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## G = C25⋊C4order 100 = 22·52

### The semidirect product of C25 and C4 acting faithfully

Aliases: C25⋊C4, C5.F5, D25.C2, SmallGroup(100,3)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C25 — C25⋊C4
 Chief series C1 — C5 — C25 — D25 — C25⋊C4
 Lower central C25 — C25⋊C4
 Upper central C1

Generators and relations for C25⋊C4
G = < a,b | a25=b4=1, bab-1=a18 >

25C2
25C4
5D5
5F5

Character table of C25⋊C4

 class 1 2 4A 4B 5 25A 25B 25C 25D 25E size 1 25 25 25 4 4 4 4 4 4 ρ1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 -1 -1 1 1 1 1 1 1 linear of order 2 ρ3 1 -1 -i i 1 1 1 1 1 1 linear of order 4 ρ4 1 -1 i -i 1 1 1 1 1 1 linear of order 4 ρ5 4 0 0 0 4 -1 -1 -1 -1 -1 orthogonal lifted from F5 ρ6 4 0 0 0 -1 ζ2516+ζ2513+ζ2512+ζ259 ζ2524+ζ2518+ζ257+ζ25 ζ2522+ζ2521+ζ254+ζ253 ζ2519+ζ2517+ζ258+ζ256 ζ2523+ζ2514+ζ2511+ζ252 orthogonal faithful ρ7 4 0 0 0 -1 ζ2524+ζ2518+ζ257+ζ25 ζ2523+ζ2514+ζ2511+ζ252 ζ2519+ζ2517+ζ258+ζ256 ζ2516+ζ2513+ζ2512+ζ259 ζ2522+ζ2521+ζ254+ζ253 orthogonal faithful ρ8 4 0 0 0 -1 ζ2519+ζ2517+ζ258+ζ256 ζ2516+ζ2513+ζ2512+ζ259 ζ2523+ζ2514+ζ2511+ζ252 ζ2522+ζ2521+ζ254+ζ253 ζ2524+ζ2518+ζ257+ζ25 orthogonal faithful ρ9 4 0 0 0 -1 ζ2522+ζ2521+ζ254+ζ253 ζ2519+ζ2517+ζ258+ζ256 ζ2524+ζ2518+ζ257+ζ25 ζ2523+ζ2514+ζ2511+ζ252 ζ2516+ζ2513+ζ2512+ζ259 orthogonal faithful ρ10 4 0 0 0 -1 ζ2523+ζ2514+ζ2511+ζ252 ζ2522+ζ2521+ζ254+ζ253 ζ2516+ζ2513+ζ2512+ζ259 ζ2524+ζ2518+ζ257+ζ25 ζ2519+ζ2517+ζ258+ζ256 orthogonal faithful

Permutation representations of C25⋊C4
On 25 points - transitive group 25T8
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)
(2 8 25 19)(3 15 24 12)(4 22 23 5)(6 11 21 16)(7 18 20 9)(10 14 17 13)```

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

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

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

`G:=TransitiveGroup(25,8);`

C25⋊C4 is a maximal subgroup of   C75⋊C4  C125⋊C4  C25⋊C20  D25.D5  C25⋊F5  C252F5
C25⋊C4 is a maximal quotient of   C25⋊C8  C75⋊C4  C125⋊C4  D25.D5  C25⋊F5  C252F5

Matrix representation of C25⋊C4 in GL4(𝔽7) generated by

 5 2 5 5 6 0 2 3 5 4 5 2 6 1 1 6
,
 0 1 3 0 0 6 1 2 2 4 3 2 2 6 3 5
`G:=sub<GL(4,GF(7))| [5,6,5,6,2,0,4,1,5,2,5,1,5,3,2,6],[0,0,2,2,1,6,4,6,3,1,3,3,0,2,2,5] >;`

C25⋊C4 in GAP, Magma, Sage, TeX

`C_{25}\rtimes C_4`
`% in TeX`

`G:=Group("C25:C4");`
`// GroupNames label`

`G:=SmallGroup(100,3);`
`// by ID`

`G=gap.SmallGroup(100,3);`
`# by ID`

`G:=PCGroup([4,-2,-2,-5,-5,8,338,582,70,643,647]);`
`// Polycyclic`

`G:=Group<a,b|a^25=b^4=1,b*a*b^-1=a^18>;`
`// generators/relations`

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