Copied to
clipboard

## G = C5⋊D5order 50 = 2·52

### The semidirect product of C5 and D5 acting via D5/C5=C2

Aliases: C5⋊D5, C522C2, SmallGroup(50,4)

Series: Derived Chief Lower central Upper central

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

Generators and relations for C5⋊D5
G = < a,b,c | a5=b5=c2=1, ab=ba, cac=a-1, cbc=b-1 >

Character table of C5⋊D5

 class 1 2 5A 5B 5C 5D 5E 5F 5G 5H 5I 5J 5K 5L size 1 25 2 2 2 2 2 2 2 2 2 2 2 2 ρ1 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 linear of order 2 ρ3 2 0 -1+√5/2 2 -1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 2 -1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 -1-√5/2 orthogonal lifted from D5 ρ4 2 0 -1+√5/2 -1+√5/2 -1+√5/2 -1-√5/2 2 -1-√5/2 -1-√5/2 2 -1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 orthogonal lifted from D5 ρ5 2 0 -1+√5/2 -1-√5/2 2 -1-√5/2 -1+√5/2 -1+√5/2 -1+√5/2 -1-√5/2 2 -1-√5/2 -1+√5/2 -1-√5/2 orthogonal lifted from D5 ρ6 2 0 -1-√5/2 -1+√5/2 2 -1+√5/2 -1-√5/2 -1-√5/2 -1-√5/2 -1+√5/2 2 -1+√5/2 -1-√5/2 -1+√5/2 orthogonal lifted from D5 ρ7 2 0 -1-√5/2 -1-√5/2 -1+√5/2 2 -1+√5/2 -1-√5/2 -1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 2 -1+√5/2 orthogonal lifted from D5 ρ8 2 0 2 -1+√5/2 -1+√5/2 -1+√5/2 -1+√5/2 -1+√5/2 -1-√5/2 -1-√5/2 -1-√5/2 -1-√5/2 -1-√5/2 2 orthogonal lifted from D5 ρ9 2 0 -1-√5/2 -1-√5/2 -1-√5/2 -1+√5/2 2 -1+√5/2 -1+√5/2 2 -1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 orthogonal lifted from D5 ρ10 2 0 -1+√5/2 -1+√5/2 -1-√5/2 2 -1-√5/2 -1+√5/2 -1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 2 -1-√5/2 orthogonal lifted from D5 ρ11 2 0 -1-√5/2 -1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 2 -1-√5/2 -1-√5/2 -1+√5/2 2 -1+√5/2 -1+√5/2 orthogonal lifted from D5 ρ12 2 0 -1-√5/2 2 -1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 2 -1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 -1+√5/2 orthogonal lifted from D5 ρ13 2 0 -1+√5/2 -1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 2 -1+√5/2 -1+√5/2 -1-√5/2 2 -1-√5/2 -1-√5/2 orthogonal lifted from D5 ρ14 2 0 2 -1-√5/2 -1-√5/2 -1-√5/2 -1-√5/2 -1-√5/2 -1+√5/2 -1+√5/2 -1+√5/2 -1+√5/2 -1+√5/2 2 orthogonal lifted from D5

Permutation representations of C5⋊D5
On 25 points - transitive group 25T5
Generators in S25
```(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)
(1 24 19 14 9)(2 25 20 15 10)(3 21 16 11 6)(4 22 17 12 7)(5 23 18 13 8)
(1 9)(2 8)(3 7)(4 6)(5 10)(11 22)(12 21)(13 25)(14 24)(15 23)(16 17)(18 20)```

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

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

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

`G:=TransitiveGroup(25,5);`

C5⋊D5 is a maximal subgroup of
C5⋊F5  C52⋊C4  D52  C52⋊C6  C5⋊D15  C52⋊C10  C25⋊D5  C53⋊C2  C5⋊D35
C5⋊D5 is a maximal quotient of
C526C4  C5⋊D15  C25⋊D5  He5⋊C2  C53⋊C2  C5⋊D35

Matrix representation of C5⋊D5 in GL4(𝔽11) generated by

 8 1 0 0 2 10 0 0 0 0 0 1 0 0 10 3
,
 10 10 0 0 9 8 0 0 0 0 0 1 0 0 10 3
,
 0 7 0 0 8 0 0 0 0 0 10 0 0 0 8 1
`G:=sub<GL(4,GF(11))| [8,2,0,0,1,10,0,0,0,0,0,10,0,0,1,3],[10,9,0,0,10,8,0,0,0,0,0,10,0,0,1,3],[0,8,0,0,7,0,0,0,0,0,10,8,0,0,0,1] >;`

C5⋊D5 in GAP, Magma, Sage, TeX

`C_5\rtimes D_5`
`% in TeX`

`G:=Group("C5:D5");`
`// GroupNames label`

`G:=SmallGroup(50,4);`
`// by ID`

`G=gap.SmallGroup(50,4);`
`# by ID`

`G:=PCGroup([3,-2,-5,-5,49,362]);`
`// Polycyclic`

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

Export

׿
×
𝔽