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G = He5⋊C2  order 250 = 2·53

2nd semidirect product of He5 and C2 acting faithfully

non-abelian, supersoluble, monomial

Aliases: He5⋊2C2, C52⋊2D5, C5.2(C5⋊D5), SmallGroup(250,8)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C5 — He5 — He5⋊C2
Chief series
C1 — C5 — C52 — He5 — He5⋊C2
Lower central
He5 — He5⋊C2
Upper central
C1 — C5

Generators and relations for He5⋊C2
 G = < a,b,c,d | a5=b5=c5=d2=1, cac-1=ab=ba, dad=a-1b-1, bc=cb, bd=db, dcd=c-1 >

C1
25C2
C5
5C5
5C5
5C5
5C5
5C5
5C5
5D5
5D5
5D5
5D5
5D5
5D5
25C10
C52
C52
C52
C52
C52
C52
5C5×D5
5C5×D5
5C5×D5
5C5×D5
5C5×D5
5C5×D5
He5
He5⋊C2

Character table of He5⋊C2

 class 125A5B5C5D5E5F5G5H5I5J5K5L5M5N5O5P10A10B10C10D
 size 125111110101010101010101010101025252525
ρ11111111111111111111111    trivial
ρ21-11111111111111111-1-1-1-1    linear of order 2
ρ3202222-1+√5/2-1-√5/2-1-√5/2-1+√5/22-1+√5/2-1+√5/22-1+√5/2-1-√5/2-1-√5/2-1-√5/20000    orthogonal lifted from D5
ρ4202222-1-√5/22-1-√5/2-1+√5/2-1+√5/2-1-√5/22-1-√5/2-1+√5/2-1+√5/2-1-√5/2-1+√5/20000    orthogonal lifted from D5
ρ5202222-1+√5/2-1+√5/2-1+√5/2-1+√5/2-1+√5/22-1-√5/2-1-√5/2-1-√5/2-1-√5/2-1-√5/220000    orthogonal lifted from D5
ρ62022222-1-√5/2-1+√5/2-1+√5/2-1-√5/2-1-√5/2-1+√5/2-1+√5/2-1-√5/22-1-√5/2-1+√5/20000    orthogonal lifted from D5
ρ7202222-1+√5/2-1-√5/22-1-√5/2-1+√5/2-1-√5/2-1+√5/2-1-√5/22-1-√5/2-1+√5/2-1+√5/20000    orthogonal lifted from D5
ρ8202222-1-√5/2-1-√5/2-1+√5/22-1+√5/2-1+√5/2-1+√5/2-1-√5/2-1-√5/2-1+√5/22-1-√5/20000    orthogonal lifted from D5
ρ9202222-1-√5/2-1+√5/22-1+√5/2-1-√5/2-1+√5/2-1-√5/2-1+√5/22-1+√5/2-1-√5/2-1-√5/20000    orthogonal lifted from D5
ρ10202222-1-√5/2-1+√5/2-1+√5/2-1-√5/22-1-√5/2-1-√5/22-1-√5/2-1+√5/2-1+√5/2-1+√5/20000    orthogonal lifted from D5
ρ11202222-1+√5/22-1+√5/2-1-√5/2-1-√5/2-1+√5/22-1+√5/2-1-√5/2-1-√5/2-1+√5/2-1-√5/20000    orthogonal lifted from D5
ρ12202222-1-√5/2-1-√5/2-1-√5/2-1-√5/2-1-√5/22-1+√5/2-1+√5/2-1+√5/2-1+√5/2-1+√5/220000    orthogonal lifted from D5
ρ13202222-1+√5/2-1+√5/2-1-√5/22-1-√5/2-1-√5/2-1-√5/2-1+√5/2-1+√5/2-1-√5/22-1+√5/20000    orthogonal lifted from D5
ρ142022222-1+√5/2-1-√5/2-1-√5/2-1+√5/2-1+√5/2-1-√5/2-1-√5/2-1+√5/22-1+√5/2-1-√5/20000    orthogonal lifted from D5
ρ155-15ζ525ζ535ζ545ζ5000000000000-ζ5-ζ52-ζ53-ζ54    complex faithful
ρ16515ζ525ζ535ζ545ζ5000000000000ζ5ζ52ζ53ζ54    complex faithful
ρ17515ζ535ζ525ζ55ζ54000000000000ζ54ζ53ζ52ζ5    complex faithful
ρ185-15ζ55ζ545ζ525ζ53000000000000-ζ53-ζ5-ζ54-ζ52    complex faithful
ρ19515ζ55ζ545ζ525ζ53000000000000ζ53ζ5ζ54ζ52    complex faithful
ρ205-15ζ545ζ55ζ535ζ52000000000000-ζ52-ζ54-ζ5-ζ53    complex faithful
ρ21515ζ545ζ55ζ535ζ52000000000000ζ52ζ54ζ5ζ53    complex faithful
ρ225-15ζ535ζ525ζ55ζ54000000000000-ζ54-ζ53-ζ52-ζ5    complex faithful

Permutation representations of He5⋊C2
►On 25 points - transitive group 25T22
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 9 19 12)(2 21 10 20 13)(3 22 6 16 14)(4 23 7 17 15)(5 24 8 18 11)

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

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

He5⋊C2 is a maximal subgroup of   C52⋊F5  He5⋊4C4  C52⋊D10
He5⋊C2 is a maximal quotient of   He5⋊6C4

Matrix representation of He5⋊C2 ►in GL5(𝔽11)

004102
50199
031029
00517
00090
,
90000
09000
00900
00090
00009
,
81000
30100
10000
00001
400103
,
18100
03300
01800
004310
04088

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

He5⋊C2 in GAP, Magma, Sage, TeX 

{\rm He}_5\rtimes C_2
 
% in TeX
 
G:=Group("He5:C2");
 
// GroupNames label
 
G:=SmallGroup(250,8);
 
// by ID
 
G=gap.SmallGroup(250,8);
 
# by ID
 
G:=PCGroup([4,-2,-5,-5,-5,65,482,366]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^5=b^5=c^5=d^2=1,c*a*c^-1=a*b=b*a,d*a*d=a^-1*b^-1,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
 
// generators/relations
 

Export

Subgroup lattice of He5⋊C2 in TeX
Character table of He5⋊C2 in TeX

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