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G = He5  order 125 = 53

Heisenberg group

p-group, metabelian, nilpotent (class 2), monomial

Aliases: He5, 5+ 1+2, C52⋊C5, C5.1C52, 5-Sylow(SL(3,5)), SmallGroup(125,3)

Series: Derived ►Chief ►Lower central ►Upper central ►Jennings

Derived series
C1 — C5 — He5
Chief series
C1 — C5 — C52 — He5
Lower central
C1 — C5 — He5
Upper central
C1 — C5 — He5
Jennings
C1 — C5 — He5

Generators and relations for He5
 G = < a,b,c | a5=b5=c5=1, cac-1=ab=ba, bc=cb >

C1
C5
5C5
5C5
5C5
5C5
5C5
5C5
C52
C52
C52
C52
C52
C52
He5

Character table of He5

 class 15A5B5C5D5E5F5G5H5I5J5K5L5M5N5O5P5Q5R5S5T5U5V5W5X5Y5Z5AA5AB
 size 11111555555555555555555555555
ρ111111111111111111111111111111    trivial
ρ211111ζ5ζ52ζ53ζ541ζ52ζ53ζ541ζ5ζ53ζ541ζ5ζ52ζ541ζ5ζ52ζ53ζ5ζ52ζ53ζ54    linear of order 5
ρ311111ζ52ζ52ζ52ζ52ζ52ζ54ζ54ζ54ζ54ζ54ζ5ζ5ζ5ζ5ζ5ζ53ζ53ζ53ζ53ζ531111    linear of order 5
ρ411111ζ521ζ53ζ5ζ54ζ54ζ521ζ53ζ5ζ5ζ54ζ521ζ53ζ53ζ5ζ54ζ521ζ53ζ5ζ54ζ52    linear of order 5
ρ511111ζ53ζ52ζ51ζ54ζ51ζ54ζ53ζ52ζ54ζ53ζ52ζ51ζ52ζ51ζ54ζ53ζ54ζ53ζ52ζ5    linear of order 5
ρ6111111ζ53ζ5ζ54ζ521ζ53ζ5ζ54ζ521ζ53ζ5ζ54ζ521ζ53ζ5ζ54ζ52ζ53ζ5ζ54ζ52    linear of order 5
ρ7111111ζ54ζ53ζ52ζ51ζ54ζ53ζ52ζ51ζ54ζ53ζ52ζ51ζ54ζ53ζ52ζ5ζ54ζ53ζ52ζ5    linear of order 5
ρ811111ζ53ζ53ζ53ζ53ζ53ζ5ζ5ζ5ζ5ζ5ζ54ζ54ζ54ζ54ζ54ζ52ζ52ζ52ζ52ζ521111    linear of order 5
ρ911111ζ53ζ541ζ5ζ52ζ5ζ52ζ53ζ541ζ541ζ5ζ52ζ53ζ52ζ53ζ541ζ5ζ5ζ52ζ53ζ54    linear of order 5
ρ1011111ζ541ζ5ζ52ζ53ζ53ζ541ζ5ζ52ζ52ζ53ζ541ζ5ζ5ζ52ζ53ζ541ζ5ζ52ζ53ζ54    linear of order 5
ρ1111111ζ54ζ521ζ53ζ5ζ53ζ5ζ54ζ521ζ521ζ53ζ5ζ54ζ5ζ54ζ521ζ53ζ53ζ5ζ54ζ52    linear of order 5
ρ1211111ζ52ζ53ζ541ζ5ζ541ζ5ζ52ζ53ζ5ζ52ζ53ζ541ζ53ζ541ζ5ζ52ζ5ζ52ζ53ζ54    linear of order 5
ρ1311111ζ52ζ51ζ54ζ53ζ54ζ53ζ52ζ51ζ51ζ54ζ53ζ52ζ53ζ52ζ51ζ54ζ54ζ53ζ52ζ5    linear of order 5
ρ1411111ζ52ζ54ζ5ζ531ζ54ζ5ζ531ζ52ζ5ζ531ζ52ζ54ζ531ζ52ζ54ζ5ζ52ζ54ζ5ζ53    linear of order 5
ρ1511111ζ51ζ54ζ53ζ52ζ52ζ51ζ54ζ53ζ53ζ52ζ51ζ54ζ54ζ53ζ52ζ51ζ54ζ53ζ52ζ5    linear of order 5
ρ1611111ζ5ζ54ζ521ζ53ζ521ζ53ζ5ζ54ζ53ζ5ζ54ζ521ζ54ζ521ζ53ζ5ζ53ζ5ζ54ζ52    linear of order 5
ρ17111111ζ5ζ52ζ53ζ541ζ5ζ52ζ53ζ541ζ5ζ52ζ53ζ541ζ5ζ52ζ53ζ54ζ5ζ52ζ53ζ54    linear of order 5
ρ1811111ζ53ζ5ζ54ζ521ζ5ζ54ζ521ζ53ζ54ζ521ζ53ζ5ζ521ζ53ζ5ζ54ζ53ζ5ζ54ζ52    linear of order 5
ρ1911111ζ531ζ52ζ54ζ5ζ5ζ531ζ52ζ54ζ54ζ5ζ531ζ52ζ52ζ54ζ5ζ531ζ52ζ54ζ5ζ53    linear of order 5
ρ20111111ζ52ζ54ζ5ζ531ζ52ζ54ζ5ζ531ζ52ζ54ζ5ζ531ζ52ζ54ζ5ζ53ζ52ζ54ζ5ζ53    linear of order 5
ρ2111111ζ5ζ531ζ52ζ54ζ52ζ54ζ5ζ531ζ531ζ52ζ54ζ5ζ54ζ5ζ531ζ52ζ52ζ54ζ5ζ53    linear of order 5
ρ2211111ζ54ζ5ζ531ζ52ζ531ζ52ζ54ζ5ζ52ζ54ζ5ζ531ζ5ζ531ζ52ζ54ζ52ζ54ζ5ζ53    linear of order 5
ρ2311111ζ54ζ53ζ52ζ51ζ53ζ52ζ51ζ54ζ52ζ51ζ54ζ53ζ51ζ54ζ53ζ52ζ54ζ53ζ52ζ5    linear of order 5
ρ2411111ζ5ζ5ζ5ζ5ζ5ζ52ζ52ζ52ζ52ζ52ζ53ζ53ζ53ζ53ζ53ζ54ζ54ζ54ζ54ζ541111    linear of order 5
ρ2511111ζ54ζ54ζ54ζ54ζ54ζ53ζ53ζ53ζ53ζ53ζ52ζ52ζ52ζ52ζ52ζ5ζ5ζ5ζ5ζ51111    linear of order 5
ρ2655ζ525ζ55ζ545ζ53000000000000000000000000    complex faithful
ρ2755ζ55ζ535ζ525ζ54000000000000000000000000    complex faithful
ρ2855ζ545ζ525ζ535ζ5000000000000000000000000    complex faithful
ρ2955ζ535ζ545ζ55ζ52000000000000000000000000    complex faithful

Permutation representations of He5
►On 25 points - transitive group 25T14
Generators in S25
(6 7 8 9 10)(11 12 13 14 15)(16 17 18 19 20)(21 22 23 24 25)

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

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

He5 is a maximal subgroup of   C52⋊C10  He5⋊C2  He5⋊C3

Matrix representation of He5 ►in GL5(𝔽11)

40040
200104
60020
53070
90980
,
40000
04000
00400
00040
00004
,
93000
02100
07010
08001
06000

G:=sub<GL(5,GF(11))| [4,2,6,5,9,0,0,0,3,0,0,0,0,0,9,4,10,2,7,8,0,4,0,0,0],[4,0,0,0,0,0,4,0,0,0,0,0,4,0,0,0,0,0,4,0,0,0,0,0,4],[9,0,0,0,0,3,2,7,8,6,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0] >;
 

He5 in GAP, Magma, Sage, TeX 

{\rm He}_5
 
% in TeX
 
G:=Group("He5");
 
// GroupNames label
 
G:=SmallGroup(125,3);
 
// by ID
 
G=gap.SmallGroup(125,3);
 
# by ID
 
G:=PCGroup([3,-5,5,-5,181]);
 
// Polycyclic
 
G:=Group<a,b,c|a^5=b^5=c^5=1,c*a*c^-1=a*b=b*a,b*c=c*b>;
 
// generators/relations
 

Export

Subgroup lattice of He5 in TeX
Character table of He5 in TeX

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