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G = C53⋊9C4order 500 = 22·53

9th semidirect product of C53 and C4 acting faithfully

Aliases: C539C4, C5210F5, C5⋊(C52⋊C4), C52(C5⋊F5), C53⋊C2.2C2, SmallGroup(500,49)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C53 — C53⋊9C4
 Chief series C1 — C5 — C52 — C53 — C53⋊C2 — C53⋊9C4
 Lower central C53 — C53⋊9C4
 Upper central C1

Generators and relations for C539C4
G = < a,b,c,d | a5=b5=c5=d4=1, ab=ba, ac=ca, dad-1=a2, bc=cb, dbd-1=b2, dcd-1=c3 >

Subgroups: 1456 in 96 conjugacy classes, 18 normal (6 characteristic)
C1, C2, C4, C5, C5, C5, D5, F5, C52, C52, C52, C5⋊D5, C5⋊F5, C52⋊C4, C53, C53⋊C2, C539C4
Quotients: C1, C2, C4, F5, C5⋊F5, C52⋊C4, C539C4

Smallest permutation representation of C539C4
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 40 34 42 30)(2 36 35 43 26)(3 37 31 44 27)(4 38 32 45 28)(5 39 33 41 29)(6 19 46 15 23)(7 20 47 11 24)(8 16 48 12 25)(9 17 49 13 21)(10 18 50 14 22)
(1 39 32 44 26)(2 40 33 45 27)(3 36 34 41 28)(4 37 35 42 29)(5 38 31 43 30)(6 24 12 49 18)(7 25 13 50 19)(8 21 14 46 20)(9 22 15 47 16)(10 23 11 48 17)
(1 7)(2 10 5 9)(3 8 4 6)(11 30 47 40)(12 28 46 37)(13 26 50 39)(14 29 49 36)(15 27 48 38)(16 45 23 31)(17 43 22 33)(18 41 21 35)(19 44 25 32)(20 42 24 34)```

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

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

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

35 conjugacy classes

 class 1 2 4A 4B 5A ··· 5AE order 1 2 4 4 5 ··· 5 size 1 125 125 125 4 ··· 4

35 irreducible representations

 dim 1 1 1 4 4 type + + + + image C1 C2 C4 F5 C52⋊C4 kernel C53⋊9C4 C53⋊C2 C53 C52 C5 # reps 1 1 2 7 24

Matrix representation of C539C4 in GL8(𝔽41)

 7 7 0 0 0 0 0 0 34 40 0 0 0 0 0 0 0 0 34 40 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
,
 34 40 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 40 34 0 0 0 0 0 0 7 7 0 0 0 0 0 0 0 0 6 1 0 0 0 0 0 0 40 0 0 0 0 0 0 0 35 34 34 35 0 0 0 0 1 7 7 0
,
 0 1 0 0 0 0 0 0 40 34 0 0 0 0 0 0 0 0 40 34 0 0 0 0 0 0 7 7 0 0 0 0 0 0 0 0 6 1 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 40 0 6 0 0 0 0 35 0 34 34
,
 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 40 0 0 0 0 0 0 0 7 1 0 0 0 0 0 0 0 0 0 0 0 0 6 1 0 0 0 0 40 40 39 40 0 0 0 0 1 7 1 0 0 0 0 0 0 34 35 0

`G:=sub<GL(8,GF(41))| [7,34,0,0,0,0,0,0,7,40,0,0,0,0,0,0,0,0,34,1,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[34,1,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,40,7,0,0,0,0,0,0,34,7,0,0,0,0,0,0,0,0,6,40,35,1,0,0,0,0,1,0,34,7,0,0,0,0,0,0,34,7,0,0,0,0,0,0,35,0],[0,40,0,0,0,0,0,0,1,34,0,0,0,0,0,0,0,0,40,7,0,0,0,0,0,0,34,7,0,0,0,0,0,0,0,0,6,40,0,35,0,0,0,0,1,0,40,0,0,0,0,0,0,0,0,34,0,0,0,0,0,0,6,34],[0,0,40,7,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,40,1,0,0,0,0,0,0,40,7,34,0,0,0,0,6,39,1,35,0,0,0,0,1,40,0,0] >;`

C539C4 in GAP, Magma, Sage, TeX

`C_5^3\rtimes_9C_4`
`% in TeX`

`G:=Group("C5^3:9C4");`
`// GroupNames label`

`G:=SmallGroup(500,49);`
`// by ID`

`G=gap.SmallGroup(500,49);`
`# by ID`

`G:=PCGroup([5,-2,-2,-5,-5,-5,10,182,127,1203,808,5004,5009]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^5=b^5=c^5=d^4=1,a*b=b*a,a*c=c*a,d*a*d^-1=a^2,b*c=c*b,d*b*d^-1=b^2,d*c*d^-1=c^3>;`
`// generators/relations`

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