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G = C25⋊C10order 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

C1C25 — C25⋊C10
C1C5C255- 1+2 — C25⋊C10
C25 — C25⋊C10
C1

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

25C2
5C5
5D5
25C10
2C25
2C25
5C5×D5

Character table of C25⋊C10

 class 125A5B5C5D5E5F10A10B10C10D25A25B25C25D25E25F25G25H25I25J
 size 1252255552525252510101010101010101010
ρ11111111111111111111111    trivial
ρ21-1111111-1-1-1-11111111111    linear of order 2
ρ31-111ζ53ζ5ζ54ζ525455253ζ54ζ53ζ51ζ54ζ5ζ52ζ521ζ53    linear of order 10
ρ41111ζ5ζ52ζ53ζ54ζ53ζ52ζ54ζ5ζ53ζ5ζ521ζ53ζ52ζ54ζ541ζ5    linear of order 5
ρ51111ζ53ζ5ζ54ζ52ζ54ζ5ζ52ζ53ζ54ζ53ζ51ζ54ζ5ζ52ζ521ζ53    linear of order 5
ρ61-111ζ54ζ53ζ52ζ55253554ζ52ζ54ζ531ζ52ζ53ζ5ζ51ζ54    linear of order 10
ρ71111ζ54ζ53ζ52ζ5ζ52ζ53ζ5ζ54ζ52ζ54ζ531ζ52ζ53ζ5ζ51ζ54    linear of order 5
ρ81111ζ52ζ54ζ5ζ53ζ5ζ54ζ53ζ52ζ5ζ52ζ541ζ5ζ54ζ53ζ531ζ52    linear of order 5
ρ91-111ζ5ζ52ζ53ζ545352545ζ53ζ5ζ521ζ53ζ52ζ54ζ541ζ5    linear of order 10
ρ101-111ζ52ζ54ζ5ζ535545352ζ5ζ52ζ541ζ5ζ54ζ53ζ531ζ52    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
ρ13202253554520000ζ53+1ζ5+1ζ52+1-1-5/2ζ525ζ5453ζ535ζ54+1-1+5/2ζ5452    complex lifted from C5×D5
ρ14202255253540000ζ5+1ζ52+1ζ54+1-1+5/2ζ5452ζ535ζ525ζ53+1-1-5/2ζ5453    complex lifted from C5×D5
ρ15202254535250000ζ54+1ζ53+1ζ5+1-1+5/2ζ535ζ5452ζ5453ζ52+1-1-5/2ζ525    complex lifted from C5×D5
ρ16202255253540000ζ5452ζ5453ζ535-1-5/2ζ5+1ζ54+1ζ53+1ζ525-1+5/2ζ52+1    complex lifted from C5×D5
ρ17202252545530000ζ5453ζ535ζ525-1+5/2ζ52+1ζ53+1ζ5+1ζ5452-1-5/2ζ54+1    complex lifted from C5×D5
ρ18202253554520000ζ525ζ5452ζ5453-1+5/2ζ53+1ζ52+1ζ54+1ζ535-1-5/2ζ5+1    complex lifted from C5×D5
ρ19202252545530000ζ52+1ζ54+1ζ53+1-1-5/2ζ5453ζ525ζ5452ζ5+1-1+5/2ζ535    complex lifted from C5×D5
ρ20202254535250000ζ535ζ525ζ5452-1-5/2ζ54+1ζ5+1ζ52+1ζ5453-1+5/2ζ53+1    complex lifted from C5×D5
ρ21100-5+55/2-5-55/2000000000000000000    orthogonal faithful
ρ22100-5-55/2-5+55/2000000000000000000    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

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

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