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## G = C53⋊6C4order 500 = 22·53

### 6th semidirect product of C53 and C4 acting faithfully

Aliases: C536C4, C527F5, C526Dic5, C5⋊D5.2D5, C51(D5.D5), C52(C52⋊C4), (C5×C5⋊D5).3C2, SmallGroup(500,46)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C53 — C53⋊6C4
 Chief series C1 — C5 — C52 — C53 — C5×C5⋊D5 — C53⋊6C4
 Lower central C53 — C53⋊6C4
 Upper central C1

Generators and relations for C536C4
G = < a,b,c,d | a5=b5=c5=d4=1, ab=ba, ac=ca, dad-1=a-1, bc=cb, dbd-1=b2, dcd-1=c3 >

25C2
2C5
2C5
4C5
4C5
4C5
4C5
4C5
4C5
125C4
5D5
5D5
10D5
10D5
25C10
2C52
2C52
4C52
4C52
4C52
4C52
4C52
4C52
25F5
25Dic5
25F5
10C5×D5
10C5×D5

Permutation representations of C536C4
On 20 points - transitive group 20T125
Generators in S20
```(1 2 3 4 5)(6 7 8 9 10)(11 12 13 14 15)(16 17 18 19 20)
(1 4 2 5 3)(6 8 10 7 9)(11 12 13 14 15)(16 20 19 18 17)
(1 3 5 2 4)(6 9 7 10 8)(11 12 13 14 15)(16 20 19 18 17)
(1 17 8 12)(2 16 9 11)(3 20 10 15)(4 19 6 14)(5 18 7 13)```

`G:=sub<Sym(20)| (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,17,18,19,20), (1,4,2,5,3)(6,8,10,7,9)(11,12,13,14,15)(16,20,19,18,17), (1,3,5,2,4)(6,9,7,10,8)(11,12,13,14,15)(16,20,19,18,17), (1,17,8,12)(2,16,9,11)(3,20,10,15)(4,19,6,14)(5,18,7,13)>;`

`G:=Group( (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,17,18,19,20), (1,4,2,5,3)(6,8,10,7,9)(11,12,13,14,15)(16,20,19,18,17), (1,3,5,2,4)(6,9,7,10,8)(11,12,13,14,15)(16,20,19,18,17), (1,17,8,12)(2,16,9,11)(3,20,10,15)(4,19,6,14)(5,18,7,13) );`

`G=PermutationGroup([[(1,2,3,4,5),(6,7,8,9,10),(11,12,13,14,15),(16,17,18,19,20)], [(1,4,2,5,3),(6,8,10,7,9),(11,12,13,14,15),(16,20,19,18,17)], [(1,3,5,2,4),(6,9,7,10,8),(11,12,13,14,15),(16,20,19,18,17)], [(1,17,8,12),(2,16,9,11),(3,20,10,15),(4,19,6,14),(5,18,7,13)]])`

`G:=TransitiveGroup(20,125);`

38 conjugacy classes

 class 1 2 4A 4B 5A 5B 5C ··· 5AF 10A 10B order 1 2 4 4 5 5 5 ··· 5 10 10 size 1 25 125 125 2 2 4 ··· 4 50 50

38 irreducible representations

 dim 1 1 1 2 2 4 4 4 4 type + + + - + + image C1 C2 C4 D5 Dic5 F5 D5.D5 C52⋊C4 C53⋊6C4 kernel C53⋊6C4 C5×C5⋊D5 C53 C5⋊D5 C52 C52 C5 C5 C1 # reps 1 1 2 2 2 2 8 4 16

Matrix representation of C536C4 in GL4(𝔽41) generated by

 16 0 0 0 0 16 0 0 33 31 18 0 31 33 0 18
,
 37 0 0 0 0 10 0 0 35 1 18 0 23 17 0 16
,
 10 0 0 0 0 37 0 0 9 13 18 0 11 2 0 16
,
 5 37 9 0 37 5 0 9 0 0 36 4 0 0 4 36
`G:=sub<GL(4,GF(41))| [16,0,33,31,0,16,31,33,0,0,18,0,0,0,0,18],[37,0,35,23,0,10,1,17,0,0,18,0,0,0,0,16],[10,0,9,11,0,37,13,2,0,0,18,0,0,0,0,16],[5,37,0,0,37,5,0,0,9,0,36,4,0,9,4,36] >;`

C536C4 in GAP, Magma, Sage, TeX

`C_5^3\rtimes_6C_4`
`% in TeX`

`G:=Group("C5^3:6C4");`
`// GroupNames label`

`G:=SmallGroup(500,46);`
`// by ID`

`G=gap.SmallGroup(500,46);`
`# by ID`

`G:=PCGroup([5,-2,-2,-5,-5,-5,10,242,1203,808,5004,5009]);`
`// 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^-1,b*c=c*b,d*b*d^-1=b^2,d*c*d^-1=c^3>;`
`// generators/relations`

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