 19 15 8)(4 16 5 9 20)
(1 17 6 10 13)(3 12 19 15 8)
(1 2 3 4)(5 6 7 8 9 10 11 12)(13 14 15 16 17 18 19 20)```

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

`G:=Group( (1,17,6,10,13)(2,11,18,14,7)(3,12,19,15,8)(4,16,5,9,20), (1,17,6,10,13)(3,12,19,15,8), (1,2,3,4)(5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20) );`

`G=PermutationGroup([[(1,17,6,10,13),(2,11,18,14,7),(3,12,19,15,8),(4,16,5,9,20)], [(1,17,6,10,13),(3,12,19,15,8)], [(1,2,3,4),(5,6,7,8,9,10,11,12),(13,14,15,16,17,18,19,20)]])`

`G:=TransitiveGroup(20,56);`

On 25 points: primitive - transitive group 25T20
Generators in S25
```(1 13 15 11 17)(2 22 16 7 21)(3 12 18 6 25)(4 9 24 23 10)(5 8 14 19 20)
(1 9 3 7 5)(2 14 15 23 18)(4 25 16 20 17)(6 22 19 11 10)(8 13 24 12 21)
(2 3 4 5 6 7 8 9)(10 11 12 13 14 15 16 17)(18 19 20 21 22 23 24 25)```

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

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

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

`G:=TransitiveGroup(25,20);`

C52⋊C8 is a maximal subgroup of   C52⋊M4(2)
C52⋊C8 is a maximal quotient of   C52⋊C16

Polynomial with Galois group C52⋊C8 over ℚ
actionf(x)Disc(f)
10T18x10+2x9-32x8-56x7+288x6+476x5-800x4-1152x3+816x2+704x-416240·36·132·178·434

Matrix representation of C52⋊C8 in GL8(ℤ)

 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 -1 -1 -1 -1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 -1 -1 -1 -1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0
,
 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 -1 -1 -1 -1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 -1 -1 -1 -1 0 0 0 0 1 0 0 0
,
 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 -1 -1 -1 -1 0 0 0 0

`G:=sub<GL(8,Integers())| [0,0,-1,1,0,0,0,0,0,0,-1,0,0,0,0,0,1,0,-1,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,-1,0,1,0,0,0,0,0,-1,0,0,0,0,0,0,1,-1,0,0],[0,0,0,-1,0,0,0,0,1,0,0,-1,0,0,0,0,0,1,0,-1,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,-1,0,0,0,0,0,1,0,-1,0,0,0,0,0,0,1,-1,0],[0,0,0,0,1,0,0,-1,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,-1,0,0,0,0,0,1,0,-1,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0] >;`

C52⋊C8 in GAP, Magma, Sage, TeX

`C_5^2\rtimes C_8`
`% in TeX`

`G:=Group("C5^2:C8");`
`// GroupNames label`

`G:=SmallGroup(200,40);`
`// by ID`

`G=gap.SmallGroup(200,40);`
`# by ID`

`G:=PCGroup([5,-2,-2,-2,-5,5,10,26,3523,168,173,3404,1009,1014]);`
`// Polycyclic`

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

Export

׿
×
𝔽