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## G = C24⋊C25order 400 = 24·52

### The semidirect product of C24 and C25 acting via C25/C5=C5

Aliases: C24⋊C25, C5.(C24⋊C5), (C23×C10).C5, SmallGroup(400,52)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C24 — C24⋊C25
 Chief series C1 — C24 — C23×C10 — C24⋊C25
 Lower central C24 — C24⋊C25
 Upper central C1 — C5

Generators and relations for C24⋊C25
G = < a,b,c,d,e | a2=b2=c2=d2=e25=1, ab=ba, ac=ca, ad=da, eae-1=abc, bc=cb, bd=db, ebe-1=bcd, ece-1=cd=dc, ede-1=a >

Smallest permutation representation of C24⋊C25
On 50 points
Generators in S50
```(2 49)(3 50)(4 26)(5 27)(7 29)(8 30)(9 31)(10 32)(12 34)(13 35)(14 36)(15 37)(17 39)(18 40)(19 41)(20 42)(22 44)(23 45)(24 46)(25 47)
(4 26)(5 27)(9 31)(10 32)(14 36)(15 37)(19 41)(20 42)(24 46)(25 47)
(1 48)(4 26)(6 28)(9 31)(11 33)(14 36)(16 38)(19 41)(21 43)(24 46)
(1 48)(3 50)(4 26)(5 27)(6 28)(8 30)(9 31)(10 32)(11 33)(13 35)(14 36)(15 37)(16 38)(18 40)(19 41)(20 42)(21 43)(23 45)(24 46)(25 47)
(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)```

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

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

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

40 conjugacy classes

 class 1 2A 2B 2C 5A 5B 5C 5D 10A ··· 10L 25A ··· 25T order 1 2 2 2 5 5 5 5 10 ··· 10 25 ··· 25 size 1 5 5 5 1 1 1 1 5 ··· 5 16 ··· 16

40 irreducible representations

 dim 1 1 1 5 5 type + + image C1 C5 C25 C24⋊C5 C24⋊C25 kernel C24⋊C25 C23×C10 C24 C5 C1 # reps 1 4 20 3 12

Matrix representation of C24⋊C25 in GL5(𝔽101)

 1 0 0 0 0 0 100 0 0 0 33 0 100 0 0 70 0 0 100 0 1 0 0 0 100
,
 1 0 0 0 0 0 100 0 0 0 33 0 100 0 0 0 52 0 1 0 0 25 0 0 1
,
 100 0 0 0 0 0 1 0 0 0 0 97 100 0 0 31 0 0 1 0 100 0 0 0 1
,
 100 0 0 0 0 0 100 0 0 0 0 0 100 0 0 0 0 0 100 0 100 25 0 0 1
,
 0 1 0 0 0 33 97 99 0 0 0 0 4 1 0 0 0 52 0 1 0 0 25 0 0

`G:=sub<GL(5,GF(101))| [1,0,33,70,1,0,100,0,0,0,0,0,100,0,0,0,0,0,100,0,0,0,0,0,100],[1,0,33,0,0,0,100,0,52,25,0,0,100,0,0,0,0,0,1,0,0,0,0,0,1],[100,0,0,31,100,0,1,97,0,0,0,0,100,0,0,0,0,0,1,0,0,0,0,0,1],[100,0,0,0,100,0,100,0,0,25,0,0,100,0,0,0,0,0,100,0,0,0,0,0,1],[0,33,0,0,0,1,97,0,0,0,0,99,4,52,25,0,0,1,0,0,0,0,0,1,0] >;`

C24⋊C25 in GAP, Magma, Sage, TeX

`C_2^4\rtimes C_{25}`
`% in TeX`

`G:=Group("C2^4:C25");`
`// GroupNames label`

`G:=SmallGroup(400,52);`
`// by ID`

`G=gap.SmallGroup(400,52);`
`# by ID`

`G:=PCGroup([6,-5,-5,-2,2,2,2,30,3602,5403,8254,13505]);`
`// Polycyclic`

`G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^2=e^25=1,a*b=b*a,a*c=c*a,a*d=d*a,e*a*e^-1=a*b*c,b*c=c*b,b*d=d*b,e*b*e^-1=b*c*d,e*c*e^-1=c*d=d*c,e*d*e^-1=a>;`
`// generators/relations`

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