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G = C25⋊C10  order 250 = 2·53

The semidirect product of C25 and C10 acting faithfully

metacyclic, supersoluble, monomial

Aliases: C25⋊C10, D25⋊C5, C52.D5, 5- 1+2⋊C2, C5.3(C5×D5), SmallGroup(250,6)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C25 — C25⋊C10
Chief series
C1 — C5 — C25 — 5- 1+2 — C25⋊C10
Lower central
C25 — C25⋊C10
Upper central
C1

Generators and relations for C25⋊C10
 G = < a,b | a25=b10=1, bab-1=a9 >

C1
25C2
C5
5C5
5D5
25C10
C25
C52
2C25
2C25
D25
5C5×D5
5- 1+2
C25⋊C10

Character table of C25⋊C10

 class 125A5B5C5D5E5F10A10B10C10D25A25B25C25D25E25F25G25H25I25J
 size 1252255552525252510101010101010101010
ρ11111111111111111111111    trivial
ρ21-1111111-1-1-1-11111111111    linear of order 2
ρ31-111ζ53ζ5ζ54ζ52-ζ54-ζ5-ζ52-ζ53ζ54ζ53ζ51ζ54ζ5ζ52ζ521ζ53    linear of order 10
ρ41111ζ5ζ52ζ53ζ54ζ53ζ52ζ54ζ5ζ53ζ5ζ521ζ53ζ52ζ54ζ541ζ5    linear of order 5
ρ51111ζ53ζ5ζ54ζ52ζ54ζ5ζ52ζ53ζ54ζ53ζ51ζ54ζ5ζ52ζ521ζ53    linear of order 5
ρ61-111ζ54ζ53ζ52ζ5-ζ52-ζ53-ζ5-ζ54ζ52ζ54ζ531ζ52ζ53ζ5ζ51ζ54    linear of order 10
ρ71111ζ54ζ53ζ52ζ5ζ52ζ53ζ5ζ54ζ52ζ54ζ531ζ52ζ53ζ5ζ51ζ54    linear of order 5
ρ81111ζ52ζ54ζ5ζ53ζ5ζ54ζ53ζ52ζ5ζ52ζ541ζ5ζ54ζ53ζ531ζ52    linear of order 5
ρ91-111ζ5ζ52ζ53ζ54-ζ53-ζ52-ζ54-ζ5ζ53ζ5ζ521ζ53ζ52ζ54ζ541ζ5    linear of order 10
ρ101-111ζ52ζ54ζ5ζ53-ζ5-ζ54-ζ53-ζ52ζ5ζ52ζ541ζ5ζ54ζ53ζ531ζ52    linear of order 10
ρ11202222220000-1+√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+√5/2    orthogonal lifted from D5
ρ12202222220000-1-√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-√5/2    orthogonal lifted from D5
ρ1320222ζ532ζ52ζ542ζ520000ζ53+1ζ5+1ζ52+1-1-√5/2ζ52+ζ5ζ54+ζ53ζ53+ζ5ζ54+1-1+√5/2ζ54+ζ52    complex lifted from C5×D5
ρ1420222ζ52ζ522ζ532ζ540000ζ5+1ζ52+1ζ54+1-1+√5/2ζ54+ζ52ζ53+ζ5ζ52+ζ5ζ53+1-1-√5/2ζ54+ζ53    complex lifted from C5×D5
ρ1520222ζ542ζ532ζ522ζ50000ζ54+1ζ53+1ζ5+1-1+√5/2ζ53+ζ5ζ54+ζ52ζ54+ζ53ζ52+1-1-√5/2ζ52+ζ5    complex lifted from C5×D5
ρ1620222ζ52ζ522ζ532ζ540000ζ54+ζ52ζ54+ζ53ζ53+ζ5-1-√5/2ζ5+1ζ54+1ζ53+1ζ52+ζ5-1+√5/2ζ52+1    complex lifted from C5×D5
ρ1720222ζ522ζ542ζ52ζ530000ζ54+ζ53ζ53+ζ5ζ52+ζ5-1+√5/2ζ52+1ζ53+1ζ5+1ζ54+ζ52-1-√5/2ζ54+1    complex lifted from C5×D5
ρ1820222ζ532ζ52ζ542ζ520000ζ52+ζ5ζ54+ζ52ζ54+ζ53-1+√5/2ζ53+1ζ52+1ζ54+1ζ53+ζ5-1-√5/2ζ5+1    complex lifted from C5×D5
ρ1920222ζ522ζ542ζ52ζ530000ζ52+1ζ54+1ζ53+1-1-√5/2ζ54+ζ53ζ52+ζ5ζ54+ζ52ζ5+1-1+√5/2ζ53+ζ5    complex lifted from C5×D5
ρ2020222ζ542ζ532ζ522ζ50000ζ53+ζ5ζ52+ζ5ζ54+ζ52-1-√5/2ζ54+1ζ5+1ζ52+1ζ54+ζ53-1+√5/2ζ53+1    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 C25⋊C10
►On 25 points - transitive group 25T25
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)

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

C25⋊C10 is a maximal subgroup of   C25⋊C20
C25⋊C10 is a maximal quotient of   C50.C10

Matrix representation of C25⋊C10 ►in GL10(𝔽101)

007979000000
0022100000000
000079790000
0000221000000
000000797900
0000002210000
000000007979
0000000022100
1002200000000
797900000000
,
1000000000
2210000000000
0000000022100
000000007979
0000001002200
0000000100
0000100000
0000221000000
007979000000
0010022000000

G:=sub<GL(10,GF(101))| [0,0,0,0,0,0,0,0,100,79,0,0,0,0,0,0,0,0,22,79,79,22,0,0,0,0,0,0,0,0,79,100,0,0,0,0,0,0,0,0,0,0,79,22,0,0,0,0,0,0,0,0,79,100,0,0,0,0,0,0,0,0,0,0,79,22,0,0,0,0,0,0,0,0,79,100,0,0,0,0,0,0,0,0,0,0,79,22,0,0,0,0,0,0,0,0,79,100,0,0],[1,22,0,0,0,0,0,0,0,0,0,100,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,79,100,0,0,0,0,0,0,0,0,79,22,0,0,0,0,0,0,1,22,0,0,0,0,0,0,0,0,0,100,0,0,0,0,0,0,100,0,0,0,0,0,0,0,0,0,22,1,0,0,0,0,0,0,22,79,0,0,0,0,0,0,0,0,100,79,0,0,0,0,0,0] >;
 

C25⋊C10 in GAP, Magma, Sage, TeX 

C_{25}\rtimes C_{10}
 
% in TeX
 
G:=Group("C25:C10");
 
// GroupNames label
 
G:=SmallGroup(250,6);
 
// by ID
 
G=gap.SmallGroup(250,6);
 
# by ID
 
G:=PCGroup([4,-2,-5,-5,-5,1082,366,250,3203]);
 
// Polycyclic
 
G:=Group<a,b|a^25=b^10=1,b*a*b^-1=a^9>;
 
// generators/relations
 

Export

Subgroup lattice of C25⋊C10 in TeX
Character table of C25⋊C10 in TeX

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