Copied to
clipboard

G = C52  order 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

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 >

C1
C5
C5
C5
C5
C5
C5
C52

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 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

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
 

Export

Subgroup lattice of C52 in TeX
Character table of C52 in TeX

׿
×
⋊
ℤ
𝔽
○
≀
ℚ
◁