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G = C24⋊2F5order 320 = 26·5

1st semidirect product of C24 and F5 acting via F5/C5=C4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22×C10 — C24⋊2F5
 Chief series C1 — C5 — C10 — C2×C10 — C22×D5 — C2×C5⋊D4 — C23⋊F5 — C24⋊2F5
 Lower central C5 — C10 — C2×C10 — C22×C10 — C24⋊2F5
 Upper central C1 — C2 — C22 — C23 — C24

Generators and relations for C242F5
G = < a,b,c,d,e,f | a2=b2=c2=d2=e5=f4=1, ab=ba, ac=ca, ad=da, ae=ea, faf-1=abcd, bc=cb, bd=db, be=eb, fbf-1=bcd, fcf-1=cd=dc, ce=ec, de=ed, df=fd, fef-1=e3 >

Subgroups: 506 in 94 conjugacy classes, 18 normal (all characteristic)
C1, C2, C2, C4, C22, C22, C5, C8, C2×C4, D4, C23, C23, D5, C10, C10, C22⋊C4, M4(2), C2×D4, C24, Dic5, F5, D10, C2×C10, C2×C10, C23⋊C4, C4.D4, C22≀C2, C5⋊C8, C2×Dic5, C2×Dic5, C5⋊D4, C2×F5, C22×D5, C22×C10, C22×C10, C2≀C4, C23.D5, C23.D5, C22.F5, C22⋊F5, C2×C5⋊D4, C2×C5⋊D4, C23×C10, C23⋊F5, C23.F5, C242D5, C242F5
Quotients: C1, C2, C4, C22, C2×C4, D4, C22⋊C4, F5, C23⋊C4, C2×F5, C2≀C4, C22⋊F5, C23⋊F5, C242F5

Character table of C242F5

 class 1 2A 2B 2C 2D 2E 2F 4A 4B 4C 4D 5 8A 8B 10A 10B 10C 10D 10E 10F 10G 10H 10I 10J 10K 10L 10M 10N 10O size 1 1 2 4 4 4 20 20 40 40 40 4 40 40 4 4 4 4 4 4 4 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 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 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 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 1 -1 -1 -1 -1 1 1 1 1 1 linear of order 2 ρ5 1 1 1 1 1 1 -1 -1 i -1 -i 1 i -i 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 linear of order 4 ρ6 1 1 1 1 1 1 -1 -1 -i -1 i 1 -i i 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 linear of order 4 ρ7 1 1 1 -1 1 -1 -1 -1 i 1 -i 1 -i i -1 -1 -1 1 -1 1 -1 -1 -1 -1 1 1 1 1 1 linear of order 4 ρ8 1 1 1 -1 1 -1 -1 -1 -i 1 i 1 i -i -1 -1 -1 1 -1 1 -1 -1 -1 -1 1 1 1 1 1 linear of order 4 ρ9 2 2 2 0 -2 0 2 -2 0 0 0 2 0 0 0 0 0 2 0 2 0 0 0 0 -2 -2 2 -2 -2 orthogonal lifted from D4 ρ10 2 2 2 0 -2 0 -2 2 0 0 0 2 0 0 0 0 0 2 0 2 0 0 0 0 -2 -2 2 -2 -2 orthogonal lifted from D4 ρ11 4 4 -4 0 0 0 0 0 0 0 0 4 0 0 0 0 0 -4 0 -4 0 0 0 0 0 0 4 0 0 orthogonal lifted from C23⋊C4 ρ12 4 -4 0 2 0 -2 0 0 0 0 0 4 0 0 2 2 2 0 2 0 -2 -2 -2 -2 0 0 -4 0 0 orthogonal lifted from C2≀C4 ρ13 4 -4 0 -2 0 2 0 0 0 0 0 4 0 0 -2 -2 -2 0 -2 0 2 2 2 2 0 0 -4 0 0 orthogonal lifted from C2≀C4 ρ14 4 4 4 -4 4 -4 0 0 0 0 0 -1 0 0 1 1 1 -1 1 -1 1 1 1 1 -1 -1 -1 -1 -1 orthogonal lifted from C2×F5 ρ15 4 4 4 4 4 4 0 0 0 0 0 -1 0 0 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 orthogonal lifted from F5 ρ16 4 4 4 0 -4 0 0 0 0 0 0 -1 0 0 √5 -√5 √5 -1 -√5 -1 -√5 √5 √5 -√5 1 1 -1 1 1 orthogonal lifted from C22⋊F5 ρ17 4 4 4 0 -4 0 0 0 0 0 0 -1 0 0 -√5 √5 -√5 -1 √5 -1 √5 -√5 -√5 √5 1 1 -1 1 1 orthogonal lifted from C22⋊F5 ρ18 4 4 -4 0 0 0 0 0 0 0 0 -1 0 0 2ζ53+2ζ5+1 2ζ52+2ζ5+1 2ζ54+2ζ52+1 1 2ζ54+2ζ53+1 1 2ζ52+2ζ5+1 2ζ54+2ζ52+1 2ζ53+2ζ5+1 2ζ54+2ζ53+1 -√5 √5 -1 √5 -√5 complex lifted from C23⋊F5 ρ19 4 -4 0 -2 0 2 0 0 0 0 0 -1 0 0 2ζ52+1 2ζ54+1 2ζ53+1 √5 2ζ5+1 -√5 -2ζ54-1 -2ζ53-1 -2ζ52-1 -2ζ5-1 2ζ54+2ζ53+1 2ζ54+2ζ52+1 1 2ζ53+2ζ5+1 2ζ52+2ζ5+1 complex faithful ρ20 4 4 -4 0 0 0 0 0 0 0 0 -1 0 0 2ζ54+2ζ53+1 2ζ53+2ζ5+1 2ζ52+2ζ5+1 1 2ζ54+2ζ52+1 1 2ζ53+2ζ5+1 2ζ52+2ζ5+1 2ζ54+2ζ53+1 2ζ54+2ζ52+1 √5 -√5 -1 -√5 √5 complex lifted from C23⋊F5 ρ21 4 -4 0 2 0 -2 0 0 0 0 0 -1 0 0 -2ζ54-1 -2ζ53-1 -2ζ5-1 -√5 -2ζ52-1 √5 2ζ53+1 2ζ5+1 2ζ54+1 2ζ52+1 2ζ53+2ζ5+1 2ζ54+2ζ53+1 1 2ζ52+2ζ5+1 2ζ54+2ζ52+1 complex faithful ρ22 4 -4 0 -2 0 2 0 0 0 0 0 -1 0 0 2ζ54+1 2ζ53+1 2ζ5+1 -√5 2ζ52+1 √5 -2ζ53-1 -2ζ5-1 -2ζ54-1 -2ζ52-1 2ζ53+2ζ5+1 2ζ54+2ζ53+1 1 2ζ52+2ζ5+1 2ζ54+2ζ52+1 complex faithful ρ23 4 -4 0 2 0 -2 0 0 0 0 0 -1 0 0 -2ζ5-1 -2ζ52-1 -2ζ54-1 -√5 -2ζ53-1 √5 2ζ52+1 2ζ54+1 2ζ5+1 2ζ53+1 2ζ54+2ζ52+1 2ζ52+2ζ5+1 1 2ζ54+2ζ53+1 2ζ53+2ζ5+1 complex faithful ρ24 4 4 -4 0 0 0 0 0 0 0 0 -1 0 0 2ζ52+2ζ5+1 2ζ54+2ζ52+1 2ζ54+2ζ53+1 1 2ζ53+2ζ5+1 1 2ζ54+2ζ52+1 2ζ54+2ζ53+1 2ζ52+2ζ5+1 2ζ53+2ζ5+1 √5 -√5 -1 -√5 √5 complex lifted from C23⋊F5 ρ25 4 4 -4 0 0 0 0 0 0 0 0 -1 0 0 2ζ54+2ζ52+1 2ζ54+2ζ53+1 2ζ53+2ζ5+1 1 2ζ52+2ζ5+1 1 2ζ54+2ζ53+1 2ζ53+2ζ5+1 2ζ54+2ζ52+1 2ζ52+2ζ5+1 -√5 √5 -1 √5 -√5 complex lifted from C23⋊F5 ρ26 4 -4 0 -2 0 2 0 0 0 0 0 -1 0 0 2ζ53+1 2ζ5+1 2ζ52+1 √5 2ζ54+1 -√5 -2ζ5-1 -2ζ52-1 -2ζ53-1 -2ζ54-1 2ζ52+2ζ5+1 2ζ53+2ζ5+1 1 2ζ54+2ζ52+1 2ζ54+2ζ53+1 complex faithful ρ27 4 -4 0 2 0 -2 0 0 0 0 0 -1 0 0 -2ζ52-1 -2ζ54-1 -2ζ53-1 √5 -2ζ5-1 -√5 2ζ54+1 2ζ53+1 2ζ52+1 2ζ5+1 2ζ54+2ζ53+1 2ζ54+2ζ52+1 1 2ζ53+2ζ5+1 2ζ52+2ζ5+1 complex faithful ρ28 4 -4 0 2 0 -2 0 0 0 0 0 -1 0 0 -2ζ53-1 -2ζ5-1 -2ζ52-1 √5 -2ζ54-1 -√5 2ζ5+1 2ζ52+1 2ζ53+1 2ζ54+1 2ζ52+2ζ5+1 2ζ53+2ζ5+1 1 2ζ54+2ζ52+1 2ζ54+2ζ53+1 complex faithful ρ29 4 -4 0 -2 0 2 0 0 0 0 0 -1 0 0 2ζ5+1 2ζ52+1 2ζ54+1 -√5 2ζ53+1 √5 -2ζ52-1 -2ζ54-1 -2ζ5-1 -2ζ53-1 2ζ54+2ζ52+1 2ζ52+2ζ5+1 1 2ζ54+2ζ53+1 2ζ53+2ζ5+1 complex faithful

Smallest permutation representation of C242F5
On 40 points
Generators in S40
```(1 6)(2 7)(3 8)(4 9)(5 10)(11 16)(12 17)(13 18)(14 19)(15 20)(21 36)(22 37)(23 38)(24 39)(25 40)(26 31)(27 32)(28 33)(29 34)(30 35)
(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)
(21 26)(22 27)(23 28)(24 29)(25 30)(31 36)(32 37)(33 38)(34 39)(35 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)
(1 23)(2 25 5 21)(3 22 4 24)(6 28)(7 30 10 26)(8 27 9 29)(11 38 16 33)(12 40 20 31)(13 37 19 34)(14 39 18 32)(15 36 17 35)```

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

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

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

Matrix representation of C242F5 in GL4(𝔽41) generated by

 40 0 0 0 0 40 0 0 0 0 23 5 0 0 1 18
,
 18 36 0 0 40 23 0 0 0 0 18 36 0 0 40 23
,
 1 0 0 0 0 1 0 0 0 0 40 0 0 0 0 40
,
 40 0 0 0 0 40 0 0 0 0 40 0 0 0 0 40
,
 40 5 0 0 1 35 0 0 0 0 6 6 0 0 34 0
,
 0 0 1 0 0 0 0 1 6 6 0 0 1 35 0 0
`G:=sub<GL(4,GF(41))| [40,0,0,0,0,40,0,0,0,0,23,1,0,0,5,18],[18,40,0,0,36,23,0,0,0,0,18,40,0,0,36,23],[1,0,0,0,0,1,0,0,0,0,40,0,0,0,0,40],[40,0,0,0,0,40,0,0,0,0,40,0,0,0,0,40],[40,1,0,0,5,35,0,0,0,0,6,34,0,0,6,0],[0,0,6,1,0,0,6,35,1,0,0,0,0,1,0,0] >;`

C242F5 in GAP, Magma, Sage, TeX

`C_2^4\rtimes_2F_5`
`% in TeX`

`G:=Group("C2^4:2F5");`
`// GroupNames label`

`G:=SmallGroup(320,272);`
`// by ID`

`G=gap.SmallGroup(320,272);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,141,219,675,297,1684,6278,3156]);`
`// Polycyclic`

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

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