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G = C6×D5order 60 = 22·3·5

Direct product of C6 and D5

direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary

Aliases: C6×D5, C10⋊C6, C302C2, C153C22, C5⋊(C2×C6), SmallGroup(60,10)

Series: Derived Chief Lower central Upper central

C1C5 — C6×D5
C1C5C15C3×D5 — C6×D5
C5 — C6×D5
C1C6

Generators and relations for C6×D5
 G = < a,b,c | a6=b5=c2=1, ab=ba, ac=ca, cbc=b-1 >

5C2
5C2
5C22
5C6
5C6
5C2×C6

Character table of C6×D5

 class 12A2B2C3A3B5A5B6A6B6C6D6E6F10A10B15A15B15C15D30A30B30C30D
 size 115511221155552222222222
ρ1111111111111111111111111    trivial
ρ21-1-111111-1-1-111-1-1-11111-1-1-1-1    linear of order 2
ρ31-11-11111-1-11-1-11-1-11111-1-1-1-1    linear of order 2
ρ411-1-1111111-1-1-1-11111111111    linear of order 2
ρ51-11-1ζ32ζ311ζ65ζ6ζ32ζ65ζ6ζ3-1-1ζ32ζ32ζ3ζ3ζ65ζ6ζ65ζ6    linear of order 6
ρ61111ζ32ζ311ζ3ζ32ζ32ζ3ζ32ζ311ζ32ζ32ζ3ζ3ζ3ζ32ζ3ζ32    linear of order 3
ρ71-1-11ζ3ζ3211ζ6ζ65ζ65ζ32ζ3ζ6-1-1ζ3ζ3ζ32ζ32ζ6ζ65ζ6ζ65    linear of order 6
ρ81-11-1ζ3ζ3211ζ6ζ65ζ3ζ6ζ65ζ32-1-1ζ3ζ3ζ32ζ32ζ6ζ65ζ6ζ65    linear of order 6
ρ91111ζ3ζ3211ζ32ζ3ζ3ζ32ζ3ζ3211ζ3ζ3ζ32ζ32ζ32ζ3ζ32ζ3    linear of order 3
ρ101-1-11ζ32ζ311ζ65ζ6ζ6ζ3ζ32ζ65-1-1ζ32ζ32ζ3ζ3ζ65ζ6ζ65ζ6    linear of order 6
ρ1111-1-1ζ32ζ311ζ3ζ32ζ6ζ65ζ6ζ6511ζ32ζ32ζ3ζ3ζ3ζ32ζ3ζ32    linear of order 6
ρ1211-1-1ζ3ζ3211ζ32ζ3ζ65ζ6ζ65ζ611ζ3ζ3ζ32ζ32ζ32ζ3ζ32ζ3    linear of order 6
ρ132-20022-1-5/2-1+5/2-2-200001-5/21+5/2-1+5/2-1-5/2-1+5/2-1-5/21-5/21+5/21+5/21-5/2    orthogonal lifted from D10
ρ14220022-1+5/2-1-5/2220000-1-5/2-1+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
ρ15220022-1-5/2-1+5/2220000-1+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
ρ162-20022-1+5/2-1-5/2-2-200001+5/21-5/2-1-5/2-1+5/2-1-5/2-1+5/21+5/21-5/21-5/21+5/2    orthogonal lifted from D10
ρ172200-1+-3-1--3-1+5/2-1-5/2-1--3-1+-30000-1-5/2-1+5/2ζ3ζ533ζ52ζ3ζ543ζ5ζ32ζ5332ζ52ζ32ζ5432ζ5ζ32ζ5332ζ52ζ3ζ543ζ5ζ32ζ5432ζ5ζ3ζ533ζ52    complex lifted from C3×D5
ρ182200-1--3-1+-3-1-5/2-1+5/2-1+-3-1--30000-1+5/2-1-5/2ζ32ζ5432ζ5ζ32ζ5332ζ52ζ3ζ543ζ5ζ3ζ533ζ52ζ3ζ543ζ5ζ32ζ5332ζ52ζ3ζ533ζ52ζ32ζ5432ζ5    complex lifted from C3×D5
ρ192200-1--3-1+-3-1+5/2-1-5/2-1+-3-1--30000-1-5/2-1+5/2ζ32ζ5332ζ52ζ32ζ5432ζ5ζ3ζ533ζ52ζ3ζ543ζ5ζ3ζ533ζ52ζ32ζ5432ζ5ζ3ζ543ζ5ζ32ζ5332ζ52    complex lifted from C3×D5
ρ202-200-1--3-1+-3-1-5/2-1+5/21--31+-300001-5/21+5/2ζ32ζ5432ζ5ζ32ζ5332ζ52ζ3ζ543ζ5ζ3ζ533ζ523ζ543ζ532ζ5332ζ523ζ533ζ5232ζ5432ζ5    complex faithful
ρ212-200-1--3-1+-3-1+5/2-1-5/21--31+-300001+5/21-5/2ζ32ζ5332ζ52ζ32ζ5432ζ5ζ3ζ533ζ52ζ3ζ543ζ53ζ533ζ5232ζ5432ζ53ζ543ζ532ζ5332ζ52    complex faithful
ρ222200-1+-3-1--3-1-5/2-1+5/2-1--3-1+-30000-1+5/2-1-5/2ζ3ζ543ζ5ζ3ζ533ζ52ζ32ζ5432ζ5ζ32ζ5332ζ52ζ32ζ5432ζ5ζ3ζ533ζ52ζ32ζ5332ζ52ζ3ζ543ζ5    complex lifted from C3×D5
ρ232-200-1+-3-1--3-1+5/2-1-5/21+-31--300001+5/21-5/2ζ3ζ533ζ52ζ3ζ543ζ5ζ32ζ5332ζ52ζ32ζ5432ζ532ζ5332ζ523ζ543ζ532ζ5432ζ53ζ533ζ52    complex faithful
ρ242-200-1+-3-1--3-1-5/2-1+5/21+-31--300001-5/21+5/2ζ3ζ543ζ5ζ3ζ533ζ52ζ32ζ5432ζ5ζ32ζ5332ζ5232ζ5432ζ53ζ533ζ5232ζ5332ζ523ζ543ζ5    complex faithful

Permutation representations of C6×D5
On 30 points - transitive group 30T5
Generators in S30
(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 26 27 28 29 30)
(1 22 25 8 14)(2 23 26 9 15)(3 24 27 10 16)(4 19 28 11 17)(5 20 29 12 18)(6 21 30 7 13)
(1 14)(2 15)(3 16)(4 17)(5 18)(6 13)(7 21)(8 22)(9 23)(10 24)(11 19)(12 20)

G:=sub<Sym(30)| (1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18)(19,20,21,22,23,24)(25,26,27,28,29,30), (1,22,25,8,14)(2,23,26,9,15)(3,24,27,10,16)(4,19,28,11,17)(5,20,29,12,18)(6,21,30,7,13), (1,14)(2,15)(3,16)(4,17)(5,18)(6,13)(7,21)(8,22)(9,23)(10,24)(11,19)(12,20)>;

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)(25,26,27,28,29,30), (1,22,25,8,14)(2,23,26,9,15)(3,24,27,10,16)(4,19,28,11,17)(5,20,29,12,18)(6,21,30,7,13), (1,14)(2,15)(3,16)(4,17)(5,18)(6,13)(7,21)(8,22)(9,23)(10,24)(11,19)(12,20) );

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),(25,26,27,28,29,30)], [(1,22,25,8,14),(2,23,26,9,15),(3,24,27,10,16),(4,19,28,11,17),(5,20,29,12,18),(6,21,30,7,13)], [(1,14),(2,15),(3,16),(4,17),(5,18),(6,13),(7,21),(8,22),(9,23),(10,24),(11,19),(12,20)])

G:=TransitiveGroup(30,5);

Matrix representation of C6×D5 in GL2(𝔽19) generated by

120
012
,
1513
918
,
10
918
G:=sub<GL(2,GF(19))| [12,0,0,12],[15,9,13,18],[1,9,0,18] >;

C6×D5 in GAP, Magma, Sage, TeX

C_6\times D_5
% in TeX

G:=Group("C6xD5");
// GroupNames label

G:=SmallGroup(60,10);
// by ID

G=gap.SmallGroup(60,10);
# by ID

G:=PCGroup([4,-2,-2,-3,-5,771]);
// Polycyclic

G:=Group<a,b,c|a^6=b^5=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

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