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G = C52order 25 = 52

Elementary abelian group of type [5,5]

direct product, p-group, elementary abelian, monomial

Aliases: C52, SmallGroup(25,2)

Series: Derived Chief Lower central Upper central Jennings

C1 — C52
C1C5 — C52
C1 — C52
C1 — C52
C1 — C52

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


Character table of C52

 class 15A5B5C5D5E5F5G5H5I5J5K5L5M5N5O5P5Q5R5S5T5U5V5W5X
 size 1111111111111111111111111
ρ11111111111111111111111111    trivial
ρ21ζ54111ζ5ζ5ζ5ζ5ζ5ζ52ζ52ζ52ζ52ζ52ζ53ζ53ζ53ζ53ζ53ζ54ζ54ζ54ζ541    linear of order 5
ρ31ζ53111ζ52ζ52ζ52ζ52ζ52ζ54ζ54ζ54ζ54ζ54ζ5ζ5ζ5ζ5ζ5ζ53ζ53ζ53ζ531    linear of order 5
ρ41ζ52111ζ53ζ53ζ53ζ53ζ53ζ5ζ5ζ5ζ5ζ5ζ54ζ54ζ54ζ54ζ54ζ52ζ52ζ52ζ521    linear of order 5
ρ51ζ5111ζ54ζ54ζ54ζ54ζ54ζ53ζ53ζ53ζ53ζ53ζ52ζ52ζ52ζ52ζ52ζ5ζ5ζ5ζ51    linear of order 5
ρ61ζ54ζ52ζ53ζ541ζ5ζ52ζ53ζ541ζ5ζ52ζ53ζ541ζ5ζ52ζ53ζ541ζ5ζ52ζ53ζ5    linear of order 5
ρ71ζ53ζ52ζ53ζ54ζ5ζ52ζ53ζ541ζ52ζ53ζ541ζ5ζ53ζ541ζ5ζ52ζ541ζ5ζ52ζ5    linear of order 5
ρ81ζ52ζ52ζ53ζ54ζ52ζ53ζ541ζ5ζ541ζ5ζ52ζ53ζ5ζ52ζ53ζ541ζ53ζ541ζ5ζ5    linear of order 5
ρ91ζ5ζ52ζ53ζ54ζ53ζ541ζ5ζ52ζ5ζ52ζ53ζ541ζ541ζ5ζ52ζ53ζ52ζ53ζ541ζ5    linear of order 5
ρ1011ζ52ζ53ζ54ζ541ζ5ζ52ζ53ζ53ζ541ζ5ζ52ζ52ζ53ζ541ζ5ζ5ζ52ζ53ζ54ζ5    linear of order 5
ρ111ζ53ζ54ζ5ζ531ζ52ζ54ζ5ζ531ζ52ζ54ζ5ζ531ζ52ζ54ζ5ζ531ζ52ζ54ζ5ζ52    linear of order 5
ρ121ζ52ζ54ζ5ζ53ζ5ζ531ζ52ζ54ζ52ζ54ζ5ζ531ζ531ζ52ζ54ζ5ζ54ζ5ζ531ζ52    linear of order 5
ρ131ζ5ζ54ζ5ζ53ζ52ζ54ζ5ζ531ζ54ζ5ζ531ζ52ζ5ζ531ζ52ζ54ζ531ζ52ζ54ζ52    linear of order 5
ρ1411ζ54ζ5ζ53ζ531ζ52ζ54ζ5ζ5ζ531ζ52ζ54ζ54ζ5ζ531ζ52ζ52ζ54ζ5ζ53ζ52    linear of order 5
ρ151ζ54ζ54ζ5ζ53ζ54ζ5ζ531ζ52ζ531ζ52ζ54ζ5ζ52ζ54ζ5ζ531ζ5ζ531ζ52ζ52    linear of order 5
ρ161ζ52ζ5ζ54ζ521ζ53ζ5ζ54ζ521ζ53ζ5ζ54ζ521ζ53ζ5ζ54ζ521ζ53ζ5ζ54ζ53    linear of order 5
ρ171ζ5ζ5ζ54ζ52ζ5ζ54ζ521ζ53ζ521ζ53ζ5ζ54ζ53ζ5ζ54ζ521ζ54ζ521ζ53ζ53    linear of order 5
ρ1811ζ5ζ54ζ52ζ521ζ53ζ5ζ54ζ54ζ521ζ53ζ5ζ5ζ54ζ521ζ53ζ53ζ5ζ54ζ52ζ53    linear of order 5
ρ191ζ54ζ5ζ54ζ52ζ53ζ5ζ54ζ521ζ5ζ54ζ521ζ53ζ54ζ521ζ53ζ5ζ521ζ53ζ5ζ53    linear of order 5
ρ201ζ53ζ5ζ54ζ52ζ54ζ521ζ53ζ5ζ53ζ5ζ54ζ521ζ521ζ53ζ5ζ54ζ5ζ54ζ521ζ53    linear of order 5
ρ211ζ5ζ53ζ52ζ51ζ54ζ53ζ52ζ51ζ54ζ53ζ52ζ51ζ54ζ53ζ52ζ51ζ54ζ53ζ52ζ54    linear of order 5
ρ2211ζ53ζ52ζ5ζ51ζ54ζ53ζ52ζ52ζ51ζ54ζ53ζ53ζ52ζ51ζ54ζ54ζ53ζ52ζ5ζ54    linear of order 5
ρ231ζ54ζ53ζ52ζ5ζ52ζ51ζ54ζ53ζ54ζ53ζ52ζ51ζ51ζ54ζ53ζ52ζ53ζ52ζ51ζ54    linear of order 5
ρ241ζ53ζ53ζ52ζ5ζ53ζ52ζ51ζ54ζ51ζ54ζ53ζ52ζ54ζ53ζ52ζ51ζ52ζ51ζ54ζ54    linear of order 5
ρ251ζ52ζ53ζ52ζ5ζ54ζ53ζ52ζ51ζ53ζ52ζ51ζ54ζ52ζ51ζ54ζ53ζ51ζ54ζ53ζ54    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 19 14 9)(2 21 20 15 10)(3 22 16 11 6)(4 23 17 12 7)(5 24 18 13 8)

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

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

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

G:=TransitiveGroup(25,2);

Matrix representation of C52 in GL2(𝔽11) generated by

40
04
,
90
05
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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