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## G = C23⋊F5order 160 = 25·5

### The semidirect product of C23 and F5 acting via F5/C5=C4

Aliases: C23⋊F5, D10.4D4, C52(C23⋊C4), C22⋊F52C2, (C22×C10)⋊3C4, (C2×Dic5)⋊2C4, C22.4(C2×F5), C10.9(C22⋊C4), C2.10(C22⋊F5), (C22×D5).15C22, (C2×C5⋊D4).7C2, (C2×C10).10(C2×C4), SmallGroup(160,86)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C23⋊F5
 Chief series C1 — C5 — C10 — D10 — C22×D5 — C22⋊F5 — C23⋊F5
 Lower central C5 — C10 — C2×C10 — C23⋊F5
 Upper central C1 — C2 — C22 — C23

Generators and relations for C23⋊F5
G = < a,b,c,d,e | a2=b2=c2=d5=e4=1, ab=ba, ac=ca, ad=da, eae-1=abc, ebe-1=bc=cb, bd=db, cd=dc, ce=ec, ede-1=d3 >

Character table of C23⋊F5

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E 5 10A 10B 10C 10D 10E 10F 10G size 1 1 2 4 10 10 20 20 20 20 20 4 4 4 4 4 4 4 4 ρ1 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 linear of order 2 ρ3 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 linear of order 2 ρ5 1 1 1 1 -1 -1 i -1 -i i -i 1 1 1 1 1 1 1 1 linear of order 4 ρ6 1 1 1 1 -1 -1 -i -1 i -i i 1 1 1 1 1 1 1 1 linear of order 4 ρ7 1 1 1 -1 -1 -1 -i 1 -i i i 1 -1 -1 1 1 1 -1 -1 linear of order 4 ρ8 1 1 1 -1 -1 -1 i 1 i -i -i 1 -1 -1 1 1 1 -1 -1 linear of order 4 ρ9 2 2 -2 0 2 -2 0 0 0 0 0 2 0 0 -2 -2 2 0 0 orthogonal lifted from D4 ρ10 2 2 -2 0 -2 2 0 0 0 0 0 2 0 0 -2 -2 2 0 0 orthogonal lifted from D4 ρ11 4 -4 0 0 0 0 0 0 0 0 0 4 0 0 0 0 -4 0 0 orthogonal lifted from C23⋊C4 ρ12 4 4 4 4 0 0 0 0 0 0 0 -1 -1 -1 -1 -1 -1 -1 -1 orthogonal lifted from F5 ρ13 4 4 4 -4 0 0 0 0 0 0 0 -1 1 1 -1 -1 -1 1 1 orthogonal lifted from C2×F5 ρ14 4 4 -4 0 0 0 0 0 0 0 0 -1 -√5 √5 1 1 -1 √5 -√5 orthogonal lifted from C22⋊F5 ρ15 4 4 -4 0 0 0 0 0 0 0 0 -1 √5 -√5 1 1 -1 -√5 √5 orthogonal lifted from C22⋊F5 ρ16 4 -4 0 0 0 0 0 0 0 0 0 -1 2ζ54+2ζ52+1 2ζ52+2ζ5+1 √5 -√5 1 2ζ54+2ζ53+1 2ζ53+2ζ5+1 complex faithful ρ17 4 -4 0 0 0 0 0 0 0 0 0 -1 2ζ54+2ζ53+1 2ζ54+2ζ52+1 -√5 √5 1 2ζ53+2ζ5+1 2ζ52+2ζ5+1 complex faithful ρ18 4 -4 0 0 0 0 0 0 0 0 0 -1 2ζ52+2ζ5+1 2ζ53+2ζ5+1 -√5 √5 1 2ζ54+2ζ52+1 2ζ54+2ζ53+1 complex faithful ρ19 4 -4 0 0 0 0 0 0 0 0 0 -1 2ζ53+2ζ5+1 2ζ54+2ζ53+1 √5 -√5 1 2ζ52+2ζ5+1 2ζ54+2ζ52+1 complex faithful

Smallest permutation representation of C23⋊F5
On 40 points
Generators in S40
```(1 21)(2 22)(3 23)(4 24)(5 25)(6 26)(7 27)(8 28)(9 29)(10 30)(11 31)(12 32)(13 33)(14 34)(15 35)(16 36)(17 37)(18 38)(19 39)(20 40)
(1 11)(2 12)(3 13)(4 14)(5 15)(6 16)(7 17)(8 18)(9 19)(10 20)(21 31)(22 32)(23 33)(24 34)(25 35)(26 36)(27 37)(28 38)(29 39)(30 40)
(1 6)(2 7)(3 8)(4 9)(5 10)(11 16)(12 17)(13 18)(14 19)(15 20)(21 26)(22 27)(23 28)(24 29)(25 30)(31 36)(32 37)(33 38)(34 39)(35 40)
(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 33 34 35)(36 37 38 39 40)
(2 3 5 4)(7 8 10 9)(11 16)(12 18 15 19)(13 20 14 17)(21 31 26 36)(22 33 30 39)(23 35 29 37)(24 32 28 40)(25 34 27 38)```

`G:=sub<Sym(40)| (1,21)(2,22)(3,23)(4,24)(5,25)(6,26)(7,27)(8,28)(9,29)(10,30)(11,31)(12,32)(13,33)(14,34)(15,35)(16,36)(17,37)(18,38)(19,39)(20,40), (1,11)(2,12)(3,13)(4,14)(5,15)(6,16)(7,17)(8,18)(9,19)(10,20)(21,31)(22,32)(23,33)(24,34)(25,35)(26,36)(27,37)(28,38)(29,39)(30,40), (1,6)(2,7)(3,8)(4,9)(5,10)(11,16)(12,17)(13,18)(14,19)(15,20)(21,26)(22,27)(23,28)(24,29)(25,30)(31,36)(32,37)(33,38)(34,39)(35,40), (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)(31,32,33,34,35)(36,37,38,39,40), (2,3,5,4)(7,8,10,9)(11,16)(12,18,15,19)(13,20,14,17)(21,31,26,36)(22,33,30,39)(23,35,29,37)(24,32,28,40)(25,34,27,38)>;`

`G:=Group( (1,21)(2,22)(3,23)(4,24)(5,25)(6,26)(7,27)(8,28)(9,29)(10,30)(11,31)(12,32)(13,33)(14,34)(15,35)(16,36)(17,37)(18,38)(19,39)(20,40), (1,11)(2,12)(3,13)(4,14)(5,15)(6,16)(7,17)(8,18)(9,19)(10,20)(21,31)(22,32)(23,33)(24,34)(25,35)(26,36)(27,37)(28,38)(29,39)(30,40), (1,6)(2,7)(3,8)(4,9)(5,10)(11,16)(12,17)(13,18)(14,19)(15,20)(21,26)(22,27)(23,28)(24,29)(25,30)(31,36)(32,37)(33,38)(34,39)(35,40), (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)(31,32,33,34,35)(36,37,38,39,40), (2,3,5,4)(7,8,10,9)(11,16)(12,18,15,19)(13,20,14,17)(21,31,26,36)(22,33,30,39)(23,35,29,37)(24,32,28,40)(25,34,27,38) );`

`G=PermutationGroup([[(1,21),(2,22),(3,23),(4,24),(5,25),(6,26),(7,27),(8,28),(9,29),(10,30),(11,31),(12,32),(13,33),(14,34),(15,35),(16,36),(17,37),(18,38),(19,39),(20,40)], [(1,11),(2,12),(3,13),(4,14),(5,15),(6,16),(7,17),(8,18),(9,19),(10,20),(21,31),(22,32),(23,33),(24,34),(25,35),(26,36),(27,37),(28,38),(29,39),(30,40)], [(1,6),(2,7),(3,8),(4,9),(5,10),(11,16),(12,17),(13,18),(14,19),(15,20),(21,26),(22,27),(23,28),(24,29),(25,30),(31,36),(32,37),(33,38),(34,39),(35,40)], [(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),(31,32,33,34,35),(36,37,38,39,40)], [(2,3,5,4),(7,8,10,9),(11,16),(12,18,15,19),(13,20,14,17),(21,31,26,36),(22,33,30,39),(23,35,29,37),(24,32,28,40),(25,34,27,38)]])`

C23⋊F5 is a maximal subgroup of
C5⋊C2≀C4  C22⋊C4⋊F5  (C22×C4)⋊F5  C242F5  C23⋊F55C2  (C2×D4)⋊7F5  (C2×D4)⋊8F5  D10.D12  C3⋊(C23⋊F5)
C23⋊F5 is a maximal quotient of
(C22×C4).F5  (C22×C4)⋊F5  C22⋊F5⋊C4  D10.SD16  (C2×D4)⋊F5  (D4×C10).C4  D10.Q16  (C2×Q8)⋊F5  (Q8×C10).C4  C24.F5  C242F5  D10.D12  C3⋊(C23⋊F5)

Matrix representation of C23⋊F5 in GL4(𝔽41) generated by

 5 10 32 19 22 27 32 13 28 9 14 19 22 9 31 36
,
 22 0 3 3 38 19 38 0 0 38 19 38 3 3 0 22
,
 40 0 0 0 0 40 0 0 0 0 40 0 0 0 0 40
,
 40 40 40 40 1 0 0 0 0 1 0 0 0 0 1 0
,
 1 0 0 0 0 0 0 1 0 1 0 0 40 40 40 40
`G:=sub<GL(4,GF(41))| [5,22,28,22,10,27,9,9,32,32,14,31,19,13,19,36],[22,38,0,3,0,19,38,3,3,38,19,0,3,0,38,22],[40,0,0,0,0,40,0,0,0,0,40,0,0,0,0,40],[40,1,0,0,40,0,1,0,40,0,0,1,40,0,0,0],[1,0,0,40,0,0,1,40,0,0,0,40,0,1,0,40] >;`

C23⋊F5 in GAP, Magma, Sage, TeX

`C_2^3\rtimes F_5`
`% in TeX`

`G:=Group("C2^3:F5");`
`// GroupNames label`

`G:=SmallGroup(160,86);`
`// by ID`

`G=gap.SmallGroup(160,86);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-5,24,121,188,579,2309,1169]);`
`// Polycyclic`

`G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^5=e^4=1,a*b=b*a,a*c=c*a,a*d=d*a,e*a*e^-1=a*b*c,e*b*e^-1=b*c=c*b,b*d=d*b,c*d=d*c,c*e=e*c,e*d*e^-1=d^3>;`
`// generators/relations`

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