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G = He5⋊C4  order 500 = 22·53

2nd semidirect product of He5 and C4 acting faithfully

non-abelian, supersoluble, monomial

Aliases: He5⋊2C4, C52⋊Dic5, C52⋊2F5, C5⋊D5.D5, C52⋊C10.2C2, C5.2(D5.D5), SmallGroup(500,21)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C5 — He5 — He5⋊C4
Chief series
C1 — C5 — C52 — He5 — C52⋊C10 — He5⋊C4
Lower central
He5 — He5⋊C4
Upper central
C1

Generators and relations for He5⋊C4
 G = < a,b,c,d | a5=b5=c5=d4=1, cac-1=ab=ba, dad-1=a-1b-1c3, bc=cb, dbd-1=b3, dcd-1=c2 >

C1
25C2
C5
5C5
5C5
20C5
125C4
5D5
25C10
25D5
C52
C52
4C52
25F5
25F5
25Dic5
C5⋊D5
5C5×D5
He5
5C52⋊C4
5D5.D5
C52⋊C10
He5⋊C4

Character table of He5⋊C4

 class 124A4B5A5B5C5D5E5F5G5H10A10B
 size 1251251254101020202020205050
ρ111111111111111    trivial
ρ211-1-11111111111    linear of order 2
ρ31-1i-i11111111-1-1    linear of order 4
ρ41-1-ii11111111-1-1    linear of order 4
ρ522002-1-√5/2-1+√5/2-1+√5/2-1+√5/2-1-√5/22-1-√5/2-1-√5/2-1+√5/2    orthogonal lifted from D5
ρ622002-1+√5/2-1-√5/2-1-√5/2-1-√5/2-1+√5/22-1+√5/2-1+√5/2-1-√5/2    orthogonal lifted from D5
ρ72-2002-1+√5/2-1-√5/2-1-√5/2-1-√5/2-1+√5/22-1+√5/21-√5/21+√5/2    symplectic lifted from Dic5, Schur index 2
ρ82-2002-1-√5/2-1+√5/2-1+√5/2-1+√5/2-1-√5/22-1-√5/21+√5/21-√5/2    symplectic lifted from Dic5, Schur index 2
ρ94000444-1-1-1-1-100    orthogonal lifted from F5
ρ1040004-1+√5-1-√5ζ53+2ζ5+12ζ54+ζ52+1ζ54+2ζ53+1-12ζ52+ζ5+100    complex lifted from D5.D5
ρ1140004-1-√5-1+√5ζ54+2ζ53+12ζ52+ζ5+12ζ54+ζ52+1-1ζ53+2ζ5+100    complex lifted from D5.D5
ρ1240004-1-√5-1+√52ζ52+ζ5+1ζ54+2ζ53+1ζ53+2ζ5+1-12ζ54+ζ52+100    complex lifted from D5.D5
ρ1340004-1+√5-1-√52ζ54+ζ52+1ζ53+2ζ5+12ζ52+ζ5+1-1ζ54+2ζ53+100    complex lifted from D5.D5
ρ1420000-5000000000    orthogonal faithful

Permutation representations of He5⋊C4
►On 25 points - transitive group 25T36
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 13 21 18 9)(2 14 22 19 10)(3 15 23 20 6)(4 11 24 16 7)(5 12 25 17 8)

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

(2 17 22 25)(3 4 15 24)(5 14)(6 16 23 7)(8 10 12 19)(9 18 13 21)(11 20)
 
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), (1,13,21,18,9)(2,14,22,19,10)(3,15,23,20,6)(4,11,24,16,7)(5,12,25,17,8), (2,10,19,22,14)(3,20,15,6,23)(4,24,7,11,16)(5,12,25,17,8), (2,17,22,25)(3,4,15,24)(5,14)(6,16,23,7)(8,10,12,19)(9,18,13,21)(11,20)>;
 
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), (1,13,21,18,9)(2,14,22,19,10)(3,15,23,20,6)(4,11,24,16,7)(5,12,25,17,8), (2,10,19,22,14)(3,20,15,6,23)(4,24,7,11,16)(5,12,25,17,8), (2,17,22,25)(3,4,15,24)(5,14)(6,16,23,7)(8,10,12,19)(9,18,13,21)(11,20) );
 
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)], [(1,13,21,18,9),(2,14,22,19,10),(3,15,23,20,6),(4,11,24,16,7),(5,12,25,17,8)], [(2,10,19,22,14),(3,20,15,6,23),(4,24,7,11,16),(5,12,25,17,8)], [(2,17,22,25),(3,4,15,24),(5,14),(6,16,23,7),(8,10,12,19),(9,18,13,21),(11,20)]])
 
G:=TransitiveGroup(25,36);
 

►On 25 points - transitive group 25T39
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 7 11 14)(3 10 23 8)(4 5 22 13)(6 21 15 18)(9 12 19 25)(16 20 17 24)
 
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), (2,7,11,14)(3,10,23,8)(4,5,22,13)(6,21,15,18)(9,12,19,25)(16,20,17,24)>;
 
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), (2,7,11,14)(3,10,23,8)(4,5,22,13)(6,21,15,18)(9,12,19,25)(16,20,17,24) );
 
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)], [(2,7,11,14),(3,10,23,8),(4,5,22,13),(6,21,15,18),(9,12,19,25),(16,20,17,24)]])
 
G:=TransitiveGroup(25,39);
 

Matrix representation of He5⋊C4 ►in GL20(ℤ)

00001000000000000000
00000100000000000000
00000010000000000000
00000001000000000000
00000000010000000000
00000000001000000000
00000000000100000000
00000000-1-1-1-100000000
00000000000000100000
00000000000000010000
000000000000-1-1-1-10000
00000000000010000000
00000000000000000001
0000000000000000-1-1-1-1
00000000000000001000
00000000000000000100
-1-1-1-10000000000000000
10000000000000000000
01000000000000000000
00100000000000000000
,
01000000000000000000
00100000000000000000
00010000000000000000
-1-1-1-10000000000000000
00000100000000000000
00000010000000000000
00000001000000000000
0000-1-1-1-1000000000000
00000000010000000000
00000000001000000000
00000000000100000000
00000000-1-1-1-100000000
00000000000001000000
00000000000000100000
00000000000000010000
000000000000-1-1-1-10000
00000000000000000100
00000000000000000010
00000000000000000001
0000000000000000-1-1-1-1
,
10000000000000000000
01000000000000000000
00100000000000000000
00010000000000000000
00000100000000000000
00000010000000000000
00000001000000000000
0000-1-1-1-1000000000000
00000000001000000000
00000000000100000000
00000000-1-1-1-100000000
00000000100000000000
00000000000000010000
000000000000-1-1-1-10000
00000000000010000000
00000000000001000000
0000000000000000-1-1-1-1
00000000000000001000
00000000000000000100
00000000000000000010
,
10000000000000000000
00010000000000000000
01000000000000000000
-1-1-1-10000000000000000
00000000000000001000
00000000000000000001
00000000000000000100
0000000000000000-1-1-1-1
00000000000010000000
00000000000000010000
00000000000001000000
000000000000-1-1-1-10000
00000000100000000000
00000000000100000000
00000000010000000000
00000000-1-1-1-100000000
00001000000000000000
00000001000000000000
00000100000000000000
0000-1-1-1-1000000000000

G:=sub<GL(20,Integers())| [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0],[0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1],[1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0],[1,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,-1,0,0,0,0,0,0,0,0,0,0,0,0] >;
 

He5⋊C4 in GAP, Magma, Sage, TeX 

{\rm He}_5\rtimes C_4
 
% in TeX
 
G:=Group("He5:C4");
 
// GroupNames label
 
G:=SmallGroup(500,21);
 
// by ID
 
G=gap.SmallGroup(500,21);
 
# by ID
 
G:=PCGroup([5,-2,-2,-5,-5,-5,10,242,1203,808,613,5004,5009]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^5=b^5=c^5=d^4=1,c*a*c^-1=a*b=b*a,d*a*d^-1=a^-1*b^-1*c^3,b*c=c*b,d*b*d^-1=b^3,d*c*d^-1=c^2>;
 
// generators/relations
 

Export

Subgroup lattice of He5⋊C4 in TeX
Character table of He5⋊C4 in TeX

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