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## G = C42⋊F5order 320 = 26·5

### 1st semidirect product of C42 and F5 acting via F5/C5=C4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C20 — C42⋊F5
 Chief series C1 — C5 — C10 — C2×C10 — C22×D5 — C2×D20 — D10.D4 — C42⋊F5
 Lower central C5 — C10 — C2×C10 — C2×C20 — C42⋊F5
 Upper central C1 — C2 — C22 — C2×C4 — C42

Generators and relations for C42⋊F5
G = < a,b,c,d | a4=b4=c5=d4=1, ab=ba, ac=ca, dad-1=a-1b-1, bc=cb, dbd-1=a2b, dcd-1=c3 >

Subgroups: 666 in 86 conjugacy classes, 18 normal (14 characteristic)
C1, C2, C2, C4, C22, C22, C5, C2×C4, C2×C4, D4, C23, D5, C10, C10, C42, C22⋊C4, C2×D4, C20, F5, D10, C2×C10, C23⋊C4, C41D4, D20, C2×C20, C2×C20, C2×F5, C22×D5, C22×D5, C42⋊C4, C4×C20, C22⋊F5, C2×D20, C2×D20, D10.D4, C204D4, C42⋊F5
Quotients: C1, C2, C4, C22, C2×C4, D4, C22⋊C4, F5, C23⋊C4, C2×F5, C42⋊C4, C22⋊F5, D10.D4, C42⋊F5

Character table of C42⋊F5

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E 4F 4G 5 10A 10B 10C 20A 20B 20C 20D 20E 20F 20G 20H 20I 20J 20K 20L size 1 1 2 20 20 40 4 4 4 40 40 40 40 4 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 1 -i i -i i 1 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 1 i i -i -i 1 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 1 -i -i i i 1 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 1 i -i i -i 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 linear of order 4 ρ9 2 2 2 2 -2 0 0 0 -2 0 0 0 0 2 2 2 2 0 0 -2 0 0 0 0 -2 -2 -2 0 0 orthogonal lifted from D4 ρ10 2 2 2 -2 2 0 0 0 -2 0 0 0 0 2 2 2 2 0 0 -2 0 0 0 0 -2 -2 -2 0 0 orthogonal lifted from D4 ρ11 4 4 -4 0 0 0 0 0 0 0 0 0 0 4 4 -4 -4 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C23⋊C4 ρ12 4 -4 0 0 0 0 -2 2 0 0 0 0 0 4 -4 0 0 -2 -2 0 2 2 2 2 0 0 0 -2 -2 orthogonal lifted from C42⋊C4 ρ13 4 4 4 0 0 0 -4 -4 4 0 0 0 0 -1 -1 -1 -1 1 1 -1 1 1 1 1 -1 -1 -1 1 1 orthogonal lifted from C2×F5 ρ14 4 -4 0 0 0 0 2 -2 0 0 0 0 0 4 -4 0 0 2 2 0 -2 -2 -2 -2 0 0 0 2 2 orthogonal lifted from C42⋊C4 ρ15 4 4 4 0 0 0 4 4 4 0 0 0 0 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 orthogonal lifted from F5 ρ16 4 4 4 0 0 0 0 0 -4 0 0 0 0 -1 -1 -1 -1 √5 -√5 1 -√5 √5 √5 -√5 1 1 1 -√5 √5 orthogonal lifted from C22⋊F5 ρ17 4 4 4 0 0 0 0 0 -4 0 0 0 0 -1 -1 -1 -1 -√5 √5 1 √5 -√5 -√5 √5 1 1 1 √5 -√5 orthogonal lifted from C22⋊F5 ρ18 4 -4 0 0 0 0 -2 2 0 0 0 0 0 -1 1 √5 -√5 ζ43ζ54-ζ43ζ5+ζ54+ζ5+1 ζ4ζ53-ζ4ζ52+ζ53+ζ52+1 2ζ4ζ54+2ζ4ζ53+ζ4 ζ4ζ53-ζ4ζ52-ζ53-ζ52-1 ζ43ζ54-ζ43ζ5-ζ54-ζ5-1 -ζ43ζ54+ζ43ζ5-ζ54-ζ5-1 -ζ4ζ53+ζ4ζ52-ζ53-ζ52-1 2ζ43ζ53+2ζ43ζ5+ζ43 2ζ43ζ54+2ζ43ζ52+ζ43 2ζ4ζ52+2ζ4ζ5+ζ4 -ζ4ζ53+ζ4ζ52+ζ53+ζ52+1 -ζ43ζ54+ζ43ζ5+ζ54+ζ5+1 orthogonal faithful ρ19 4 4 -4 0 0 0 0 0 0 0 0 0 0 -1 -1 1 1 2ζ43ζ53+2ζ43ζ5+ζ43 2ζ4ζ52+2ζ4ζ5+ζ4 -√5 2ζ4ζ54+2ζ4ζ53+ζ4 2ζ43ζ54+2ζ43ζ52+ζ43 2ζ43ζ53+2ζ43ζ5+ζ43 2ζ4ζ52+2ζ4ζ5+ζ4 √5 √5 -√5 2ζ4ζ54+2ζ4ζ53+ζ4 2ζ43ζ54+2ζ43ζ52+ζ43 orthogonal lifted from D10.D4 ρ20 4 -4 0 0 0 0 -2 2 0 0 0 0 0 -1 1 √5 -√5 -ζ43ζ54+ζ43ζ5+ζ54+ζ5+1 -ζ4ζ53+ζ4ζ52+ζ53+ζ52+1 2ζ4ζ52+2ζ4ζ5+ζ4 -ζ4ζ53+ζ4ζ52-ζ53-ζ52-1 -ζ43ζ54+ζ43ζ5-ζ54-ζ5-1 ζ43ζ54-ζ43ζ5-ζ54-ζ5-1 ζ4ζ53-ζ4ζ52-ζ53-ζ52-1 2ζ43ζ54+2ζ43ζ52+ζ43 2ζ43ζ53+2ζ43ζ5+ζ43 2ζ4ζ54+2ζ4ζ53+ζ4 ζ4ζ53-ζ4ζ52+ζ53+ζ52+1 ζ43ζ54-ζ43ζ5+ζ54+ζ5+1 orthogonal faithful ρ21 4 4 -4 0 0 0 0 0 0 0 0 0 0 -1 -1 1 1 2ζ43ζ54+2ζ43ζ52+ζ43 2ζ4ζ54+2ζ4ζ53+ζ4 -√5 2ζ4ζ52+2ζ4ζ5+ζ4 2ζ43ζ53+2ζ43ζ5+ζ43 2ζ43ζ54+2ζ43ζ52+ζ43 2ζ4ζ54+2ζ4ζ53+ζ4 √5 √5 -√5 2ζ4ζ52+2ζ4ζ5+ζ4 2ζ43ζ53+2ζ43ζ5+ζ43 orthogonal lifted from D10.D4 ρ22 4 4 -4 0 0 0 0 0 0 0 0 0 0 -1 -1 1 1 2ζ4ζ54+2ζ4ζ53+ζ4 2ζ43ζ53+2ζ43ζ5+ζ43 √5 2ζ43ζ54+2ζ43ζ52+ζ43 2ζ4ζ52+2ζ4ζ5+ζ4 2ζ4ζ54+2ζ4ζ53+ζ4 2ζ43ζ53+2ζ43ζ5+ζ43 -√5 -√5 √5 2ζ43ζ54+2ζ43ζ52+ζ43 2ζ4ζ52+2ζ4ζ5+ζ4 orthogonal lifted from D10.D4 ρ23 4 -4 0 0 0 0 -2 2 0 0 0 0 0 -1 1 -√5 √5 -ζ4ζ53+ζ4ζ52+ζ53+ζ52+1 ζ43ζ54-ζ43ζ5+ζ54+ζ5+1 2ζ43ζ54+2ζ43ζ52+ζ43 ζ43ζ54-ζ43ζ5-ζ54-ζ5-1 -ζ4ζ53+ζ4ζ52-ζ53-ζ52-1 ζ4ζ53-ζ4ζ52-ζ53-ζ52-1 -ζ43ζ54+ζ43ζ5-ζ54-ζ5-1 2ζ4ζ54+2ζ4ζ53+ζ4 2ζ4ζ52+2ζ4ζ5+ζ4 2ζ43ζ53+2ζ43ζ5+ζ43 -ζ43ζ54+ζ43ζ5+ζ54+ζ5+1 ζ4ζ53-ζ4ζ52+ζ53+ζ52+1 orthogonal faithful ρ24 4 -4 0 0 0 0 2 -2 0 0 0 0 0 -1 1 -√5 √5 -ζ4ζ53+ζ4ζ52-ζ53-ζ52-1 ζ43ζ54-ζ43ζ5-ζ54-ζ5-1 2ζ43ζ53+2ζ43ζ5+ζ43 ζ43ζ54-ζ43ζ5+ζ54+ζ5+1 -ζ4ζ53+ζ4ζ52+ζ53+ζ52+1 ζ4ζ53-ζ4ζ52+ζ53+ζ52+1 -ζ43ζ54+ζ43ζ5+ζ54+ζ5+1 2ζ4ζ52+2ζ4ζ5+ζ4 2ζ4ζ54+2ζ4ζ53+ζ4 2ζ43ζ54+2ζ43ζ52+ζ43 -ζ43ζ54+ζ43ζ5-ζ54-ζ5-1 ζ4ζ53-ζ4ζ52-ζ53-ζ52-1 orthogonal faithful ρ25 4 -4 0 0 0 0 2 -2 0 0 0 0 0 -1 1 √5 -√5 ζ43ζ54-ζ43ζ5-ζ54-ζ5-1 ζ4ζ53-ζ4ζ52-ζ53-ζ52-1 2ζ4ζ52+2ζ4ζ5+ζ4 ζ4ζ53-ζ4ζ52+ζ53+ζ52+1 ζ43ζ54-ζ43ζ5+ζ54+ζ5+1 -ζ43ζ54+ζ43ζ5+ζ54+ζ5+1 -ζ4ζ53+ζ4ζ52+ζ53+ζ52+1 2ζ43ζ54+2ζ43ζ52+ζ43 2ζ43ζ53+2ζ43ζ5+ζ43 2ζ4ζ54+2ζ4ζ53+ζ4 -ζ4ζ53+ζ4ζ52-ζ53-ζ52-1 -ζ43ζ54+ζ43ζ5-ζ54-ζ5-1 orthogonal faithful ρ26 4 -4 0 0 0 0 -2 2 0 0 0 0 0 -1 1 -√5 √5 ζ4ζ53-ζ4ζ52+ζ53+ζ52+1 -ζ43ζ54+ζ43ζ5+ζ54+ζ5+1 2ζ43ζ53+2ζ43ζ5+ζ43 -ζ43ζ54+ζ43ζ5-ζ54-ζ5-1 ζ4ζ53-ζ4ζ52-ζ53-ζ52-1 -ζ4ζ53+ζ4ζ52-ζ53-ζ52-1 ζ43ζ54-ζ43ζ5-ζ54-ζ5-1 2ζ4ζ52+2ζ4ζ5+ζ4 2ζ4ζ54+2ζ4ζ53+ζ4 2ζ43ζ54+2ζ43ζ52+ζ43 ζ43ζ54-ζ43ζ5+ζ54+ζ5+1 -ζ4ζ53+ζ4ζ52+ζ53+ζ52+1 orthogonal faithful ρ27 4 4 -4 0 0 0 0 0 0 0 0 0 0 -1 -1 1 1 2ζ4ζ52+2ζ4ζ5+ζ4 2ζ43ζ54+2ζ43ζ52+ζ43 √5 2ζ43ζ53+2ζ43ζ5+ζ43 2ζ4ζ54+2ζ4ζ53+ζ4 2ζ4ζ52+2ζ4ζ5+ζ4 2ζ43ζ54+2ζ43ζ52+ζ43 -√5 -√5 √5 2ζ43ζ53+2ζ43ζ5+ζ43 2ζ4ζ54+2ζ4ζ53+ζ4 orthogonal lifted from D10.D4 ρ28 4 -4 0 0 0 0 2 -2 0 0 0 0 0 -1 1 √5 -√5 -ζ43ζ54+ζ43ζ5-ζ54-ζ5-1 -ζ4ζ53+ζ4ζ52-ζ53-ζ52-1 2ζ4ζ54+2ζ4ζ53+ζ4 -ζ4ζ53+ζ4ζ52+ζ53+ζ52+1 -ζ43ζ54+ζ43ζ5+ζ54+ζ5+1 ζ43ζ54-ζ43ζ5+ζ54+ζ5+1 ζ4ζ53-ζ4ζ52+ζ53+ζ52+1 2ζ43ζ53+2ζ43ζ5+ζ43 2ζ43ζ54+2ζ43ζ52+ζ43 2ζ4ζ52+2ζ4ζ5+ζ4 ζ4ζ53-ζ4ζ52-ζ53-ζ52-1 ζ43ζ54-ζ43ζ5-ζ54-ζ5-1 orthogonal faithful ρ29 4 -4 0 0 0 0 2 -2 0 0 0 0 0 -1 1 -√5 √5 ζ4ζ53-ζ4ζ52-ζ53-ζ52-1 -ζ43ζ54+ζ43ζ5-ζ54-ζ5-1 2ζ43ζ54+2ζ43ζ52+ζ43 -ζ43ζ54+ζ43ζ5+ζ54+ζ5+1 ζ4ζ53-ζ4ζ52+ζ53+ζ52+1 -ζ4ζ53+ζ4ζ52+ζ53+ζ52+1 ζ43ζ54-ζ43ζ5+ζ54+ζ5+1 2ζ4ζ54+2ζ4ζ53+ζ4 2ζ4ζ52+2ζ4ζ5+ζ4 2ζ43ζ53+2ζ43ζ5+ζ43 ζ43ζ54-ζ43ζ5-ζ54-ζ5-1 -ζ4ζ53+ζ4ζ52-ζ53-ζ52-1 orthogonal faithful

Smallest permutation representation of C42⋊F5
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 31 26 36)(22 32 27 37)(23 33 28 38)(24 34 29 39)(25 35 30 40)
(1 16 6 11)(2 17 7 12)(3 18 8 13)(4 19 9 14)(5 20 10 15)(21 31 26 36)(22 32 27 37)(23 33 28 38)(24 34 29 39)(25 35 30 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 33 11 23)(2 35 15 21)(3 32 14 24)(4 34 13 22)(5 31 12 25)(6 38 16 28)(7 40 20 26)(8 37 19 29)(9 39 18 27)(10 36 17 30)```

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

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

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

Matrix representation of C42⋊F5 in GL4(𝔽41) generated by

 40 0 0 0 0 40 0 0 20 24 39 13 26 30 28 2
,
 30 28 0 0 22 11 0 0 37 2 39 13 21 23 28 2
,
 40 40 0 0 8 7 0 0 37 0 40 35 36 36 6 35
,
 11 9 20 20 7 18 1 3 27 13 28 18 0 8 20 25
`G:=sub<GL(4,GF(41))| [40,0,20,26,0,40,24,30,0,0,39,28,0,0,13,2],[30,22,37,21,28,11,2,23,0,0,39,28,0,0,13,2],[40,8,37,36,40,7,0,36,0,0,40,6,0,0,35,35],[11,7,27,0,9,18,13,8,20,1,28,20,20,3,18,25] >;`

C42⋊F5 in GAP, Magma, Sage, TeX

`C_4^2\rtimes F_5`
`% in TeX`

`G:=Group("C4^2:F5");`
`// GroupNames label`

`G:=SmallGroup(320,191);`
`// by ID`

`G=gap.SmallGroup(320,191);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,141,120,219,1571,297,136,1684,6278,3156]);`
`// Polycyclic`

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

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