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## G = C25⋊F5order 500 = 22·53

### 1st semidirect product of C25 and F5 acting via F5/C5=C4

Aliases: C251F5, C52.4F5, (C5×C25)⋊5C4, C51(C25⋊C4), C25⋊D5.1C2, C5.(C5⋊F5), SmallGroup(500,22)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C5×C25 — C25⋊F5
 Chief series C1 — C5 — C52 — C5×C25 — C25⋊D5 — C25⋊F5
 Lower central C5×C25 — C25⋊F5
 Upper central C1

Generators and relations for C25⋊F5
G = < a,b,c | a25=b5=c4=1, ab=ba, cac-1=a18, cbc-1=b3 >

125C2
125C4
25D5
25D5
25D5
25D5
25D5
25D5
25F5
25F5
25F5
25F5
25F5
25F5
5D25
5D25
5D25
5D25
5D25

Smallest permutation representation of C25⋊F5
On 125 points
Generators in S125
```(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)(101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125)
(1 84 57 28 106)(2 85 58 29 107)(3 86 59 30 108)(4 87 60 31 109)(5 88 61 32 110)(6 89 62 33 111)(7 90 63 34 112)(8 91 64 35 113)(9 92 65 36 114)(10 93 66 37 115)(11 94 67 38 116)(12 95 68 39 117)(13 96 69 40 118)(14 97 70 41 119)(15 98 71 42 120)(16 99 72 43 121)(17 100 73 44 122)(18 76 74 45 123)(19 77 75 46 124)(20 78 51 47 125)(21 79 52 48 101)(22 80 53 49 102)(23 81 54 50 103)(24 82 55 26 104)(25 83 56 27 105)
(2 8 25 19)(3 15 24 12)(4 22 23 5)(6 11 21 16)(7 18 20 9)(10 14 17 13)(26 95 59 120)(27 77 58 113)(28 84 57 106)(29 91 56 124)(30 98 55 117)(31 80 54 110)(32 87 53 103)(33 94 52 121)(34 76 51 114)(35 83 75 107)(36 90 74 125)(37 97 73 118)(38 79 72 111)(39 86 71 104)(40 93 70 122)(41 100 69 115)(42 82 68 108)(43 89 67 101)(44 96 66 119)(45 78 65 112)(46 85 64 105)(47 92 63 123)(48 99 62 116)(49 81 61 109)(50 88 60 102)```

`G:=sub<Sym(125)| (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)(101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125), (1,84,57,28,106)(2,85,58,29,107)(3,86,59,30,108)(4,87,60,31,109)(5,88,61,32,110)(6,89,62,33,111)(7,90,63,34,112)(8,91,64,35,113)(9,92,65,36,114)(10,93,66,37,115)(11,94,67,38,116)(12,95,68,39,117)(13,96,69,40,118)(14,97,70,41,119)(15,98,71,42,120)(16,99,72,43,121)(17,100,73,44,122)(18,76,74,45,123)(19,77,75,46,124)(20,78,51,47,125)(21,79,52,48,101)(22,80,53,49,102)(23,81,54,50,103)(24,82,55,26,104)(25,83,56,27,105), (2,8,25,19)(3,15,24,12)(4,22,23,5)(6,11,21,16)(7,18,20,9)(10,14,17,13)(26,95,59,120)(27,77,58,113)(28,84,57,106)(29,91,56,124)(30,98,55,117)(31,80,54,110)(32,87,53,103)(33,94,52,121)(34,76,51,114)(35,83,75,107)(36,90,74,125)(37,97,73,118)(38,79,72,111)(39,86,71,104)(40,93,70,122)(41,100,69,115)(42,82,68,108)(43,89,67,101)(44,96,66,119)(45,78,65,112)(46,85,64,105)(47,92,63,123)(48,99,62,116)(49,81,61,109)(50,88,60,102)>;`

`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)(101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125), (1,84,57,28,106)(2,85,58,29,107)(3,86,59,30,108)(4,87,60,31,109)(5,88,61,32,110)(6,89,62,33,111)(7,90,63,34,112)(8,91,64,35,113)(9,92,65,36,114)(10,93,66,37,115)(11,94,67,38,116)(12,95,68,39,117)(13,96,69,40,118)(14,97,70,41,119)(15,98,71,42,120)(16,99,72,43,121)(17,100,73,44,122)(18,76,74,45,123)(19,77,75,46,124)(20,78,51,47,125)(21,79,52,48,101)(22,80,53,49,102)(23,81,54,50,103)(24,82,55,26,104)(25,83,56,27,105), (2,8,25,19)(3,15,24,12)(4,22,23,5)(6,11,21,16)(7,18,20,9)(10,14,17,13)(26,95,59,120)(27,77,58,113)(28,84,57,106)(29,91,56,124)(30,98,55,117)(31,80,54,110)(32,87,53,103)(33,94,52,121)(34,76,51,114)(35,83,75,107)(36,90,74,125)(37,97,73,118)(38,79,72,111)(39,86,71,104)(40,93,70,122)(41,100,69,115)(42,82,68,108)(43,89,67,101)(44,96,66,119)(45,78,65,112)(46,85,64,105)(47,92,63,123)(48,99,62,116)(49,81,61,109)(50,88,60,102) );`

`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),(101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125)], [(1,84,57,28,106),(2,85,58,29,107),(3,86,59,30,108),(4,87,60,31,109),(5,88,61,32,110),(6,89,62,33,111),(7,90,63,34,112),(8,91,64,35,113),(9,92,65,36,114),(10,93,66,37,115),(11,94,67,38,116),(12,95,68,39,117),(13,96,69,40,118),(14,97,70,41,119),(15,98,71,42,120),(16,99,72,43,121),(17,100,73,44,122),(18,76,74,45,123),(19,77,75,46,124),(20,78,51,47,125),(21,79,52,48,101),(22,80,53,49,102),(23,81,54,50,103),(24,82,55,26,104),(25,83,56,27,105)], [(2,8,25,19),(3,15,24,12),(4,22,23,5),(6,11,21,16),(7,18,20,9),(10,14,17,13),(26,95,59,120),(27,77,58,113),(28,84,57,106),(29,91,56,124),(30,98,55,117),(31,80,54,110),(32,87,53,103),(33,94,52,121),(34,76,51,114),(35,83,75,107),(36,90,74,125),(37,97,73,118),(38,79,72,111),(39,86,71,104),(40,93,70,122),(41,100,69,115),(42,82,68,108),(43,89,67,101),(44,96,66,119),(45,78,65,112),(46,85,64,105),(47,92,63,123),(48,99,62,116),(49,81,61,109),(50,88,60,102)]])`

35 conjugacy classes

 class 1 2 4A 4B 5A ··· 5F 25A ··· 25Y order 1 2 4 4 5 ··· 5 25 ··· 25 size 1 125 125 125 4 ··· 4 4 ··· 4

35 irreducible representations

 dim 1 1 1 4 4 4 type + + + + + image C1 C2 C4 F5 F5 C25⋊C4 kernel C25⋊F5 C25⋊D5 C5×C25 C25 C52 C5 # reps 1 1 2 5 1 25

Matrix representation of C25⋊F5 in GL8(𝔽101)

 100 100 100 100 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 78 4 70 32 0 0 0 0 69 46 73 38 0 0 0 0 63 31 8 35 0 0 0 0 66 28 97 74
,
 100 100 100 100 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
,
 0 0 100 100 0 0 0 0 100 100 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 100 100 0 0 0 0 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 0 0 0 0 100 100 100 100

`G:=sub<GL(8,GF(101))| [100,1,0,0,0,0,0,0,100,0,1,0,0,0,0,0,100,0,0,1,0,0,0,0,100,0,0,0,0,0,0,0,0,0,0,0,78,69,63,66,0,0,0,0,4,46,31,28,0,0,0,0,70,73,8,97,0,0,0,0,32,38,35,74],[100,1,0,0,0,0,0,0,100,0,1,0,0,0,0,0,100,0,0,1,0,0,0,0,100,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[0,100,1,0,0,0,0,0,0,100,1,100,0,0,0,0,100,0,1,100,0,0,0,0,100,0,0,0,0,0,0,0,0,0,0,0,1,0,0,100,0,0,0,0,0,0,1,100,0,0,0,0,0,0,0,100,0,0,0,0,0,1,0,100] >;`

C25⋊F5 in GAP, Magma, Sage, TeX

`C_{25}\rtimes F_5`
`% in TeX`

`G:=Group("C25:F5");`
`// GroupNames label`

`G:=SmallGroup(500,22);`
`// by ID`

`G=gap.SmallGroup(500,22);`
`# by ID`

`G:=PCGroup([5,-2,-2,-5,-5,-5,10,1622,3127,387,803,808,5004,5009]);`
`// Polycyclic`

`G:=Group<a,b,c|a^25=b^5=c^4=1,a*b=b*a,c*a*c^-1=a^18,c*b*c^-1=b^3>;`
`// generators/relations`

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