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G = C52⋊C10  order 250 = 2·53

The semidirect product of C52 and C10 acting faithfully

metabelian, supersoluble, monomial

Aliases: C52⋊C10, He5⋊1C2, C52⋊1D5, C5⋊D5⋊C5, C5.2(C5×D5), SmallGroup(250,5)

Series: Derived ►Chief ►Lower central ►Upper central

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

Generators and relations for C52⋊C10
 G = < a,b,c | a5=b5=c10=1, ab=ba, cac-1=a-1b3, cbc-1=b-1 >

C1
25C2
C5
5C5
5C5
10C5
10C5
5D5
25C10
25D5
C52
C52
2C52
2C52
C5⋊D5
5C5×D5
He5
C52⋊C10

Character table of C52⋊C10

 class 125A5B5C5D5E5F5G5H5I5J5K5L5M5N5O5P10A10B10C10D
 size 1252255551010101010101010101025252525
ρ11111111111111111111111    trivial
ρ21-11111111111111111-1-1-1-1    linear of order 2
ρ31-111ζ5ζ52ζ53ζ541ζ54ζ541ζ5ζ5ζ52ζ53ζ53ζ52-ζ52-ζ54-ζ5-ζ53    linear of order 10
ρ41111ζ5ζ52ζ53ζ541ζ54ζ541ζ5ζ5ζ52ζ53ζ53ζ52ζ52ζ54ζ5ζ53    linear of order 5
ρ51111ζ54ζ53ζ52ζ51ζ5ζ51ζ54ζ54ζ53ζ52ζ52ζ53ζ53ζ5ζ54ζ52    linear of order 5
ρ61-111ζ52ζ54ζ5ζ531ζ53ζ531ζ52ζ52ζ54ζ5ζ5ζ54-ζ54-ζ53-ζ52-ζ5    linear of order 10
ρ71-111ζ54ζ53ζ52ζ51ζ5ζ51ζ54ζ54ζ53ζ52ζ52ζ53-ζ53-ζ5-ζ54-ζ52    linear of order 10
ρ81111ζ52ζ54ζ5ζ531ζ53ζ531ζ52ζ52ζ54ζ5ζ5ζ54ζ54ζ53ζ52ζ5    linear of order 5
ρ91111ζ53ζ5ζ54ζ521ζ52ζ521ζ53ζ53ζ5ζ54ζ54ζ5ζ5ζ52ζ53ζ54    linear of order 5
ρ101-111ζ53ζ5ζ54ζ521ζ52ζ521ζ53ζ53ζ5ζ54ζ54ζ5-ζ5-ζ52-ζ53-ζ54    linear of order 10
ρ1120222222-1-√5/2-1+√5/2-1-√5/2-1+√5/2-1+√5/2-1-√5/2-1+√5/2-1+√5/2-1-√5/2-1-√5/20000    orthogonal lifted from D5
ρ1220222222-1+√5/2-1-√5/2-1+√5/2-1-√5/2-1-√5/2-1+√5/2-1-√5/2-1-√5/2-1+√5/2-1+√5/20000    orthogonal lifted from D5
ρ1320222ζ522ζ542ζ52ζ53-1+√5/2ζ5+1ζ54+ζ52-1-√5/2ζ54+1ζ53+ζ5ζ52+ζ5ζ54+ζ53ζ52+1ζ53+10000    complex lifted from C5×D5
ρ1420222ζ522ζ542ζ52ζ53-1-√5/2ζ54+ζ52ζ5+1-1+√5/2ζ53+ζ5ζ54+1ζ53+1ζ52+1ζ54+ζ53ζ52+ζ50000    complex lifted from C5×D5
ρ1520222ζ52ζ522ζ532ζ54-1-√5/2ζ53+1ζ52+ζ5-1+√5/2ζ52+1ζ54+ζ53ζ53+ζ5ζ54+ζ52ζ5+1ζ54+10000    complex lifted from C5×D5
ρ1620222ζ542ζ532ζ522ζ5-1+√5/2ζ54+ζ53ζ52+1-1-√5/2ζ52+ζ5ζ53+1ζ5+1ζ54+1ζ53+ζ5ζ54+ζ520000    complex lifted from C5×D5
ρ1720222ζ532ζ52ζ542ζ52-1+√5/2ζ54+1ζ53+ζ5-1-√5/2ζ5+1ζ54+ζ52ζ54+ζ53ζ52+ζ5ζ53+1ζ52+10000    complex lifted from C5×D5
ρ1820222ζ52ζ522ζ532ζ54-1+√5/2ζ52+ζ5ζ53+1-1-√5/2ζ54+ζ53ζ52+1ζ54+1ζ5+1ζ54+ζ52ζ53+ζ50000    complex lifted from C5×D5
ρ1920222ζ532ζ52ζ542ζ52-1-√5/2ζ53+ζ5ζ54+1-1+√5/2ζ54+ζ52ζ5+1ζ52+1ζ53+1ζ52+ζ5ζ54+ζ530000    complex lifted from C5×D5
ρ2020222ζ542ζ532ζ522ζ5-1-√5/2ζ52+1ζ54+ζ53-1+√5/2ζ53+1ζ52+ζ5ζ54+ζ52ζ53+ζ5ζ54+1ζ5+10000    complex lifted from C5×D5
ρ21100-5+5√5/2-5-5√5/2000000000000000000    orthogonal faithful
ρ22100-5-5√5/2-5+5√5/2000000000000000000    orthogonal faithful

Permutation representations of C52⋊C10
►On 25 points - transitive group 25T23
Generators in S25
(1 16 6 11 21)(2 24 10 7 23)(3 18 12 15 19)(4 22 14 13 25)(5 20 8 9 17)

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

►On 25 points - transitive group 25T24
Generators in S25
(1 15 25 20 10)(2 16 11 6 21)(4 23 8 13 18)(5 14 24 19 9)

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

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

C52⋊C10 is a maximal subgroup of   C52⋊C20  He5⋊C4  C52⋊D10
C52⋊C10 is a maximal quotient of   He5⋊5C4

Matrix representation of C52⋊C10 ►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

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] >;
 

C52⋊C10 in GAP, Magma, Sage, TeX 

C_5^2\rtimes C_{10}
 
% in TeX
 
G:=Group("C5^2:C10");
 
// GroupNames label
 
G:=SmallGroup(250,5);
 
// by ID
 
G=gap.SmallGroup(250,5);
 
# by ID
 
G:=PCGroup([4,-2,-5,-5,-5,482,366,3203]);
 
// Polycyclic
 
G:=Group<a,b,c|a^5=b^5=c^10=1,a*b=b*a,c*a*c^-1=a^-1*b^3,c*b*c^-1=b^-1>;
 
// generators/relations
 

Export

Subgroup lattice of C52⋊C10 in TeX
Character table of C52⋊C10 in TeX

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