direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C6×D5, C10⋊C6, C30⋊2C2, C15⋊3C22, C5⋊(C2×C6), SmallGroup(60,10)
Series: Derived ►Chief ►Lower central ►Upper central
| C5 — C6×D5 |
Generators and relations for C6×D5
G = < a,b,c | a6=b5=c2=1, ab=ba, ac=ca, cbc=b-1 >
Character table of C6×D5
| class | 1 | 2A | 2B | 2C | 3A | 3B | 5A | 5B | 6A | 6B | 6C | 6D | 6E | 6F | 10A | 10B | 15A | 15B | 15C | 15D | 30A | 30B | 30C | 30D | |
| size | 1 | 1 | 5 | 5 | 1 | 1 | 2 | 2 | 1 | 1 | 5 | 5 | 5 | 5 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | linear of order 2 |
| ρ3 | 1 | -1 | 1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | -1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | linear of order 2 |
| ρ4 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 2 |
| ρ5 | 1 | -1 | 1 | -1 | ζ32 | ζ3 | 1 | 1 | ζ65 | ζ6 | ζ32 | ζ65 | ζ6 | ζ3 | -1 | -1 | ζ32 | ζ32 | ζ3 | ζ3 | ζ65 | ζ6 | ζ65 | ζ6 | linear of order 6 |
| ρ6 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | 1 | 1 | ζ3 | ζ32 | ζ32 | ζ3 | ζ32 | ζ3 | 1 | 1 | ζ32 | ζ32 | ζ3 | ζ3 | ζ3 | ζ32 | ζ3 | ζ32 | linear of order 3 |
| ρ7 | 1 | -1 | -1 | 1 | ζ3 | ζ32 | 1 | 1 | ζ6 | ζ65 | ζ65 | ζ32 | ζ3 | ζ6 | -1 | -1 | ζ3 | ζ3 | ζ32 | ζ32 | ζ6 | ζ65 | ζ6 | ζ65 | linear of order 6 |
| ρ8 | 1 | -1 | 1 | -1 | ζ3 | ζ32 | 1 | 1 | ζ6 | ζ65 | ζ3 | ζ6 | ζ65 | ζ32 | -1 | -1 | ζ3 | ζ3 | ζ32 | ζ32 | ζ6 | ζ65 | ζ6 | ζ65 | linear of order 6 |
| ρ9 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | 1 | 1 | ζ32 | ζ3 | ζ3 | ζ32 | ζ3 | ζ32 | 1 | 1 | ζ3 | ζ3 | ζ32 | ζ32 | ζ32 | ζ3 | ζ32 | ζ3 | linear of order 3 |
| ρ10 | 1 | -1 | -1 | 1 | ζ32 | ζ3 | 1 | 1 | ζ65 | ζ6 | ζ6 | ζ3 | ζ32 | ζ65 | -1 | -1 | ζ32 | ζ32 | ζ3 | ζ3 | ζ65 | ζ6 | ζ65 | ζ6 | linear of order 6 |
| ρ11 | 1 | 1 | -1 | -1 | ζ32 | ζ3 | 1 | 1 | ζ3 | ζ32 | ζ6 | ζ65 | ζ6 | ζ65 | 1 | 1 | ζ32 | ζ32 | ζ3 | ζ3 | ζ3 | ζ32 | ζ3 | ζ32 | linear of order 6 |
| ρ12 | 1 | 1 | -1 | -1 | ζ3 | ζ32 | 1 | 1 | ζ32 | ζ3 | ζ65 | ζ6 | ζ65 | ζ6 | 1 | 1 | ζ3 | ζ3 | ζ32 | ζ32 | ζ32 | ζ3 | ζ32 | ζ3 | linear of order 6 |
| ρ13 | 2 | -2 | 0 | 0 | 2 | 2 | -1-√5/2 | -1+√5/2 | -2 | -2 | 0 | 0 | 0 | 0 | 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 D10 |
| ρ14 | 2 | 2 | 0 | 0 | 2 | 2 | -1+√5/2 | -1-√5/2 | 2 | 2 | 0 | 0 | 0 | 0 | -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 |
| ρ15 | 2 | 2 | 0 | 0 | 2 | 2 | -1-√5/2 | -1+√5/2 | 2 | 2 | 0 | 0 | 0 | 0 | -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 |
| ρ16 | 2 | -2 | 0 | 0 | 2 | 2 | -1+√5/2 | -1-√5/2 | -2 | -2 | 0 | 0 | 0 | 0 | 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 D10 |
| ρ17 | 2 | 2 | 0 | 0 | -1+√-3 | -1-√-3 | -1+√5/2 | -1-√5/2 | -1-√-3 | -1+√-3 | 0 | 0 | 0 | 0 | -1-√5/2 | -1+√5/2 | ζ3ζ53+ζ3ζ52 | ζ3ζ54+ζ3ζ5 | ζ32ζ53+ζ32ζ52 | ζ32ζ54+ζ32ζ5 | ζ32ζ53+ζ32ζ52 | ζ3ζ54+ζ3ζ5 | ζ32ζ54+ζ32ζ5 | ζ3ζ53+ζ3ζ52 | complex lifted from C3×D5 |
| ρ18 | 2 | 2 | 0 | 0 | -1-√-3 | -1+√-3 | -1-√5/2 | -1+√5/2 | -1+√-3 | -1-√-3 | 0 | 0 | 0 | 0 | -1+√5/2 | -1-√5/2 | ζ32ζ54+ζ32ζ5 | ζ32ζ53+ζ32ζ52 | ζ3ζ54+ζ3ζ5 | ζ3ζ53+ζ3ζ52 | ζ3ζ54+ζ3ζ5 | ζ32ζ53+ζ32ζ52 | ζ3ζ53+ζ3ζ52 | ζ32ζ54+ζ32ζ5 | complex lifted from C3×D5 |
| ρ19 | 2 | 2 | 0 | 0 | -1-√-3 | -1+√-3 | -1+√5/2 | -1-√5/2 | -1+√-3 | -1-√-3 | 0 | 0 | 0 | 0 | -1-√5/2 | -1+√5/2 | ζ32ζ53+ζ32ζ52 | ζ32ζ54+ζ32ζ5 | ζ3ζ53+ζ3ζ52 | ζ3ζ54+ζ3ζ5 | ζ3ζ53+ζ3ζ52 | ζ32ζ54+ζ32ζ5 | ζ3ζ54+ζ3ζ5 | ζ32ζ53+ζ32ζ52 | complex lifted from C3×D5 |
| ρ20 | 2 | -2 | 0 | 0 | -1-√-3 | -1+√-3 | -1-√5/2 | -1+√5/2 | 1-√-3 | 1+√-3 | 0 | 0 | 0 | 0 | 1-√5/2 | 1+√5/2 | ζ32ζ54+ζ32ζ5 | ζ32ζ53+ζ32ζ52 | ζ3ζ54+ζ3ζ5 | ζ3ζ53+ζ3ζ52 | -ζ3ζ54-ζ3ζ5 | -ζ32ζ53-ζ32ζ52 | -ζ3ζ53-ζ3ζ52 | -ζ32ζ54-ζ32ζ5 | complex faithful |
| ρ21 | 2 | -2 | 0 | 0 | -1-√-3 | -1+√-3 | -1+√5/2 | -1-√5/2 | 1-√-3 | 1+√-3 | 0 | 0 | 0 | 0 | 1+√5/2 | 1-√5/2 | ζ32ζ53+ζ32ζ52 | ζ32ζ54+ζ32ζ5 | ζ3ζ53+ζ3ζ52 | ζ3ζ54+ζ3ζ5 | -ζ3ζ53-ζ3ζ52 | -ζ32ζ54-ζ32ζ5 | -ζ3ζ54-ζ3ζ5 | -ζ32ζ53-ζ32ζ52 | complex faithful |
| ρ22 | 2 | 2 | 0 | 0 | -1+√-3 | -1-√-3 | -1-√5/2 | -1+√5/2 | -1-√-3 | -1+√-3 | 0 | 0 | 0 | 0 | -1+√5/2 | -1-√5/2 | ζ3ζ54+ζ3ζ5 | ζ3ζ53+ζ3ζ52 | ζ32ζ54+ζ32ζ5 | ζ32ζ53+ζ32ζ52 | ζ32ζ54+ζ32ζ5 | ζ3ζ53+ζ3ζ52 | ζ32ζ53+ζ32ζ52 | ζ3ζ54+ζ3ζ5 | complex lifted from C3×D5 |
| ρ23 | 2 | -2 | 0 | 0 | -1+√-3 | -1-√-3 | -1+√5/2 | -1-√5/2 | 1+√-3 | 1-√-3 | 0 | 0 | 0 | 0 | 1+√5/2 | 1-√5/2 | ζ3ζ53+ζ3ζ52 | ζ3ζ54+ζ3ζ5 | ζ32ζ53+ζ32ζ52 | ζ32ζ54+ζ32ζ5 | -ζ32ζ53-ζ32ζ52 | -ζ3ζ54-ζ3ζ5 | -ζ32ζ54-ζ32ζ5 | -ζ3ζ53-ζ3ζ52 | complex faithful |
| ρ24 | 2 | -2 | 0 | 0 | -1+√-3 | -1-√-3 | -1-√5/2 | -1+√5/2 | 1+√-3 | 1-√-3 | 0 | 0 | 0 | 0 | 1-√5/2 | 1+√5/2 | ζ3ζ54+ζ3ζ5 | ζ3ζ53+ζ3ζ52 | ζ32ζ54+ζ32ζ5 | ζ32ζ53+ζ32ζ52 | -ζ32ζ54-ζ32ζ5 | -ζ3ζ53-ζ3ζ52 | -ζ32ζ53-ζ32ζ52 | -ζ3ζ54-ζ3ζ5 | complex faithful |
►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 10 29 14)(2 23 11 30 15)(3 24 12 25 16)(4 19 7 26 17)(5 20 8 27 18)(6 21 9 28 13)
(1 14)(2 15)(3 16)(4 17)(5 18)(6 13)(19 26)(20 27)(21 28)(22 29)(23 30)(24 25)
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), (1,22,10,29,14)(2,23,11,30,15)(3,24,12,25,16)(4,19,7,26,17)(5,20,8,27,18)(6,21,9,28,13), (1,14)(2,15)(3,16)(4,17)(5,18)(6,13)(19,26)(20,27)(21,28)(22,29)(23,30)(24,25)>;
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), (1,22,10,29,14)(2,23,11,30,15)(3,24,12,25,16)(4,19,7,26,17)(5,20,8,27,18)(6,21,9,28,13), (1,14)(2,15)(3,16)(4,17)(5,18)(6,13)(19,26)(20,27)(21,28)(22,29)(23,30)(24,25) );
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)], [(1,22,10,29,14),(2,23,11,30,15),(3,24,12,25,16),(4,19,7,26,17),(5,20,8,27,18),(6,21,9,28,13)], [(1,14),(2,15),(3,16),(4,17),(5,18),(6,13),(19,26),(20,27),(21,28),(22,29),(23,30),(24,25)]])
G:=TransitiveGroup(30,5);
C6×D5 is a maximal subgroup of
C15⋊D4 C3⋊D20
Matrix representation of C6×D5 ►in GL2(𝔽19) generated by
| 12 | 0 |
| 0 | 12 |
| 15 | 13 |
| 9 | 18 |
| 1 | 0 |
| 9 | 18 |
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
Export
Subgroup lattice of C6×D5 in TeX
Character table of C6×D5 in TeX