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G = C52order 25 = 52

Elementary abelian group of type [5,5]

Aliases: C52, SmallGroup(25,2)

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C52
 Chief series C1 — C5 — C52
 Lower central C1 — C52
 Upper central C1 — C52
 Jennings C1 — C52

Generators and relations for C52
G = < a,b | a5=b5=1, ab=ba >

Character table of C52

 class 1 5A 5B 5C 5D 5E 5F 5G 5H 5I 5J 5K 5L 5M 5N 5O 5P 5Q 5R 5S 5T 5U 5V 5W 5X size 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 ζ54 1 1 1 ζ5 ζ5 ζ5 ζ5 ζ5 ζ52 ζ52 ζ52 ζ52 ζ52 ζ53 ζ53 ζ53 ζ53 ζ53 ζ54 ζ54 ζ54 ζ54 1 linear of order 5 ρ3 1 ζ53 1 1 1 ζ52 ζ52 ζ52 ζ52 ζ52 ζ54 ζ54 ζ54 ζ54 ζ54 ζ5 ζ5 ζ5 ζ5 ζ5 ζ53 ζ53 ζ53 ζ53 1 linear of order 5 ρ4 1 ζ52 1 1 1 ζ53 ζ53 ζ53 ζ53 ζ53 ζ5 ζ5 ζ5 ζ5 ζ5 ζ54 ζ54 ζ54 ζ54 ζ54 ζ52 ζ52 ζ52 ζ52 1 linear of order 5 ρ5 1 ζ5 1 1 1 ζ54 ζ54 ζ54 ζ54 ζ54 ζ53 ζ53 ζ53 ζ53 ζ53 ζ52 ζ52 ζ52 ζ52 ζ52 ζ5 ζ5 ζ5 ζ5 1 linear of order 5 ρ6 1 ζ54 ζ52 ζ53 ζ54 1 ζ5 ζ52 ζ53 ζ54 1 ζ5 ζ52 ζ53 ζ54 1 ζ5 ζ52 ζ53 ζ54 1 ζ5 ζ52 ζ53 ζ5 linear of order 5 ρ7 1 ζ53 ζ52 ζ53 ζ54 ζ5 ζ52 ζ53 ζ54 1 ζ52 ζ53 ζ54 1 ζ5 ζ53 ζ54 1 ζ5 ζ52 ζ54 1 ζ5 ζ52 ζ5 linear of order 5 ρ8 1 ζ52 ζ52 ζ53 ζ54 ζ52 ζ53 ζ54 1 ζ5 ζ54 1 ζ5 ζ52 ζ53 ζ5 ζ52 ζ53 ζ54 1 ζ53 ζ54 1 ζ5 ζ5 linear of order 5 ρ9 1 ζ5 ζ52 ζ53 ζ54 ζ53 ζ54 1 ζ5 ζ52 ζ5 ζ52 ζ53 ζ54 1 ζ54 1 ζ5 ζ52 ζ53 ζ52 ζ53 ζ54 1 ζ5 linear of order 5 ρ10 1 1 ζ52 ζ53 ζ54 ζ54 1 ζ5 ζ52 ζ53 ζ53 ζ54 1 ζ5 ζ52 ζ52 ζ53 ζ54 1 ζ5 ζ5 ζ52 ζ53 ζ54 ζ5 linear of order 5 ρ11 1 ζ53 ζ54 ζ5 ζ53 1 ζ52 ζ54 ζ5 ζ53 1 ζ52 ζ54 ζ5 ζ53 1 ζ52 ζ54 ζ5 ζ53 1 ζ52 ζ54 ζ5 ζ52 linear of order 5 ρ12 1 ζ52 ζ54 ζ5 ζ53 ζ5 ζ53 1 ζ52 ζ54 ζ52 ζ54 ζ5 ζ53 1 ζ53 1 ζ52 ζ54 ζ5 ζ54 ζ5 ζ53 1 ζ52 linear of order 5 ρ13 1 ζ5 ζ54 ζ5 ζ53 ζ52 ζ54 ζ5 ζ53 1 ζ54 ζ5 ζ53 1 ζ52 ζ5 ζ53 1 ζ52 ζ54 ζ53 1 ζ52 ζ54 ζ52 linear of order 5 ρ14 1 1 ζ54 ζ5 ζ53 ζ53 1 ζ52 ζ54 ζ5 ζ5 ζ53 1 ζ52 ζ54 ζ54 ζ5 ζ53 1 ζ52 ζ52 ζ54 ζ5 ζ53 ζ52 linear of order 5 ρ15 1 ζ54 ζ54 ζ5 ζ53 ζ54 ζ5 ζ53 1 ζ52 ζ53 1 ζ52 ζ54 ζ5 ζ52 ζ54 ζ5 ζ53 1 ζ5 ζ53 1 ζ52 ζ52 linear of order 5 ρ16 1 ζ52 ζ5 ζ54 ζ52 1 ζ53 ζ5 ζ54 ζ52 1 ζ53 ζ5 ζ54 ζ52 1 ζ53 ζ5 ζ54 ζ52 1 ζ53 ζ5 ζ54 ζ53 linear of order 5 ρ17 1 ζ5 ζ5 ζ54 ζ52 ζ5 ζ54 ζ52 1 ζ53 ζ52 1 ζ53 ζ5 ζ54 ζ53 ζ5 ζ54 ζ52 1 ζ54 ζ52 1 ζ53 ζ53 linear of order 5 ρ18 1 1 ζ5 ζ54 ζ52 ζ52 1 ζ53 ζ5 ζ54 ζ54 ζ52 1 ζ53 ζ5 ζ5 ζ54 ζ52 1 ζ53 ζ53 ζ5 ζ54 ζ52 ζ53 linear of order 5 ρ19 1 ζ54 ζ5 ζ54 ζ52 ζ53 ζ5 ζ54 ζ52 1 ζ5 ζ54 ζ52 1 ζ53 ζ54 ζ52 1 ζ53 ζ5 ζ52 1 ζ53 ζ5 ζ53 linear of order 5 ρ20 1 ζ53 ζ5 ζ54 ζ52 ζ54 ζ52 1 ζ53 ζ5 ζ53 ζ5 ζ54 ζ52 1 ζ52 1 ζ53 ζ5 ζ54 ζ5 ζ54 ζ52 1 ζ53 linear of order 5 ρ21 1 ζ5 ζ53 ζ52 ζ5 1 ζ54 ζ53 ζ52 ζ5 1 ζ54 ζ53 ζ52 ζ5 1 ζ54 ζ53 ζ52 ζ5 1 ζ54 ζ53 ζ52 ζ54 linear of order 5 ρ22 1 1 ζ53 ζ52 ζ5 ζ5 1 ζ54 ζ53 ζ52 ζ52 ζ5 1 ζ54 ζ53 ζ53 ζ52 ζ5 1 ζ54 ζ54 ζ53 ζ52 ζ5 ζ54 linear of order 5 ρ23 1 ζ54 ζ53 ζ52 ζ5 ζ52 ζ5 1 ζ54 ζ53 ζ54 ζ53 ζ52 ζ5 1 ζ5 1 ζ54 ζ53 ζ52 ζ53 ζ52 ζ5 1 ζ54 linear of order 5 ρ24 1 ζ53 ζ53 ζ52 ζ5 ζ53 ζ52 ζ5 1 ζ54 ζ5 1 ζ54 ζ53 ζ52 ζ54 ζ53 ζ52 ζ5 1 ζ52 ζ5 1 ζ54 ζ54 linear of order 5 ρ25 1 ζ52 ζ53 ζ52 ζ5 ζ54 ζ53 ζ52 ζ5 1 ζ53 ζ52 ζ5 1 ζ54 ζ52 ζ5 1 ζ54 ζ53 ζ5 1 ζ54 ζ53 ζ54 linear of order 5

Permutation representations of C52
Regular action on 25 points - transitive group 25T2
Generators in S25
```(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)
(1 25 14 9 19)(2 21 15 10 20)(3 22 11 6 16)(4 23 12 7 17)(5 24 13 8 18)```

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

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

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

`G:=TransitiveGroup(25,2);`

C52 is a maximal subgroup of   C5⋊D5  C52⋊C3  He5  5- 1+2
C52 is a maximal quotient of   He5  5- 1+2

Matrix representation of C52 in GL2(𝔽11) generated by

 4 0 0 4
,
 9 0 0 5
`G:=sub<GL(2,GF(11))| [4,0,0,4],[9,0,0,5] >;`

C52 in GAP, Magma, Sage, TeX

`C_5^2`
`% in TeX`

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

`G:=SmallGroup(25,2);`
`// by ID`

`G=gap.SmallGroup(25,2);`
`# by ID`

`G:=PCGroup([2,-5,5]:ExponentLimit:=1);`
`// Polycyclic`

`G:=Group<a,b|a^5=b^5=1,a*b=b*a>;`
`// generators/relations`

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