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## G = C23.F5order 160 = 25·5

### The non-split extension by C23 of F5 acting via F5/C5=C4

Aliases: C23.F5, Dic5.4D4, C5⋊(C4.D4), C22.F51C2, C22.5(C2×F5), (C22×C10).4C4, (C22×D5).2C4, C2.11(C22⋊F5), C10.11(C22⋊C4), (C2×Dic5).24C22, (C2×C5⋊D4).8C2, (C2×C10).12(C2×C4), SmallGroup(160,88)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C23.F5
 Chief series C1 — C5 — C10 — Dic5 — C2×Dic5 — 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=1, e4=c, 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 4A 4B 5 8A 8B 8C 8D 10A 10B 10C 10D 10E 10F 10G size 1 1 2 4 20 10 10 4 20 20 20 20 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 -1 1 -i i i -i -1 -1 1 1 1 -1 -1 linear of order 4 ρ6 1 1 1 -1 1 -1 -1 1 i -i -i i -1 -1 1 1 1 -1 -1 linear of order 4 ρ7 1 1 1 1 -1 -1 -1 1 -i -i i i 1 1 1 1 1 1 1 linear of order 4 ρ8 1 1 1 1 -1 -1 -1 1 i i -i -i 1 1 1 1 1 1 1 linear of order 4 ρ9 2 2 -2 0 0 2 -2 2 0 0 0 0 0 0 -2 -2 2 0 0 orthogonal lifted from D4 ρ10 2 2 -2 0 0 -2 2 2 0 0 0 0 0 0 -2 -2 2 0 0 orthogonal lifted from D4 ρ11 4 -4 0 0 0 0 0 4 0 0 0 0 0 0 0 0 -4 0 0 orthogonal lifted from C4.D4 ρ12 4 4 4 -4 0 0 0 -1 0 0 0 0 1 1 -1 -1 -1 1 1 orthogonal lifted from C2×F5 ρ13 4 4 4 4 0 0 0 -1 0 0 0 0 -1 -1 -1 -1 -1 -1 -1 orthogonal lifted from F5 ρ14 4 4 -4 0 0 0 0 -1 0 0 0 0 √5 -√5 1 1 -1 -√5 √5 orthogonal lifted from C22⋊F5 ρ15 4 4 -4 0 0 0 0 -1 0 0 0 0 -√5 √5 1 1 -1 √5 -√5 orthogonal lifted from C22⋊F5 ρ16 4 -4 0 0 0 0 0 -1 0 0 0 0 2ζ53+2ζ5+1 2ζ54+2ζ53+1 √5 -√5 1 2ζ52+2ζ5+1 2ζ54+2ζ52+1 complex faithful ρ17 4 -4 0 0 0 0 0 -1 0 0 0 0 2ζ52+2ζ5+1 2ζ53+2ζ5+1 -√5 √5 1 2ζ54+2ζ52+1 2ζ54+2ζ53+1 complex faithful ρ18 4 -4 0 0 0 0 0 -1 0 0 0 0 2ζ54+2ζ53+1 2ζ54+2ζ52+1 -√5 √5 1 2ζ53+2ζ5+1 2ζ52+2ζ5+1 complex faithful ρ19 4 -4 0 0 0 0 0 -1 0 0 0 0 2ζ54+2ζ52+1 2ζ52+2ζ5+1 √5 -√5 1 2ζ54+2ζ53+1 2ζ53+2ζ5+1 complex faithful

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

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

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

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

C23.F5 is a maximal subgroup of
C5⋊C2≀C4  C22⋊C4⋊F5  (C22×C4)⋊F5  C242F5  (C4×D5).D4  (C2×D4).9F5  D5⋊(C4.D4)  Dic5.D12  C5⋊(C12.D4)
C23.F5 is a maximal quotient of
C5⋊(C23⋊C8)  C22.F5⋊C4  (C2×D4).F5  Dic5.SD16  (C2×Q8).F5  Dic5.Q16  C24.F5  Dic5.D12  C5⋊(C12.D4)

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

 18 35 0 0 6 23 0 0 23 20 18 35 35 18 6 23
,
 1 0 0 0 0 1 0 0 1 0 40 0 35 35 0 40
,
 40 0 0 0 0 40 0 0 0 0 40 0 0 0 0 40
,
 6 40 0 0 1 0 0 0 29 2 35 35 14 0 6 40
,
 1 0 39 0 35 35 0 39 12 3 40 0 5 27 6 6
`G:=sub<GL(4,GF(41))| [18,6,23,35,35,23,20,18,0,0,18,6,0,0,35,23],[1,0,1,35,0,1,0,35,0,0,40,0,0,0,0,40],[40,0,0,0,0,40,0,0,0,0,40,0,0,0,0,40],[6,1,29,14,40,0,2,0,0,0,35,6,0,0,35,40],[1,35,12,5,0,35,3,27,39,0,40,6,0,39,0,6] >;`

C23.F5 in GAP, Magma, Sage, TeX

`C_2^3.F_5`
`% in TeX`

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

`G:=SmallGroup(160,88);`
`// by ID`

`G=gap.SmallGroup(160,88);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-5,24,121,188,86,579,2309,1169]);`
`// Polycyclic`

`G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^5=1,e^4=c,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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