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## G = C20⋊F5order 400 = 24·52

### 1st semidirect product of C20 and F5 acting via F5/C5=C4

Series: Derived Chief Lower central Upper central

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

Generators and relations for C20⋊F5
G = < a,b,c | a20=b5=c4=1, ab=ba, cac-1=a3, cbc-1=b3 >

Subgroups: 696 in 104 conjugacy classes, 32 normal (12 characteristic)
C1, C2, C2, C4, C4, C22, C5, C2×C4, D5, C10, C4⋊C4, Dic5, C20, F5, D10, C52, C4×D5, C2×F5, C5⋊D5, C5×C10, C4⋊F5, C526C4, C5×C20, C5⋊F5, C2×C5⋊D5, C4×C5⋊D5, C2×C5⋊F5, C20⋊F5
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C4⋊C4, F5, C2×F5, C4⋊F5, C5⋊F5, C2×C5⋊F5, C20⋊F5

Smallest permutation representation of C20⋊F5
On 100 points
Generators in S100
```(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)(41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60)(61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80)(81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100)
(1 30 70 84 44)(2 31 71 85 45)(3 32 72 86 46)(4 33 73 87 47)(5 34 74 88 48)(6 35 75 89 49)(7 36 76 90 50)(8 37 77 91 51)(9 38 78 92 52)(10 39 79 93 53)(11 40 80 94 54)(12 21 61 95 55)(13 22 62 96 56)(14 23 63 97 57)(15 24 64 98 58)(16 25 65 99 59)(17 26 66 100 60)(18 27 67 81 41)(19 28 68 82 42)(20 29 69 83 43)
(2 8 10 4)(3 15 19 7)(5 9 17 13)(6 16)(12 18 20 14)(21 67 43 97)(22 74 52 100)(23 61 41 83)(24 68 50 86)(25 75 59 89)(26 62 48 92)(27 69 57 95)(28 76 46 98)(29 63 55 81)(30 70 44 84)(31 77 53 87)(32 64 42 90)(33 71 51 93)(34 78 60 96)(35 65 49 99)(36 72 58 82)(37 79 47 85)(38 66 56 88)(39 73 45 91)(40 80 54 94)```

`G:=sub<Sym(100)| (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)(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80)(81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100), (1,30,70,84,44)(2,31,71,85,45)(3,32,72,86,46)(4,33,73,87,47)(5,34,74,88,48)(6,35,75,89,49)(7,36,76,90,50)(8,37,77,91,51)(9,38,78,92,52)(10,39,79,93,53)(11,40,80,94,54)(12,21,61,95,55)(13,22,62,96,56)(14,23,63,97,57)(15,24,64,98,58)(16,25,65,99,59)(17,26,66,100,60)(18,27,67,81,41)(19,28,68,82,42)(20,29,69,83,43), (2,8,10,4)(3,15,19,7)(5,9,17,13)(6,16)(12,18,20,14)(21,67,43,97)(22,74,52,100)(23,61,41,83)(24,68,50,86)(25,75,59,89)(26,62,48,92)(27,69,57,95)(28,76,46,98)(29,63,55,81)(30,70,44,84)(31,77,53,87)(32,64,42,90)(33,71,51,93)(34,78,60,96)(35,65,49,99)(36,72,58,82)(37,79,47,85)(38,66,56,88)(39,73,45,91)(40,80,54,94)>;`

`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)(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80)(81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100), (1,30,70,84,44)(2,31,71,85,45)(3,32,72,86,46)(4,33,73,87,47)(5,34,74,88,48)(6,35,75,89,49)(7,36,76,90,50)(8,37,77,91,51)(9,38,78,92,52)(10,39,79,93,53)(11,40,80,94,54)(12,21,61,95,55)(13,22,62,96,56)(14,23,63,97,57)(15,24,64,98,58)(16,25,65,99,59)(17,26,66,100,60)(18,27,67,81,41)(19,28,68,82,42)(20,29,69,83,43), (2,8,10,4)(3,15,19,7)(5,9,17,13)(6,16)(12,18,20,14)(21,67,43,97)(22,74,52,100)(23,61,41,83)(24,68,50,86)(25,75,59,89)(26,62,48,92)(27,69,57,95)(28,76,46,98)(29,63,55,81)(30,70,44,84)(31,77,53,87)(32,64,42,90)(33,71,51,93)(34,78,60,96)(35,65,49,99)(36,72,58,82)(37,79,47,85)(38,66,56,88)(39,73,45,91)(40,80,54,94) );`

`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),(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60),(61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80),(81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100)], [(1,30,70,84,44),(2,31,71,85,45),(3,32,72,86,46),(4,33,73,87,47),(5,34,74,88,48),(6,35,75,89,49),(7,36,76,90,50),(8,37,77,91,51),(9,38,78,92,52),(10,39,79,93,53),(11,40,80,94,54),(12,21,61,95,55),(13,22,62,96,56),(14,23,63,97,57),(15,24,64,98,58),(16,25,65,99,59),(17,26,66,100,60),(18,27,67,81,41),(19,28,68,82,42),(20,29,69,83,43)], [(2,8,10,4),(3,15,19,7),(5,9,17,13),(6,16),(12,18,20,14),(21,67,43,97),(22,74,52,100),(23,61,41,83),(24,68,50,86),(25,75,59,89),(26,62,48,92),(27,69,57,95),(28,76,46,98),(29,63,55,81),(30,70,44,84),(31,77,53,87),(32,64,42,90),(33,71,51,93),(34,78,60,96),(35,65,49,99),(36,72,58,82),(37,79,47,85),(38,66,56,88),(39,73,45,91),(40,80,54,94)]])`

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 type + + + + - + + image C1 C2 C2 C4 C4 D4 Q8 F5 C2×F5 C4⋊F5 kernel C20⋊F5 C4×C5⋊D5 C2×C5⋊F5 C52⋊6C4 C5×C20 C5⋊D5 C5⋊D5 C20 C10 C5 # reps 1 1 2 2 2 1 1 6 6 12

Matrix representation of C20⋊F5 in GL8(𝔽41)

 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 0 27 7 34 0 0 0 0 14 27 34 0 0 0 0 0 7 0 34 27 0 0 0 0 14 34 7 27
,
 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 0 0 0 40 0 0 0 0 1 0 0 40 0 0 0 0 0 1 0 40 0 0 0 0 0 0 1 40
,
 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 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0

`G:=sub<GL(8,GF(41))| [40,1,0,0,0,0,0,0,40,0,1,0,0,0,0,0,40,0,0,1,0,0,0,0,40,0,0,0,0,0,0,0,0,0,0,0,0,14,7,14,0,0,0,0,27,27,0,34,0,0,0,0,7,34,34,7,0,0,0,0,34,0,27,27],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,40,40,40,40],[1,0,0,40,0,0,0,0,0,0,1,40,0,0,0,0,0,0,0,40,0,0,0,0,0,1,0,40,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0] >;`

C20⋊F5 in GAP, Magma, Sage, TeX

`C_{20}\rtimes F_5`
`% in TeX`

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

`G:=SmallGroup(400,152);`
`// by ID`

`G=gap.SmallGroup(400,152);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-5,-5,24,121,55,964,496,5765,2897]);`
`// Polycyclic`

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

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