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## G = C4○D4⋊F5order 320 = 26·5

### 2nd semidirect product of C4○D4 and F5 acting via F5/D5=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — C4○D4⋊F5
 Chief series C1 — C5 — C10 — D10 — C4×D5 — C4⋊F5 — C2×C4⋊F5 — C4○D4⋊F5
 Lower central C5 — C10 — C20 — C4○D4⋊F5
 Upper central C1 — C2 — C2×C4 — C4○D4

Generators and relations for C4○D4⋊F5
G = < a,b,c,d,e | a4=c2=d5=e4=1, b2=a2, ab=ba, ac=ca, ad=da, eae-1=a-1, cbc=a2b, bd=db, ebe-1=abc, cd=dc, ce=ec, ede-1=d3 >

Subgroups: 730 in 162 conjugacy classes, 50 normal (42 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, D5, D5, C10, C10, C4⋊C4, C2×C8, M4(2), C22×C4, C2×D4, C2×Q8, C4○D4, C4○D4, Dic5, Dic5, C20, C20, F5, D10, D10, C2×C10, C2×C10, D4⋊C4, Q8⋊C4, C2×C4⋊C4, C2×M4(2), C2×C4○D4, C5⋊C8, Dic10, Dic10, C4×D5, C4×D5, D20, D20, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C5×D4, C5×D4, C5×Q8, C2×F5, C22×D5, C22×D5, C23.36D4, D5⋊C8, C4.F5, C4⋊F5, C4⋊F5, C22.F5, C2×C4×D5, C2×C4×D5, C4○D20, C4○D20, D4×D5, D4×D5, D42D5, D42D5, Q8×D5, Q82D5, C5×C4○D4, C22×F5, D20⋊C4, Q8⋊F5, D5⋊M4(2), C2×C4⋊F5, D5×C4○D4, C4○D4⋊F5
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, C22×C4, C2×D4, F5, C2×C22⋊C4, C8⋊C22, C8.C22, C2×F5, C23.36D4, C22⋊F5, C22×F5, C2×C22⋊F5, C4○D4⋊F5

Smallest permutation representation of C4○D4⋊F5
On 40 points
Generators in S40
```(1 11 6 16)(2 12 7 17)(3 13 8 18)(4 14 9 19)(5 15 10 20)(21 31 26 36)(22 32 27 37)(23 33 28 38)(24 34 29 39)(25 35 30 40)
(1 26 6 21)(2 27 7 22)(3 28 8 23)(4 29 9 24)(5 30 10 25)(11 36 16 31)(12 37 17 32)(13 38 18 33)(14 39 19 34)(15 40 20 35)
(1 6)(2 7)(3 8)(4 9)(5 10)(11 16)(12 17)(13 18)(14 19)(15 20)
(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)(22 33 25 34)(23 35 24 32)(26 36)(27 38 30 39)(28 40 29 37)```

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

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

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

32 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 4A 4B 4C 4D 4E 4F ··· 4J 5 8A 8B 8C 8D 10A 10B 10C 10D 20A 20B 20C 20D 20E order 1 2 2 2 2 2 2 2 4 4 4 4 4 4 ··· 4 5 8 8 8 8 10 10 10 10 20 20 20 20 20 size 1 1 2 4 5 5 10 20 2 2 4 10 10 20 ··· 20 4 20 20 20 20 4 8 8 8 4 4 8 8 8

32 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 4 4 4 4 4 4 4 4 8 type + + + + + + + + + + + - + + + + + image C1 C2 C2 C2 C2 C2 C4 C4 C4 C4 D4 D4 D4 F5 C8⋊C22 C8.C22 C2×F5 C2×F5 C2×F5 C22⋊F5 C22⋊F5 C4○D4⋊F5 kernel C4○D4⋊F5 D20⋊C4 Q8⋊F5 D5⋊M4(2) C2×C4⋊F5 D5×C4○D4 C4○D20 D4⋊2D5 Q8⋊2D5 C5×C4○D4 C4×D5 C2×Dic5 C22×D5 C4○D4 D5 D5 C2×C4 D4 Q8 C4 C22 C1 # reps 1 2 2 1 1 1 2 2 2 2 2 1 1 1 1 1 1 1 1 2 2 2

Matrix representation of C4○D4⋊F5 in GL8(𝔽41)

 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 40 36 0 0 0 0 0 0 25 1 0 0 0 0 0 0 0 1 0 40 0 0 0 0 25 1 1 0
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 5 0 0 0 0 0 0 0 40 1 0 0 0 0 16 0 40 0 0 0 0 0 16 40 40 0
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 25 0 1 0 0 0 0 0 25 0 0 1
,
 40 40 40 40 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 40 40 40 40 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 16 40 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0

`G:=sub<GL(8,GF(41))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,40,25,0,25,0,0,0,0,36,1,1,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,16,16,0,0,0,0,0,0,0,40,0,0,0,0,5,40,40,40,0,0,0,0,0,1,0,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,40,0,25,25,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[40,1,0,0,0,0,0,0,40,0,1,0,0,0,0,0,40,0,0,1,0,0,0,0,40,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,40,0,0,0,0,0,0,1,40,0,0,0,0,0,0,0,40,0,0,0,0,0,1,0,40,0,0,0,0,0,0,0,0,1,16,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0] >;`

C4○D4⋊F5 in GAP, Magma, Sage, TeX

`C_4\circ D_4\rtimes F_5`
`% in TeX`

`G:=Group("C4oD4:F5");`
`// GroupNames label`

`G:=SmallGroup(320,1131);`
`// by ID`

`G=gap.SmallGroup(320,1131);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,422,387,1684,438,102,6278,1595]);`
`// Polycyclic`

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

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