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## G = C25⋊2F5order 500 = 22·53

### 2nd semidirect product of C25 and F5 acting via F5/C5=C4

Aliases: C252F5, C52.5F5, (C5×C25)⋊6C4, C52(C25⋊C4), C25⋊D5.2C2, C5.(C52⋊C4), SmallGroup(500,24)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C5×C25 — C25⋊2F5
 Chief series C1 — C5 — C52 — C5×C25 — C25⋊D5 — C25⋊2F5
 Lower central C5×C25 — C25⋊2F5
 Upper central C1

Generators and relations for C252F5
G = < a,b,c | a25=b5=c4=1, ab=ba, cac-1=a7, cbc-1=b3 >

125C2
2C5
2C5
125C4
25D5
25D5
50D5
50D5
2C25
2C25
25F5
25F5
5D25
10D25
10D25

Smallest permutation representation of C252F5
On 50 points
Generators in S50
```(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)
(1 6 11 16 21)(2 7 12 17 22)(3 8 13 18 23)(4 9 14 19 24)(5 10 15 20 25)(26 46 41 36 31)(27 47 42 37 32)(28 48 43 38 33)(29 49 44 39 34)(30 50 45 40 35)
(1 26)(2 44 25 33)(3 37 24 40)(4 30 23 47)(5 48 22 29)(6 41 21 36)(7 34 20 43)(8 27 19 50)(9 45 18 32)(10 38 17 39)(11 31 16 46)(12 49 15 28)(13 42 14 35)```

`G:=sub<Sym(50)| (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), (1,6,11,16,21)(2,7,12,17,22)(3,8,13,18,23)(4,9,14,19,24)(5,10,15,20,25)(26,46,41,36,31)(27,47,42,37,32)(28,48,43,38,33)(29,49,44,39,34)(30,50,45,40,35), (1,26)(2,44,25,33)(3,37,24,40)(4,30,23,47)(5,48,22,29)(6,41,21,36)(7,34,20,43)(8,27,19,50)(9,45,18,32)(10,38,17,39)(11,31,16,46)(12,49,15,28)(13,42,14,35)>;`

`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), (1,6,11,16,21)(2,7,12,17,22)(3,8,13,18,23)(4,9,14,19,24)(5,10,15,20,25)(26,46,41,36,31)(27,47,42,37,32)(28,48,43,38,33)(29,49,44,39,34)(30,50,45,40,35), (1,26)(2,44,25,33)(3,37,24,40)(4,30,23,47)(5,48,22,29)(6,41,21,36)(7,34,20,43)(8,27,19,50)(9,45,18,32)(10,38,17,39)(11,31,16,46)(12,49,15,28)(13,42,14,35) );`

`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)], [(1,6,11,16,21),(2,7,12,17,22),(3,8,13,18,23),(4,9,14,19,24),(5,10,15,20,25),(26,46,41,36,31),(27,47,42,37,32),(28,48,43,38,33),(29,49,44,39,34),(30,50,45,40,35)], [(1,26),(2,44,25,33),(3,37,24,40),(4,30,23,47),(5,48,22,29),(6,41,21,36),(7,34,20,43),(8,27,19,50),(9,45,18,32),(10,38,17,39),(11,31,16,46),(12,49,15,28),(13,42,14,35)]])`

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 4 4 type + + + + + + + image C1 C2 C4 F5 F5 C25⋊C4 C52⋊C4 C25⋊2F5 kernel C25⋊2F5 C25⋊D5 C5×C25 C25 C52 C5 C5 C1 # reps 1 1 2 1 1 5 4 20

Matrix representation of C252F5 in GL4(𝔽101) generated by

 99 50 0 0 51 88 0 0 0 0 11 32 0 0 69 8
,
 100 22 0 0 79 79 0 0 0 0 22 100 0 0 1 0
,
 0 0 1 0 0 0 0 1 32 93 0 0 90 69 0 0
`G:=sub<GL(4,GF(101))| [99,51,0,0,50,88,0,0,0,0,11,69,0,0,32,8],[100,79,0,0,22,79,0,0,0,0,22,1,0,0,100,0],[0,0,32,90,0,0,93,69,1,0,0,0,0,1,0,0] >;`

C252F5 in GAP, Magma, Sage, TeX

`C_{25}\rtimes_2F_5`
`% in TeX`

`G:=Group("C25:2F5");`
`// GroupNames label`

`G:=SmallGroup(500,24);`
`// by ID`

`G=gap.SmallGroup(500,24);`
`# by ID`

`G:=PCGroup([5,-2,-2,-5,-5,-5,10,1682,2377,762,803,808,7504,5009]);`
`// Polycyclic`

`G:=Group<a,b,c|a^25=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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