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G = C52⋊D10  order 500 = 22·53

The semidirect product of C52 and D10 acting faithfully

non-abelian, supersoluble, monomial

Aliases: C52⋊D10, He5⋊C22, C5⋊D5⋊D5, C5.2D52, C52⋊C10⋊C2, He5⋊C2⋊C2, SmallGroup(500,27)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C5 — He5 — C52⋊D10
Chief series
C1 — C5 — C52 — He5 — C52⋊C10 — C52⋊D10
Lower central
He5 — C52⋊D10
Upper central
C1

Generators and relations for C52⋊D10
 G = < a,b,c,d | a5=b5=c10=d2=1, ab=ba, cac-1=dad=a-1b3, cbc-1=b-1, bd=db, dcd=c-1 >

C1
25C2
25C2
25C2
C5
5C5
5C5
10C5
10C5
125C22
5D5
5D5
5D5
5D5
10D5
10D5
25C10
25D5
25C10
25D5
25C10
C52
C52
2C52
2C52
25D10
25D10
25D10
C5⋊D5
C5⋊D5
5C5×D5
5C5×D5
5C5×D5
5C5×D5
10C5×D5
10C5×D5
He5
5D52
5D52
C52⋊C10
He5⋊C2
C52⋊C10
C52⋊D10

Character table of C52⋊D10

 class 12A2B2C5A5B5C5D5E5F5G5H5I5J10A10B10C10D10E10F
 size 1252525221010101020202020505050505050
ρ111111111111111111111    trivial
ρ21-1-111111111111-111-1-1-1    linear of order 2
ρ311-1-11111111111-1-1-1-111    linear of order 2
ρ41-11-111111111111-1-11-1-1    linear of order 2
ρ5202022-1-√5/222-1+√5/2-1-√5/2-1+√5/2-1+√5/2-1-√5/2-1+√5/200-1-√5/200    orthogonal lifted from D5
ρ6202022-1+√5/222-1-√5/2-1+√5/2-1-√5/2-1-√5/2-1+√5/2-1-√5/200-1+√5/200    orthogonal lifted from D5
ρ72002222-1-√5/2-1+√5/22-1+√5/2-1-√5/2-1+√5/2-1-√5/20-1-√5/2-1+√5/2000    orthogonal lifted from D5
ρ82002222-1+√5/2-1-√5/22-1-√5/2-1+√5/2-1-√5/2-1+√5/20-1+√5/2-1-√5/2000    orthogonal lifted from D5
ρ9200-2222-1+√5/2-1-√5/22-1-√5/2-1+√5/2-1-√5/2-1+√5/201-√5/21+√5/2000    orthogonal lifted from D10
ρ10200-2222-1-√5/2-1+√5/22-1+√5/2-1-√5/2-1+√5/2-1-√5/201+√5/21-√5/2000    orthogonal lifted from D10
ρ1120-2022-1+√5/222-1-√5/2-1+√5/2-1-√5/2-1-√5/2-1+√5/21+√5/2001-√5/200    orthogonal lifted from D10
ρ1220-2022-1-√5/222-1+√5/2-1-√5/2-1+√5/2-1+√5/2-1-√5/21-√5/2001+√5/200    orthogonal lifted from D10
ρ13400044-1+√5-1+√5-1-√5-1-√5-1-13+√5/23-√5/2000000    orthogonal lifted from D52
ρ14400044-1+√5-1-√5-1+√5-1-√53-√5/23+√5/2-1-1000000    orthogonal lifted from D52
ρ15400044-1-√5-1+√5-1-√5-1+√53+√5/23-√5/2-1-1000000    orthogonal lifted from D52
ρ16400044-1-√5-1-√5-1+√5-1+√5-1-13-√5/23+√5/2000000    orthogonal lifted from D52
ρ1710-200-5-5√5/2-5+5√5/20000000000001-√5/21+√5/2    orthogonal faithful
ρ1810200-5+5√5/2-5-5√5/2000000000000-1-√5/2-1+√5/2    orthogonal faithful
ρ1910200-5-5√5/2-5+5√5/2000000000000-1+√5/2-1-√5/2    orthogonal faithful
ρ2010-200-5+5√5/2-5-5√5/20000000000001+√5/21-√5/2    orthogonal faithful

Permutation representations of C52⋊D10
►On 25 points - transitive group 25T38
Generators in S25
(1 20 8 13 25)(2 18 12 9 17)(3 22 14 7 23)(4 16 6 15 19)(5 24 10 11 21)

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

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

(6 9)(7 8)(10 15)(11 14)(12 13)(16 23)(17 22)(18 21)(19 20)(24 25)
 
G:=sub<Sym(25)| (1,20,8,13,25)(2,18,12,9,17)(3,22,14,7,23)(4,16,6,15,19)(5,24,10,11,21), (1,3,5,4,2)(6,12,8,14,10)(7,11,15,9,13)(16,18,20,22,24)(17,25,23,21,19), (2,3)(4,5)(6,7,8,9,10,11,12,13,14,15)(16,17,18,19,20,21,22,23,24,25), (6,9)(7,8)(10,15)(11,14)(12,13)(16,23)(17,22)(18,21)(19,20)(24,25)>;
 
G:=Group( (1,20,8,13,25)(2,18,12,9,17)(3,22,14,7,23)(4,16,6,15,19)(5,24,10,11,21), (1,3,5,4,2)(6,12,8,14,10)(7,11,15,9,13)(16,18,20,22,24)(17,25,23,21,19), (2,3)(4,5)(6,7,8,9,10,11,12,13,14,15)(16,17,18,19,20,21,22,23,24,25), (6,9)(7,8)(10,15)(11,14)(12,13)(16,23)(17,22)(18,21)(19,20)(24,25) );
 
G=PermutationGroup([[(1,20,8,13,25),(2,18,12,9,17),(3,22,14,7,23),(4,16,6,15,19),(5,24,10,11,21)], [(1,3,5,4,2),(6,12,8,14,10),(7,11,15,9,13),(16,18,20,22,24),(17,25,23,21,19)], [(2,3),(4,5),(6,7,8,9,10,11,12,13,14,15),(16,17,18,19,20,21,22,23,24,25)], [(6,9),(7,8),(10,15),(11,14),(12,13),(16,23),(17,22),(18,21),(19,20),(24,25)]])
 
G:=TransitiveGroup(25,38);
 

Matrix representation of C52⋊D10 ►in GL10(𝔽11)

0010000000
0001000000
0000100000
0000010000
0000001000
0000000100
0000000010
0000000001
1000000000
0100000000
,
0100000000
10300000000
0001000000
00103000000
0000010000
00001030000
0000000100
00000010300
0000000001
00000000103
,
1000000000
31000000000
0000000088
00000000103
0000000100
0000001000
00003100000
0000880000
00103000000
0001000000
,
1000000000
0100000000
0000000088
00000000310
0000000100
00000010300
00003100000
0000100000
00103000000
0088000000

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

C52⋊D10 in GAP, Magma, Sage, TeX 

C_5^2\rtimes D_{10}
 
% in TeX
 
G:=Group("C5^2:D10");
 
// GroupNames label
 
G:=SmallGroup(500,27);
 
// by ID
 
G=gap.SmallGroup(500,27);
 
# by ID
 
G:=PCGroup([5,-2,-2,-5,-5,-5,127,1603,613,10004,5009]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^5=b^5=c^10=d^2=1,a*b=b*a,c*a*c^-1=d*a*d=a^-1*b^3,c*b*c^-1=b^-1,b*d=d*b,d*c*d=c^-1>;
 
// generators/relations
 

Export

Subgroup lattice of C52⋊D10 in TeX
Character table of C52⋊D10 in TeX

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