Copied to
clipboard

G = C5×D5order 50 = 2·52

Direct product of C5 and D5

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

Aliases: C5×D5, C5C2, AΣL1(𝔽25), C5⋊C10, C521C2, SmallGroup(50,3)

Series: Derived Chief Lower central Upper central

C1C5 — C5×D5
C1C5C52 — C5×D5
C5 — C5×D5
C1C5

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

5C2
2C5
2C5
5C10

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
ρ41-1ζ53ζ5ζ52ζ54ζ5ζ54ζ541ζ52ζ5ζ52ζ53ζ5315254553    linear of order 10
ρ511ζ5ζ52ζ54ζ53ζ52ζ53ζ531ζ54ζ52ζ54ζ5ζ51ζ54ζ53ζ52ζ5    linear of order 5
ρ611ζ52ζ54ζ53ζ5ζ54ζ5ζ51ζ53ζ54ζ53ζ52ζ521ζ53ζ5ζ54ζ52    linear of order 5
ρ71-1ζ52ζ54ζ53ζ5ζ54ζ5ζ51ζ53ζ54ζ53ζ52ζ5215355452    linear of order 10
ρ811ζ54ζ53ζ5ζ52ζ53ζ52ζ521ζ5ζ53ζ5ζ54ζ541ζ5ζ52ζ53ζ54    linear of order 5
ρ91-1ζ54ζ53ζ5ζ52ζ53ζ52ζ521ζ5ζ53ζ5ζ54ζ5415525354    linear of order 10
ρ101-1ζ5ζ52ζ54ζ53ζ52ζ53ζ531ζ54ζ52ζ54ζ5ζ515453525    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
ρ13205525453ζ535ζ5452ζ5+1-1+5/2ζ53+1ζ54+1ζ525ζ52+1ζ5453-1-5/20000    complex faithful
ρ14205355254ζ5453ζ525ζ53+1-1-5/2ζ54+1ζ52+1ζ535ζ5+1ζ5452-1+5/20000    complex faithful
ρ15205525453ζ54+1ζ5+1ζ5452-1-5/2ζ525ζ535ζ53+1ζ5453ζ52+1-1+5/20000    complex faithful
ρ16205254535ζ525ζ5453ζ52+1-1-5/2ζ5+1ζ53+1ζ5452ζ54+1ζ535-1+5/20000    complex faithful
ρ17205254535ζ53+1ζ52+1ζ5453-1+5/2ζ5452ζ525ζ5+1ζ535ζ54+1-1-5/20000    complex faithful
ρ18205355254ζ52+1ζ53+1ζ525-1+5/2ζ535ζ5453ζ54+1ζ5452ζ5+1-1-5/20000    complex faithful
ρ19205453552ζ5452ζ535ζ54+1-1+5/2ζ52+1ζ5+1ζ5453ζ53+1ζ525-1-5/20000    complex faithful
ρ20205453552ζ5+1ζ54+1ζ535-1-5/2ζ5453ζ5452ζ52+1ζ525ζ53+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 9 12 19)(2 21 10 13 20)(3 22 6 14 16)(4 23 7 15 17)(5 24 8 11 18)
(1 19)(2 20)(3 16)(4 17)(5 18)(11 24)(12 25)(13 21)(14 22)(15 23)

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

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

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

G:=TransitiveGroup(25,3);

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

׿
×
𝔽