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G = 5- 1+2  order 125 = 53

Extraspecial group

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

Aliases: 5- 1+2, C25⋊C5, C52.C5, C5.2C52, 5-Sylow(Sz(32).5), SmallGroup(125,4)

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

Derived series
C1 — C5 — 5- 1+2
Chief series
C1 — C5 — C52 — 5- 1+2
Lower central
C1 — C5 — 5- 1+2
Upper central
C1 — C5 — 5- 1+2
Jennings
C1 — C5 — C5 — C5 — C5 — 5- 1+2

Generators and relations for 5- 1+2
 G = < a,b | a25=b5=1, bab-1=a6 >

C1
C5
5C5
C25
C25
C25
C25
C25
C52
5- 1+2

Character table of 5- 1+2

 class 15A5B5C5D5E5F5G5H25A25B25C25D25E25F25G25H25I25J25K25L25M25N25O25P25Q25R25S25T
 size 11111555555555555555555555555
ρ111111111111111111111111111111    trivial
ρ211111ζ52ζ53ζ5ζ54ζ54ζ521ζ53ζ5ζ54ζ53ζ5ζ5ζ521ζ54ζ521ζ53ζ5ζ54ζ521ζ53    linear of order 5
ρ311111ζ54ζ5ζ52ζ531ζ52ζ53ζ541ζ5ζ53ζ54ζ5ζ5ζ52ζ541ζ5ζ52ζ53ζ52ζ53ζ541    linear of order 5
ρ411111ζ5ζ54ζ53ζ52ζ531ζ54ζ53ζ52ζ51ζ541ζ52ζ51ζ54ζ53ζ52ζ5ζ54ζ53ζ52ζ5    linear of order 5
ρ511111ζ53ζ52ζ54ζ5ζ541ζ52ζ54ζ5ζ531ζ521ζ5ζ531ζ52ζ54ζ5ζ53ζ52ζ54ζ5ζ53    linear of order 5
ρ611111ζ5ζ54ζ53ζ52ζ52ζ51ζ54ζ53ζ52ζ54ζ53ζ53ζ51ζ52ζ51ζ54ζ53ζ52ζ51ζ54    linear of order 5
ρ711111ζ52ζ53ζ5ζ54ζ52ζ54ζ521ζ53ζ5ζ5ζ54ζ521ζ53ζ53ζ5ζ54ζ5211ζ53ζ5ζ54    linear of order 5
ρ8111111111ζ52ζ53ζ53ζ53ζ53ζ53ζ52ζ52ζ54ζ52ζ52ζ5ζ5ζ5ζ5ζ5ζ54ζ54ζ54ζ54    linear of order 5
ρ911111ζ52ζ53ζ5ζ541ζ5ζ54ζ521ζ53ζ54ζ52ζ53ζ53ζ5ζ521ζ53ζ5ζ54ζ5ζ54ζ521    linear of order 5
ρ1011111ζ53ζ52ζ54ζ5ζ53ζ5ζ531ζ52ζ54ζ54ζ5ζ531ζ52ζ52ζ54ζ5ζ5311ζ52ζ54ζ5    linear of order 5
ρ1111111ζ53ζ52ζ54ζ5ζ52ζ52ζ54ζ5ζ531ζ531ζ5ζ54ζ5ζ54ζ5ζ531ζ52ζ531ζ52ζ54    linear of order 5
ρ1211111ζ52ζ53ζ5ζ54ζ53ζ53ζ5ζ54ζ521ζ521ζ54ζ5ζ54ζ5ζ54ζ521ζ53ζ521ζ53ζ5    linear of order 5
ρ1311111ζ54ζ5ζ52ζ53ζ53ζ541ζ5ζ52ζ53ζ5ζ52ζ52ζ541ζ53ζ541ζ5ζ52ζ53ζ541ζ5    linear of order 5
ρ14111111111ζ5ζ54ζ54ζ54ζ54ζ54ζ5ζ5ζ52ζ5ζ5ζ53ζ53ζ53ζ53ζ53ζ52ζ52ζ52ζ52    linear of order 5
ρ1511111ζ52ζ53ζ5ζ54ζ51ζ53ζ5ζ54ζ521ζ531ζ54ζ521ζ53ζ5ζ54ζ52ζ53ζ5ζ54ζ52    linear of order 5
ρ16111111111ζ54ζ5ζ5ζ5ζ5ζ5ζ54ζ54ζ53ζ54ζ54ζ52ζ52ζ52ζ52ζ52ζ53ζ53ζ53ζ53    linear of order 5
ρ1711111ζ54ζ5ζ52ζ53ζ521ζ5ζ52ζ53ζ541ζ51ζ53ζ541ζ5ζ52ζ53ζ54ζ5ζ52ζ53ζ54    linear of order 5
ρ1811111ζ53ζ52ζ54ζ5ζ5ζ531ζ52ζ54ζ5ζ52ζ54ζ54ζ531ζ5ζ531ζ52ζ54ζ5ζ531ζ52    linear of order 5
ρ1911111ζ53ζ52ζ54ζ51ζ54ζ5ζ531ζ52ζ5ζ53ζ52ζ52ζ54ζ531ζ52ζ54ζ5ζ54ζ5ζ531    linear of order 5
ρ2011111ζ54ζ5ζ52ζ53ζ5ζ5ζ52ζ53ζ541ζ541ζ53ζ52ζ53ζ52ζ53ζ541ζ5ζ541ζ5ζ52    linear of order 5
ρ2111111ζ5ζ54ζ53ζ52ζ54ζ54ζ53ζ52ζ51ζ51ζ52ζ53ζ52ζ53ζ52ζ51ζ54ζ51ζ54ζ53    linear of order 5
ρ2211111ζ5ζ54ζ53ζ52ζ5ζ52ζ51ζ54ζ53ζ53ζ52ζ51ζ54ζ54ζ53ζ52ζ511ζ54ζ53ζ52    linear of order 5
ρ2311111ζ54ζ5ζ52ζ53ζ54ζ53ζ541ζ5ζ52ζ52ζ53ζ541ζ5ζ5ζ52ζ53ζ5411ζ5ζ52ζ53    linear of order 5
ρ2411111ζ5ζ54ζ53ζ521ζ53ζ52ζ51ζ54ζ52ζ5ζ54ζ54ζ53ζ51ζ54ζ53ζ52ζ53ζ52ζ51    linear of order 5
ρ25111111111ζ53ζ52ζ52ζ52ζ52ζ52ζ53ζ53ζ5ζ53ζ53ζ54ζ54ζ54ζ54ζ54ζ5ζ5ζ5ζ5    linear of order 5
ρ2655ζ525ζ55ζ545ζ53000000000000000000000000    complex faithful
ρ2755ζ55ζ535ζ525ζ54000000000000000000000000    complex faithful
ρ2855ζ545ζ525ζ535ζ5000000000000000000000000    complex faithful
ρ2955ζ535ζ545ζ55ζ52000000000000000000000000    complex faithful

Permutation representations of 5- 1+2
►On 25 points - transitive group 25T13
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)

(2 22 17 12 7)(3 18 8 23 13)(4 14 24 9 19)(5 10 15 20 25)
 
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), (2,22,17,12,7)(3,18,8,23,13)(4,14,24,9,19)(5,10,15,20,25)>;
 
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), (2,22,17,12,7)(3,18,8,23,13)(4,14,24,9,19)(5,10,15,20,25) );
 
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)], [(2,22,17,12,7),(3,18,8,23,13),(4,14,24,9,19),(5,10,15,20,25)]])
 
G:=TransitiveGroup(25,13);
 

5- 1+2 is a maximal subgroup of   C25⋊C10

Matrix representation of 5- 1+2 ►in GL5(𝔽101)

9586000
3668700
81650950
20140084
017000
,
10000
9587000
3709500
6400840
600036

G:=sub<GL(5,GF(101))| [95,36,81,20,0,86,6,65,14,17,0,87,0,0,0,0,0,95,0,0,0,0,0,84,0],[1,95,37,64,6,0,87,0,0,0,0,0,95,0,0,0,0,0,84,0,0,0,0,0,36] >;
 

5- 1+2 in GAP, Magma, Sage, TeX 

5_-^{1+2}
 
% in TeX
 
G:=Group("ES-(5,1)");
 
// GroupNames label
 
G:=SmallGroup(125,4);
 
// by ID
 
G=gap.SmallGroup(125,4);
 
# by ID
 
G:=PCGroup([3,-5,5,-5,75,181]);
 
// Polycyclic
 
G:=Group<a,b|a^25=b^5=1,b*a*b^-1=a^6>;
 
// generators/relations
 

Export

Subgroup lattice of 5- 1+2 in TeX
Character table of 5- 1+2 in TeX

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