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G = C20⋊2F5order 400 = 24·52

2nd semidirect product of C20 and F5 acting via F5/C5=C4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C5×C10 — C20⋊2F5
 Chief series C1 — C5 — C52 — C5⋊D5 — C2×C5⋊D5 — C2×C52⋊C4 — C20⋊2F5
 Lower central C52 — C5×C10 — C20⋊2F5
 Upper central C1 — C2 — C4

Generators and relations for C202F5
G = < a,b,c | a20=b5=c4=1, ab=ba, cac-1=a7, cbc-1=b3 >

Subgroups: 556 in 68 conjugacy classes, 20 normal (12 characteristic)
C1, C2, C2, C4, C4, C22, C5, C5, C2×C4, D5, C10, C10, C4⋊C4, Dic5, C20, C20, F5, D10, C52, C4×D5, C2×F5, C5⋊D5, C5×C10, C4⋊F5, C526C4, C5×C20, C52⋊C4, C2×C5⋊D5, C4×C5⋊D5, C2×C52⋊C4, C202F5
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C4⋊C4, F5, C2×F5, C4⋊F5, C52⋊C4, C2×C52⋊C4, C202F5

Smallest permutation representation of C202F5
On 40 points
Generators in S40
```(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 5 9 13 17)(2 6 10 14 18)(3 7 11 15 19)(4 8 12 16 20)(21 37 33 29 25)(22 38 34 30 26)(23 39 35 31 27)(24 40 36 32 28)
(1 29 11 39)(2 32 20 26)(3 35 9 33)(4 38 18 40)(5 21 7 27)(6 24 16 34)(8 30 14 28)(10 36 12 22)(13 25 19 23)(15 31 17 37)```

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

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

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

34 conjugacy classes

 class 1 2A 2B 2C 4A 4B ··· 4F 5A ··· 5F 10A ··· 10F 20A ··· 20L order 1 2 2 2 4 4 ··· 4 5 ··· 5 10 ··· 10 20 ··· 20 size 1 1 25 25 2 50 ··· 50 4 ··· 4 4 ··· 4 4 ··· 4

34 irreducible representations

 dim 1 1 1 1 1 2 2 4 4 4 4 4 4 type + + + + - + + + + image C1 C2 C2 C4 C4 D4 Q8 F5 C2×F5 C4⋊F5 C52⋊C4 C2×C52⋊C4 C20⋊2F5 kernel C20⋊2F5 C4×C5⋊D5 C2×C52⋊C4 C52⋊6C4 C5×C20 C5⋊D5 C5⋊D5 C20 C10 C5 C4 C2 C1 # reps 1 1 2 2 2 1 1 2 2 4 4 4 8

Matrix representation of C202F5 in GL4(𝔽41) generated by

 0 9 0 0 32 22 0 0 0 0 32 22 0 0 19 22
,
 34 1 0 0 40 0 0 0 0 0 40 7 0 0 34 7
,
 0 0 40 7 0 0 0 1 1 0 0 0 0 1 0 0
`G:=sub<GL(4,GF(41))| [0,32,0,0,9,22,0,0,0,0,32,19,0,0,22,22],[34,40,0,0,1,0,0,0,0,0,40,34,0,0,7,7],[0,0,1,0,0,0,0,1,40,0,0,0,7,1,0,0] >;`

C202F5 in GAP, Magma, Sage, TeX

`C_{20}\rtimes_2F_5`
`% in TeX`

`G:=Group("C20:2F5");`
`// GroupNames label`

`G:=SmallGroup(400,159);`
`// by ID`

`G=gap.SmallGroup(400,159);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-5,-5,24,121,55,1444,496,5765,2897]);`
`// Polycyclic`

`G:=Group<a,b,c|a^20=b^5=c^4=1,a*b=b*a,c*a*c^-1=a^7,c*b*c^-1=b^3>;`
`// generators/relations`

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