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G = C5×D5  order 50 = 2·52

Direct product of C5 and D5

direct product, metacyclic, supersoluble, monomial, A-group

Aliases: C5×D5, C5≀C2, AΣL1(𝔽25), C5⋊C10, C52⋊1C2, SmallGroup(50,3)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C5 — C5×D5
Chief series
C1 — C5 — C52 — C5×D5
Lower central
C5 — C5×D5
Upper central
C1 — C5

Generators and relations for C5×D5
 G = < a,b,c | a5=b5=c2=1, ab=ba, ac=ca, cbc=b-1 >

C1
5C2
C5
C5
2C5
2C5
D5
5C10
C52
C5×D5

Character table of C5×D5

 class 125A5B5C5D5E5F5G5H5I5J5K5L5M5N10A10B10C10D
 size 15111122222222225555
ρ111111111111111111111    trivial
ρ21-111111111111111-1-1-1-1    linear of order 2
ρ311ζ53ζ5ζ52ζ54ζ5ζ54ζ541ζ52ζ5ζ52ζ53ζ531ζ52ζ54ζ5ζ53    linear of order 5
ρ41-1ζ53ζ5ζ52ζ54ζ5ζ54ζ541ζ52ζ5ζ52ζ53ζ531-ζ52-ζ54-ζ5-ζ53    linear of order 10
ρ511ζ5ζ52ζ54ζ53ζ52ζ53ζ531ζ54ζ52ζ54ζ5ζ51ζ54ζ53ζ52ζ5    linear of order 5
ρ611ζ52ζ54ζ53ζ5ζ54ζ5ζ51ζ53ζ54ζ53ζ52ζ521ζ53ζ5ζ54ζ52    linear of order 5
ρ71-1ζ52ζ54ζ53ζ5ζ54ζ5ζ51ζ53ζ54ζ53ζ52ζ521-ζ53-ζ5-ζ54-ζ52    linear of order 10
ρ811ζ54ζ53ζ5ζ52ζ53ζ52ζ521ζ5ζ53ζ5ζ54ζ541ζ5ζ52ζ53ζ54    linear of order 5
ρ91-1ζ54ζ53ζ5ζ52ζ53ζ52ζ521ζ5ζ53ζ5ζ54ζ541-ζ5-ζ52-ζ53-ζ54    linear of order 10
ρ101-1ζ5ζ52ζ54ζ53ζ52ζ53ζ531ζ54ζ52ζ54ζ5ζ51-ζ54-ζ53-ζ52-ζ5    linear of order 10
ρ11202222-1-√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
ρ12202222-1+√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
ρ13202ζ52ζ522ζ542ζ53ζ53+ζ5ζ54+ζ52ζ5+1-1+√5/2ζ53+1ζ54+1ζ52+ζ5ζ52+1ζ54+ζ53-1-√5/20000    complex faithful
ρ14202ζ532ζ52ζ522ζ54ζ54+ζ53ζ52+ζ5ζ53+1-1-√5/2ζ54+1ζ52+1ζ53+ζ5ζ5+1ζ54+ζ52-1+√5/20000    complex faithful
ρ15202ζ52ζ522ζ542ζ53ζ54+1ζ5+1ζ54+ζ52-1-√5/2ζ52+ζ5ζ53+ζ5ζ53+1ζ54+ζ53ζ52+1-1+√5/20000    complex faithful
ρ16202ζ522ζ542ζ532ζ5ζ52+ζ5ζ54+ζ53ζ52+1-1-√5/2ζ5+1ζ53+1ζ54+ζ52ζ54+1ζ53+ζ5-1+√5/20000    complex faithful
ρ17202ζ522ζ542ζ532ζ5ζ53+1ζ52+1ζ54+ζ53-1+√5/2ζ54+ζ52ζ52+ζ5ζ5+1ζ53+ζ5ζ54+1-1-√5/20000    complex faithful
ρ18202ζ532ζ52ζ522ζ54ζ52+1ζ53+1ζ52+ζ5-1+√5/2ζ53+ζ5ζ54+ζ53ζ54+1ζ54+ζ52ζ5+1-1-√5/20000    complex faithful
ρ19202ζ542ζ532ζ52ζ52ζ54+ζ52ζ53+ζ5ζ54+1-1+√5/2ζ52+1ζ5+1ζ54+ζ53ζ53+1ζ52+ζ5-1-√5/20000    complex faithful
ρ20202ζ542ζ532ζ52ζ52ζ5+1ζ54+1ζ53+ζ5-1-√5/2ζ54+ζ53ζ54+ζ52ζ52+1ζ52+ζ5ζ53+1-1+√5/20000    complex faithful

Permutation representations of C5×D5
►On 10 points - transitive group 10T6
Generators in S10
(1 2 3 4 5)(6 7 8 9 10)

(1 5 4 3 2)(6 7 8 9 10)

(1 10)(2 6)(3 7)(4 8)(5 9)
 
G:=sub<Sym(10)| (1,2,3,4,5)(6,7,8,9,10), (1,5,4,3,2)(6,7,8,9,10), (1,10)(2,6)(3,7)(4,8)(5,9)>;
 
G:=Group( (1,2,3,4,5)(6,7,8,9,10), (1,5,4,3,2)(6,7,8,9,10), (1,10)(2,6)(3,7)(4,8)(5,9) );
 
G=PermutationGroup([[(1,2,3,4,5),(6,7,8,9,10)], [(1,5,4,3,2),(6,7,8,9,10)], [(1,10),(2,6),(3,7),(4,8),(5,9)]])
 
G:=TransitiveGroup(10,6);
 

►On 25 points - transitive group 25T3
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)

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

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

C5×D5 is a maximal subgroup of   D5.D5  C52⋊S3  C52⋊C10  C25⋊C10  He5⋊C2
C5×D5 is a maximal quotient of   C52⋊C10  C25⋊C10

Polynomial with Galois group C5×D5 over ℚ
actionf(x)Disc(f)
10T6x10-15x8-10x7+55x6+53x5-40x4-50x3-5x2+5x+1513·114·3072

Matrix representation of C5×D5 ►in GL2(𝔽11) generated by

40
04
,
50
09
,
09
50
G:=sub<GL(2,GF(11))| [4,0,0,4],[5,0,0,9],[0,5,9,0] >;
 

C5×D5 in GAP, Magma, Sage, TeX 

C_5\times D_5
 
% in TeX
 
G:=Group("C5xD5");
 
// GroupNames label
 
G:=SmallGroup(50,3);
 
// by ID
 
G=gap.SmallGroup(50,3);
 
# by ID
 
G:=PCGroup([3,-2,-5,-5,362]);
 
// Polycyclic
 
G:=Group<a,b,c|a^5=b^5=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
 
// generators/relations
 

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Subgroup lattice of C5×D5 in TeX
Character table of C5×D5 in TeX

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