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## G = C52⋊A4order 300 = 22·3·52

### The semidirect product of C52 and A4 acting via A4/C22=C3

Aliases: C52⋊A4, C1023C3, C22⋊(C52⋊C3), SmallGroup(300,43)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C102 — C52⋊A4
 Chief series C1 — C52 — C102 — C52⋊A4
 Lower central C102 — C52⋊A4
 Upper central C1

Generators and relations for C52⋊A4
G = < a,b,c,d,e | a5=b5=c2=d2=e3=1, ebe-1=ab=ba, ac=ca, ad=da, eae-1=a3b2, bc=cb, bd=db, ece-1=cd=dc, ede-1=c >

3C2
100C3
3C5
3C5
3C10
3C10
3C10
3C10
3C10
3C10
25A4

Permutation representations of C52⋊A4
On 30 points - transitive group 30T70
Generators in S30
```(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)
(1 5 4 3 2)(6 10 9 8 7)(11 13 15 12 14)(16 18 20 17 19)
(11 16)(12 17)(13 18)(14 19)(15 20)(21 26)(22 27)(23 28)(24 29)(25 30)
(1 6)(2 7)(3 8)(4 9)(5 10)(11 16)(12 17)(13 18)(14 19)(15 20)
(1 21 12)(2 22 14)(3 23 11)(4 24 13)(5 25 15)(6 26 17)(7 27 19)(8 28 16)(9 29 18)(10 30 20)```

`G:=sub<Sym(30)| (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), (1,5,4,3,2)(6,10,9,8,7)(11,13,15,12,14)(16,18,20,17,19), (11,16)(12,17)(13,18)(14,19)(15,20)(21,26)(22,27)(23,28)(24,29)(25,30), (1,6)(2,7)(3,8)(4,9)(5,10)(11,16)(12,17)(13,18)(14,19)(15,20), (1,21,12)(2,22,14)(3,23,11)(4,24,13)(5,25,15)(6,26,17)(7,27,19)(8,28,16)(9,29,18)(10,30,20)>;`

`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), (1,5,4,3,2)(6,10,9,8,7)(11,13,15,12,14)(16,18,20,17,19), (11,16)(12,17)(13,18)(14,19)(15,20)(21,26)(22,27)(23,28)(24,29)(25,30), (1,6)(2,7)(3,8)(4,9)(5,10)(11,16)(12,17)(13,18)(14,19)(15,20), (1,21,12)(2,22,14)(3,23,11)(4,24,13)(5,25,15)(6,26,17)(7,27,19)(8,28,16)(9,29,18)(10,30,20) );`

`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)], [(1,5,4,3,2),(6,10,9,8,7),(11,13,15,12,14),(16,18,20,17,19)], [(11,16),(12,17),(13,18),(14,19),(15,20),(21,26),(22,27),(23,28),(24,29),(25,30)], [(1,6),(2,7),(3,8),(4,9),(5,10),(11,16),(12,17),(13,18),(14,19),(15,20)], [(1,21,12),(2,22,14),(3,23,11),(4,24,13),(5,25,15),(6,26,17),(7,27,19),(8,28,16),(9,29,18),(10,30,20)]])`

`G:=TransitiveGroup(30,70);`

36 conjugacy classes

 class 1 2 3A 3B 5A ··· 5H 10A ··· 10X order 1 2 3 3 5 ··· 5 10 ··· 10 size 1 3 100 100 3 ··· 3 3 ··· 3

36 irreducible representations

 dim 1 1 3 3 3 type + + image C1 C3 A4 C52⋊C3 C52⋊A4 kernel C52⋊A4 C102 C52 C22 C1 # reps 1 2 1 8 24

Matrix representation of C52⋊A4 in GL3(𝔽11) generated by

 9 0 0 0 9 0 0 0 3
,
 5 0 0 0 1 0 0 0 9
,
 10 0 0 0 1 0 0 0 10
,
 10 0 0 0 10 0 0 0 1
,
 0 0 1 1 0 0 0 1 0
`G:=sub<GL(3,GF(11))| [9,0,0,0,9,0,0,0,3],[5,0,0,0,1,0,0,0,9],[10,0,0,0,1,0,0,0,10],[10,0,0,0,10,0,0,0,1],[0,1,0,0,0,1,1,0,0] >;`

C52⋊A4 in GAP, Magma, Sage, TeX

`C_5^2\rtimes A_4`
`% in TeX`

`G:=Group("C5^2:A4");`
`// GroupNames label`

`G:=SmallGroup(300,43);`
`// by ID`

`G=gap.SmallGroup(300,43);`
`# by ID`

`G:=PCGroup([5,-3,-2,2,-5,5,61,137,3843,5704]);`
`// Polycyclic`

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

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