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G = C52⋊D10order 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

C1C5He5 — C52⋊D10
C1C5C52He5C52⋊C10 — C52⋊D10
He5 — C52⋊D10
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 >

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

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
ρ1710-200-5-55/2-5+55/20000000000001-5/21+5/2    orthogonal faithful
ρ1810200-5+55/2-5-55/2000000000000-1-5/2-1+5/2    orthogonal faithful
ρ1910200-5-55/2-5+55/2000000000000-1+5/2-1-5/2    orthogonal faithful
ρ2010-200-5+55/2-5-55/20000000000001+5/21-5/2    orthogonal faithful

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

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

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

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

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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