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## G = C4⋊F5order 80 = 24·5

### The semidirect product of C4 and F5 acting via F5/D5=C2

Aliases: C4⋊F5, C201C4, D5.Q8, D5.1D4, Dic53C4, D10.5C22, C5⋊(C4⋊C4), (C2×F5).C2, C2.5(C2×F5), C10.4(C2×C4), (C4×D5).4C2, SmallGroup(80,31)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — C4⋊F5
 Chief series C1 — C5 — D5 — D10 — C2×F5 — C4⋊F5
 Lower central C5 — C10 — C4⋊F5
 Upper central C1 — C2 — C4

Generators and relations for C4⋊F5
G = < a,b,c | a4=b5=c4=1, ab=ba, cac-1=a-1, cbc-1=b3 >

Character table of C4⋊F5

 class 1 2A 2B 2C 4A 4B 4C 4D 4E 4F 5 10 20A 20B size 1 1 5 5 2 10 10 10 10 10 4 4 4 4 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 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 linear of order 2 ρ5 1 1 -1 -1 -1 i i -i -i 1 1 1 -1 -1 linear of order 4 ρ6 1 1 -1 -1 1 -i i -i i -1 1 1 1 1 linear of order 4 ρ7 1 1 -1 -1 1 i -i i -i -1 1 1 1 1 linear of order 4 ρ8 1 1 -1 -1 -1 -i -i i i 1 1 1 -1 -1 linear of order 4 ρ9 2 -2 2 -2 0 0 0 0 0 0 2 -2 0 0 orthogonal lifted from D4 ρ10 2 -2 -2 2 0 0 0 0 0 0 2 -2 0 0 symplectic lifted from Q8, Schur index 2 ρ11 4 4 0 0 -4 0 0 0 0 0 -1 -1 1 1 orthogonal lifted from C2×F5 ρ12 4 4 0 0 4 0 0 0 0 0 -1 -1 -1 -1 orthogonal lifted from F5 ρ13 4 -4 0 0 0 0 0 0 0 0 -1 1 -√-5 √-5 complex faithful ρ14 4 -4 0 0 0 0 0 0 0 0 -1 1 √-5 -√-5 complex faithful

Permutation representations of C4⋊F5
On 20 points - transitive group 20T18
Generators in S20
```(1 16 6 11)(2 17 7 12)(3 18 8 13)(4 19 9 14)(5 20 10 15)
(1 2 3 4 5)(6 7 8 9 10)(11 12 13 14 15)(16 17 18 19 20)
(2 3 5 4)(7 8 10 9)(11 16)(12 18 15 19)(13 20 14 17)```

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

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

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

`G:=TransitiveGroup(20,18);`

C4⋊F5 is a maximal subgroup of
C40⋊C4  D5.D8  D20⋊C4  Q8⋊F5  D10.C23  D4×F5  Q8×F5  Dic3⋊F5  C60⋊C4  C100⋊C4  D5.Dic10  Dic5⋊F5  C205F5  C20⋊F5  C202F5  C4⋊S5  A5⋊Q8
C4⋊F5 is a maximal quotient of
C40⋊C4  D5.D8  C40.C4  D10.Q8  C20⋊C8  Dic5⋊C8  D10.3Q8  Dic3⋊F5  C60⋊C4  C100⋊C4  D5.Dic10  Dic5⋊F5  C205F5  C20⋊F5  C202F5

Matrix representation of C4⋊F5 in GL4(𝔽3) generated by

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

C4⋊F5 in GAP, Magma, Sage, TeX

`C_4\rtimes F_5`
`% in TeX`

`G:=Group("C4:F5");`
`// GroupNames label`

`G:=SmallGroup(80,31);`
`// by ID`

`G=gap.SmallGroup(80,31);`
`# by ID`

`G:=PCGroup([5,-2,-2,-2,-2,-5,20,101,46,804,414]);`
`// Polycyclic`

`G:=Group<a,b,c|a^4=b^5=c^4=1,a*b=b*a,c*a*c^-1=a^-1,c*b*c^-1=b^3>;`
`// generators/relations`

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