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G = He5⋊C2order 250 = 2·53

2nd semidirect product of He5 and C2 acting faithfully

non-abelian, supersoluble, monomial

Aliases: He52C2, C522D5, C5.2(C5⋊D5), SmallGroup(250,8)

Series: Derived Chief Lower central Upper central

C1C5He5 — He5⋊C2
C1C5C52He5 — He5⋊C2
He5 — He5⋊C2
C1C5

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 >

25C2
5C5
5C5
5C5
5C5
5C5
5C5
5D5
5D5
5D5
5D5
5D5
5D5
25C10
5C5×D5
5C5×D5
5C5×D5
5C5×D5
5C5×D5
5C5×D5

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
ρ155-152535450000000000005525354    complex faithful
ρ16515253545000000000000ζ5ζ52ζ53ζ54    complex faithful
ρ17515352554000000000000ζ54ζ53ζ52ζ5    complex faithful
ρ185-155452530000000000005355452    complex faithful
ρ19515545253000000000000ζ53ζ5ζ54ζ52    complex faithful
ρ205-154553520000000000005254553    complex faithful
ρ21515455352000000000000ζ52ζ54ζ5ζ53    complex faithful
ρ225-153525540000000000005453525    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 17 12)(2 21 10 18 13)(3 22 6 19 14)(4 23 7 20 15)(5 24 8 16 11)
(1 2 14 7 11)(3 20 5 25 21)(4 8 17 18 6)(9 10 22 15 24)(12 13 19 23 16)
(2 11)(3 20)(4 6)(5 21)(7 14)(8 18)(10 24)(13 16)(15 22)(19 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), (1,25,9,17,12)(2,21,10,18,13)(3,22,6,19,14)(4,23,7,20,15)(5,24,8,16,11), (1,2,14,7,11)(3,20,5,25,21)(4,8,17,18,6)(9,10,22,15,24)(12,13,19,23,16), (2,11)(3,20)(4,6)(5,21)(7,14)(8,18)(10,24)(13,16)(15,22)(19,23)>;

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

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

G:=TransitiveGroup(25,22);

He5⋊C2 is a maximal subgroup of   C52⋊F5  He54C4  C52⋊D10
He5⋊C2 is a maximal quotient of   He56C4

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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