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## G = He5⋊C2order 250 = 2·53

### 2nd semidirect product of He5 and C2 acting faithfully

Aliases: He52C2, C522D5, 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 >

Character table of He5⋊C2

 class 1 2 5A 5B 5C 5D 5E 5F 5G 5H 5I 5J 5K 5L 5M 5N 5O 5P 10A 10B 10C 10D size 1 25 1 1 1 1 10 10 10 10 10 10 10 10 10 10 10 10 25 25 25 25 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 -1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 linear of order 2 ρ3 2 0 2 2 2 2 -1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 2 -1+√5/2 -1+√5/2 2 -1+√5/2 -1-√5/2 -1-√5/2 -1-√5/2 0 0 0 0 orthogonal lifted from D5 ρ4 2 0 2 2 2 2 -1-√5/2 2 -1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 2 -1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 -1+√5/2 0 0 0 0 orthogonal lifted from D5 ρ5 2 0 2 2 2 2 -1+√5/2 -1+√5/2 -1+√5/2 -1+√5/2 -1+√5/2 2 -1-√5/2 -1-√5/2 -1-√5/2 -1-√5/2 -1-√5/2 2 0 0 0 0 orthogonal lifted from D5 ρ6 2 0 2 2 2 2 2 -1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 2 -1-√5/2 -1+√5/2 0 0 0 0 orthogonal lifted from D5 ρ7 2 0 2 2 2 2 -1+√5/2 -1-√5/2 2 -1-√5/2 -1+√5/2 -1-√5/2 -1+√5/2 -1-√5/2 2 -1-√5/2 -1+√5/2 -1+√5/2 0 0 0 0 orthogonal lifted from D5 ρ8 2 0 2 2 2 2 -1-√5/2 -1-√5/2 -1+√5/2 2 -1+√5/2 -1+√5/2 -1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 2 -1-√5/2 0 0 0 0 orthogonal lifted from D5 ρ9 2 0 2 2 2 2 -1-√5/2 -1+√5/2 2 -1+√5/2 -1-√5/2 -1+√5/2 -1-√5/2 -1+√5/2 2 -1+√5/2 -1-√5/2 -1-√5/2 0 0 0 0 orthogonal lifted from D5 ρ10 2 0 2 2 2 2 -1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 2 -1-√5/2 -1-√5/2 2 -1-√5/2 -1+√5/2 -1+√5/2 -1+√5/2 0 0 0 0 orthogonal lifted from D5 ρ11 2 0 2 2 2 2 -1+√5/2 2 -1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 2 -1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 -1-√5/2 0 0 0 0 orthogonal lifted from D5 ρ12 2 0 2 2 2 2 -1-√5/2 -1-√5/2 -1-√5/2 -1-√5/2 -1-√5/2 2 -1+√5/2 -1+√5/2 -1+√5/2 -1+√5/2 -1+√5/2 2 0 0 0 0 orthogonal lifted from D5 ρ13 2 0 2 2 2 2 -1+√5/2 -1+√5/2 -1-√5/2 2 -1-√5/2 -1-√5/2 -1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 2 -1+√5/2 0 0 0 0 orthogonal lifted from D5 ρ14 2 0 2 2 2 2 2 -1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 2 -1+√5/2 -1-√5/2 0 0 0 0 orthogonal lifted from D5 ρ15 5 -1 5ζ52 5ζ53 5ζ54 5ζ5 0 0 0 0 0 0 0 0 0 0 0 0 -ζ5 -ζ52 -ζ53 -ζ54 complex faithful ρ16 5 1 5ζ52 5ζ53 5ζ54 5ζ5 0 0 0 0 0 0 0 0 0 0 0 0 ζ5 ζ52 ζ53 ζ54 complex faithful ρ17 5 1 5ζ53 5ζ52 5ζ5 5ζ54 0 0 0 0 0 0 0 0 0 0 0 0 ζ54 ζ53 ζ52 ζ5 complex faithful ρ18 5 -1 5ζ5 5ζ54 5ζ52 5ζ53 0 0 0 0 0 0 0 0 0 0 0 0 -ζ53 -ζ5 -ζ54 -ζ52 complex faithful ρ19 5 1 5ζ5 5ζ54 5ζ52 5ζ53 0 0 0 0 0 0 0 0 0 0 0 0 ζ53 ζ5 ζ54 ζ52 complex faithful ρ20 5 -1 5ζ54 5ζ5 5ζ53 5ζ52 0 0 0 0 0 0 0 0 0 0 0 0 -ζ52 -ζ54 -ζ5 -ζ53 complex faithful ρ21 5 1 5ζ54 5ζ5 5ζ53 5ζ52 0 0 0 0 0 0 0 0 0 0 0 0 ζ52 ζ54 ζ5 ζ53 complex faithful ρ22 5 -1 5ζ53 5ζ52 5ζ5 5ζ54 0 0 0 0 0 0 0 0 0 0 0 0 -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  He54C4  C52⋊D10
He5⋊C2 is a maximal quotient of   He56C4

Matrix representation of He5⋊C2 in GL5(𝔽11)

 0 0 4 10 2 5 0 1 9 9 0 3 10 2 9 0 0 5 1 7 0 0 0 9 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 1 0 0 0 3 0 1 0 0 1 0 0 0 0 0 0 0 0 1 4 0 0 10 3
,
 1 8 1 0 0 0 3 3 0 0 0 1 8 0 0 0 0 4 3 10 0 4 0 8 8

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

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