direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C6×D7, C42⋊2C2, C14⋊3C6, C21⋊3C22, C7⋊3(C2×C6), SmallGroup(84,12)
Series: Derived ►Chief ►Lower central ►Upper central
| C7 — C6×D7 |
Generators and relations for C6×D7
G = < a,b,c | a6=b7=c2=1, ab=ba, ac=ca, cbc=b-1 >
Character table of C6×D7
| class | 1 | 2A | 2B | 2C | 3A | 3B | 6A | 6B | 6C | 6D | 6E | 6F | 7A | 7B | 7C | 14A | 14B | 14C | 21A | 21B | 21C | 21D | 21E | 21F | 42A | 42B | 42C | 42D | 42E | 42F | |
| size | 1 | 1 | 7 | 7 | 1 | 1 | 1 | 1 | 7 | 7 | 7 | 7 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 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 | 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 | -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 | -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 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 2 |
| ρ5 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | ζ3 | ζ32 | ζ32 | ζ3 | ζ3 | ζ32 | 1 | 1 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | ζ3 | ζ32 | ζ32 | ζ3 | ζ3 | ζ3 | ζ3 | ζ32 | ζ32 | ζ32 | linear of order 3 |
| ρ6 | 1 | -1 | -1 | 1 | ζ32 | ζ3 | ζ65 | ζ6 | ζ6 | ζ3 | ζ65 | ζ32 | 1 | 1 | 1 | -1 | -1 | -1 | ζ32 | ζ3 | ζ3 | ζ32 | ζ32 | ζ3 | ζ65 | ζ65 | ζ65 | ζ6 | ζ6 | ζ6 | linear of order 6 |
| ρ7 | 1 | 1 | -1 | -1 | ζ32 | ζ3 | ζ3 | ζ32 | ζ6 | ζ65 | ζ65 | ζ6 | 1 | 1 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | ζ3 | ζ32 | ζ32 | ζ3 | ζ3 | ζ3 | ζ3 | ζ32 | ζ32 | ζ32 | linear of order 6 |
| ρ8 | 1 | -1 | 1 | -1 | ζ32 | ζ3 | ζ65 | ζ6 | ζ32 | ζ65 | ζ3 | ζ6 | 1 | 1 | 1 | -1 | -1 | -1 | ζ32 | ζ3 | ζ3 | ζ32 | ζ32 | ζ3 | ζ65 | ζ65 | ζ65 | ζ6 | ζ6 | ζ6 | linear of order 6 |
| ρ9 | 1 | -1 | 1 | -1 | ζ3 | ζ32 | ζ6 | ζ65 | ζ3 | ζ6 | ζ32 | ζ65 | 1 | 1 | 1 | -1 | -1 | -1 | ζ3 | ζ32 | ζ32 | ζ3 | ζ3 | ζ32 | ζ6 | ζ6 | ζ6 | ζ65 | ζ65 | ζ65 | linear of order 6 |
| ρ10 | 1 | 1 | -1 | -1 | ζ3 | ζ32 | ζ32 | ζ3 | ζ65 | ζ6 | ζ6 | ζ65 | 1 | 1 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | ζ32 | ζ3 | ζ3 | ζ32 | ζ32 | ζ32 | ζ32 | ζ3 | ζ3 | ζ3 | linear of order 6 |
| ρ11 | 1 | -1 | -1 | 1 | ζ3 | ζ32 | ζ6 | ζ65 | ζ65 | ζ32 | ζ6 | ζ3 | 1 | 1 | 1 | -1 | -1 | -1 | ζ3 | ζ32 | ζ32 | ζ3 | ζ3 | ζ32 | ζ6 | ζ6 | ζ6 | ζ65 | ζ65 | ζ65 | linear of order 6 |
| ρ12 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | ζ32 | ζ3 | ζ3 | ζ32 | ζ32 | ζ3 | 1 | 1 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | ζ32 | ζ3 | ζ3 | ζ32 | ζ32 | ζ32 | ζ32 | ζ3 | ζ3 | ζ3 | linear of order 3 |
| ρ13 | 2 | -2 | 0 | 0 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | ζ76+ζ7 | ζ75+ζ72 | ζ74+ζ73 | -ζ76-ζ7 | -ζ75-ζ72 | -ζ74-ζ73 | ζ76+ζ7 | ζ76+ζ7 | ζ75+ζ72 | ζ75+ζ72 | ζ74+ζ73 | ζ74+ζ73 | -ζ74-ζ73 | -ζ76-ζ7 | -ζ75-ζ72 | -ζ75-ζ72 | -ζ74-ζ73 | -ζ76-ζ7 | orthogonal lifted from D14 |
| ρ14 | 2 | -2 | 0 | 0 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | ζ74+ζ73 | ζ76+ζ7 | ζ75+ζ72 | -ζ74-ζ73 | -ζ76-ζ7 | -ζ75-ζ72 | ζ74+ζ73 | ζ74+ζ73 | ζ76+ζ7 | ζ76+ζ7 | ζ75+ζ72 | ζ75+ζ72 | -ζ75-ζ72 | -ζ74-ζ73 | -ζ76-ζ7 | -ζ76-ζ7 | -ζ75-ζ72 | -ζ74-ζ73 | orthogonal lifted from D14 |
| ρ15 | 2 | 2 | 0 | 0 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | ζ74+ζ73 | ζ76+ζ7 | ζ75+ζ72 | ζ74+ζ73 | ζ76+ζ7 | ζ75+ζ72 | ζ74+ζ73 | ζ74+ζ73 | ζ76+ζ7 | ζ76+ζ7 | ζ75+ζ72 | ζ75+ζ72 | ζ75+ζ72 | ζ74+ζ73 | ζ76+ζ7 | ζ76+ζ7 | ζ75+ζ72 | ζ74+ζ73 | orthogonal lifted from D7 |
| ρ16 | 2 | -2 | 0 | 0 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | ζ75+ζ72 | ζ74+ζ73 | ζ76+ζ7 | -ζ75-ζ72 | -ζ74-ζ73 | -ζ76-ζ7 | ζ75+ζ72 | ζ75+ζ72 | ζ74+ζ73 | ζ74+ζ73 | ζ76+ζ7 | ζ76+ζ7 | -ζ76-ζ7 | -ζ75-ζ72 | -ζ74-ζ73 | -ζ74-ζ73 | -ζ76-ζ7 | -ζ75-ζ72 | orthogonal lifted from D14 |
| ρ17 | 2 | 2 | 0 | 0 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | ζ75+ζ72 | ζ74+ζ73 | ζ76+ζ7 | ζ75+ζ72 | ζ74+ζ73 | ζ76+ζ7 | ζ75+ζ72 | ζ75+ζ72 | ζ74+ζ73 | ζ74+ζ73 | ζ76+ζ7 | ζ76+ζ7 | ζ76+ζ7 | ζ75+ζ72 | ζ74+ζ73 | ζ74+ζ73 | ζ76+ζ7 | ζ75+ζ72 | orthogonal lifted from D7 |
| ρ18 | 2 | 2 | 0 | 0 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | ζ76+ζ7 | ζ75+ζ72 | ζ74+ζ73 | ζ76+ζ7 | ζ75+ζ72 | ζ74+ζ73 | ζ76+ζ7 | ζ76+ζ7 | ζ75+ζ72 | ζ75+ζ72 | ζ74+ζ73 | ζ74+ζ73 | ζ74+ζ73 | ζ76+ζ7 | ζ75+ζ72 | ζ75+ζ72 | ζ74+ζ73 | ζ76+ζ7 | orthogonal lifted from D7 |
| ρ19 | 2 | 2 | 0 | 0 | -1-√-3 | -1+√-3 | -1+√-3 | -1-√-3 | 0 | 0 | 0 | 0 | ζ75+ζ72 | ζ74+ζ73 | ζ76+ζ7 | ζ75+ζ72 | ζ74+ζ73 | ζ76+ζ7 | ζ32ζ75+ζ32ζ72 | ζ3ζ75+ζ3ζ72 | ζ3ζ74+ζ3ζ73 | ζ32ζ74+ζ32ζ73 | ζ32ζ76+ζ32ζ7 | ζ3ζ76+ζ3ζ7 | ζ3ζ76+ζ3ζ7 | ζ3ζ75+ζ3ζ72 | ζ3ζ74+ζ3ζ73 | ζ32ζ74+ζ32ζ73 | ζ32ζ76+ζ32ζ7 | ζ32ζ75+ζ32ζ72 | complex lifted from C3×D7 |
| ρ20 | 2 | -2 | 0 | 0 | -1-√-3 | -1+√-3 | 1-√-3 | 1+√-3 | 0 | 0 | 0 | 0 | ζ75+ζ72 | ζ74+ζ73 | ζ76+ζ7 | -ζ75-ζ72 | -ζ74-ζ73 | -ζ76-ζ7 | ζ32ζ75+ζ32ζ72 | ζ3ζ75+ζ3ζ72 | ζ3ζ74+ζ3ζ73 | ζ32ζ74+ζ32ζ73 | ζ32ζ76+ζ32ζ7 | ζ3ζ76+ζ3ζ7 | -ζ3ζ76-ζ3ζ7 | -ζ3ζ75-ζ3ζ72 | -ζ3ζ74-ζ3ζ73 | -ζ32ζ74-ζ32ζ73 | -ζ32ζ76-ζ32ζ7 | -ζ32ζ75-ζ32ζ72 | complex faithful |
| ρ21 | 2 | -2 | 0 | 0 | -1-√-3 | -1+√-3 | 1-√-3 | 1+√-3 | 0 | 0 | 0 | 0 | ζ74+ζ73 | ζ76+ζ7 | ζ75+ζ72 | -ζ74-ζ73 | -ζ76-ζ7 | -ζ75-ζ72 | ζ32ζ74+ζ32ζ73 | ζ3ζ74+ζ3ζ73 | ζ3ζ76+ζ3ζ7 | ζ32ζ76+ζ32ζ7 | ζ32ζ75+ζ32ζ72 | ζ3ζ75+ζ3ζ72 | -ζ3ζ75-ζ3ζ72 | -ζ3ζ74-ζ3ζ73 | -ζ3ζ76-ζ3ζ7 | -ζ32ζ76-ζ32ζ7 | -ζ32ζ75-ζ32ζ72 | -ζ32ζ74-ζ32ζ73 | complex faithful |
| ρ22 | 2 | 2 | 0 | 0 | -1+√-3 | -1-√-3 | -1-√-3 | -1+√-3 | 0 | 0 | 0 | 0 | ζ74+ζ73 | ζ76+ζ7 | ζ75+ζ72 | ζ74+ζ73 | ζ76+ζ7 | ζ75+ζ72 | ζ3ζ74+ζ3ζ73 | ζ32ζ74+ζ32ζ73 | ζ32ζ76+ζ32ζ7 | ζ3ζ76+ζ3ζ7 | ζ3ζ75+ζ3ζ72 | ζ32ζ75+ζ32ζ72 | ζ32ζ75+ζ32ζ72 | ζ32ζ74+ζ32ζ73 | ζ32ζ76+ζ32ζ7 | ζ3ζ76+ζ3ζ7 | ζ3ζ75+ζ3ζ72 | ζ3ζ74+ζ3ζ73 | complex lifted from C3×D7 |
| ρ23 | 2 | 2 | 0 | 0 | -1-√-3 | -1+√-3 | -1+√-3 | -1-√-3 | 0 | 0 | 0 | 0 | ζ74+ζ73 | ζ76+ζ7 | ζ75+ζ72 | ζ74+ζ73 | ζ76+ζ7 | ζ75+ζ72 | ζ32ζ74+ζ32ζ73 | ζ3ζ74+ζ3ζ73 | ζ3ζ76+ζ3ζ7 | ζ32ζ76+ζ32ζ7 | ζ32ζ75+ζ32ζ72 | ζ3ζ75+ζ3ζ72 | ζ3ζ75+ζ3ζ72 | ζ3ζ74+ζ3ζ73 | ζ3ζ76+ζ3ζ7 | ζ32ζ76+ζ32ζ7 | ζ32ζ75+ζ32ζ72 | ζ32ζ74+ζ32ζ73 | complex lifted from C3×D7 |
| ρ24 | 2 | 2 | 0 | 0 | -1+√-3 | -1-√-3 | -1-√-3 | -1+√-3 | 0 | 0 | 0 | 0 | ζ76+ζ7 | ζ75+ζ72 | ζ74+ζ73 | ζ76+ζ7 | ζ75+ζ72 | ζ74+ζ73 | ζ3ζ76+ζ3ζ7 | ζ32ζ76+ζ32ζ7 | ζ32ζ75+ζ32ζ72 | ζ3ζ75+ζ3ζ72 | ζ3ζ74+ζ3ζ73 | ζ32ζ74+ζ32ζ73 | ζ32ζ74+ζ32ζ73 | ζ32ζ76+ζ32ζ7 | ζ32ζ75+ζ32ζ72 | ζ3ζ75+ζ3ζ72 | ζ3ζ74+ζ3ζ73 | ζ3ζ76+ζ3ζ7 | complex lifted from C3×D7 |
| ρ25 | 2 | -2 | 0 | 0 | -1+√-3 | -1-√-3 | 1+√-3 | 1-√-3 | 0 | 0 | 0 | 0 | ζ74+ζ73 | ζ76+ζ7 | ζ75+ζ72 | -ζ74-ζ73 | -ζ76-ζ7 | -ζ75-ζ72 | ζ3ζ74+ζ3ζ73 | ζ32ζ74+ζ32ζ73 | ζ32ζ76+ζ32ζ7 | ζ3ζ76+ζ3ζ7 | ζ3ζ75+ζ3ζ72 | ζ32ζ75+ζ32ζ72 | -ζ32ζ75-ζ32ζ72 | -ζ32ζ74-ζ32ζ73 | -ζ32ζ76-ζ32ζ7 | -ζ3ζ76-ζ3ζ7 | -ζ3ζ75-ζ3ζ72 | -ζ3ζ74-ζ3ζ73 | complex faithful |
| ρ26 | 2 | -2 | 0 | 0 | -1-√-3 | -1+√-3 | 1-√-3 | 1+√-3 | 0 | 0 | 0 | 0 | ζ76+ζ7 | ζ75+ζ72 | ζ74+ζ73 | -ζ76-ζ7 | -ζ75-ζ72 | -ζ74-ζ73 | ζ32ζ76+ζ32ζ7 | ζ3ζ76+ζ3ζ7 | ζ3ζ75+ζ3ζ72 | ζ32ζ75+ζ32ζ72 | ζ32ζ74+ζ32ζ73 | ζ3ζ74+ζ3ζ73 | -ζ3ζ74-ζ3ζ73 | -ζ3ζ76-ζ3ζ7 | -ζ3ζ75-ζ3ζ72 | -ζ32ζ75-ζ32ζ72 | -ζ32ζ74-ζ32ζ73 | -ζ32ζ76-ζ32ζ7 | complex faithful |
| ρ27 | 2 | -2 | 0 | 0 | -1+√-3 | -1-√-3 | 1+√-3 | 1-√-3 | 0 | 0 | 0 | 0 | ζ76+ζ7 | ζ75+ζ72 | ζ74+ζ73 | -ζ76-ζ7 | -ζ75-ζ72 | -ζ74-ζ73 | ζ3ζ76+ζ3ζ7 | ζ32ζ76+ζ32ζ7 | ζ32ζ75+ζ32ζ72 | ζ3ζ75+ζ3ζ72 | ζ3ζ74+ζ3ζ73 | ζ32ζ74+ζ32ζ73 | -ζ32ζ74-ζ32ζ73 | -ζ32ζ76-ζ32ζ7 | -ζ32ζ75-ζ32ζ72 | -ζ3ζ75-ζ3ζ72 | -ζ3ζ74-ζ3ζ73 | -ζ3ζ76-ζ3ζ7 | complex faithful |
| ρ28 | 2 | 2 | 0 | 0 | -1-√-3 | -1+√-3 | -1+√-3 | -1-√-3 | 0 | 0 | 0 | 0 | ζ76+ζ7 | ζ75+ζ72 | ζ74+ζ73 | ζ76+ζ7 | ζ75+ζ72 | ζ74+ζ73 | ζ32ζ76+ζ32ζ7 | ζ3ζ76+ζ3ζ7 | ζ3ζ75+ζ3ζ72 | ζ32ζ75+ζ32ζ72 | ζ32ζ74+ζ32ζ73 | ζ3ζ74+ζ3ζ73 | ζ3ζ74+ζ3ζ73 | ζ3ζ76+ζ3ζ7 | ζ3ζ75+ζ3ζ72 | ζ32ζ75+ζ32ζ72 | ζ32ζ74+ζ32ζ73 | ζ32ζ76+ζ32ζ7 | complex lifted from C3×D7 |
| ρ29 | 2 | 2 | 0 | 0 | -1+√-3 | -1-√-3 | -1-√-3 | -1+√-3 | 0 | 0 | 0 | 0 | ζ75+ζ72 | ζ74+ζ73 | ζ76+ζ7 | ζ75+ζ72 | ζ74+ζ73 | ζ76+ζ7 | ζ3ζ75+ζ3ζ72 | ζ32ζ75+ζ32ζ72 | ζ32ζ74+ζ32ζ73 | ζ3ζ74+ζ3ζ73 | ζ3ζ76+ζ3ζ7 | ζ32ζ76+ζ32ζ7 | ζ32ζ76+ζ32ζ7 | ζ32ζ75+ζ32ζ72 | ζ32ζ74+ζ32ζ73 | ζ3ζ74+ζ3ζ73 | ζ3ζ76+ζ3ζ7 | ζ3ζ75+ζ3ζ72 | complex lifted from C3×D7 |
| ρ30 | 2 | -2 | 0 | 0 | -1+√-3 | -1-√-3 | 1+√-3 | 1-√-3 | 0 | 0 | 0 | 0 | ζ75+ζ72 | ζ74+ζ73 | ζ76+ζ7 | -ζ75-ζ72 | -ζ74-ζ73 | -ζ76-ζ7 | ζ3ζ75+ζ3ζ72 | ζ32ζ75+ζ32ζ72 | ζ32ζ74+ζ32ζ73 | ζ3ζ74+ζ3ζ73 | ζ3ζ76+ζ3ζ7 | ζ32ζ76+ζ32ζ7 | -ζ32ζ76-ζ32ζ7 | -ζ32ζ75-ζ32ζ72 | -ζ32ζ74-ζ32ζ73 | -ζ3ζ74-ζ3ζ73 | -ζ3ζ76-ζ3ζ7 | -ζ3ζ75-ζ3ζ72 | complex faithful |
►On 42 points
Generators in S42
(1 34 20 27 13 41)(2 35 21 28 14 42)(3 29 15 22 8 36)(4 30 16 23 9 37)(5 31 17 24 10 38)(6 32 18 25 11 39)(7 33 19 26 12 40)
(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 31 32 33 34 35)(36 37 38 39 40 41 42)
(1 26)(2 25)(3 24)(4 23)(5 22)(6 28)(7 27)(8 31)(9 30)(10 29)(11 35)(12 34)(13 33)(14 32)(15 38)(16 37)(17 36)(18 42)(19 41)(20 40)(21 39)
G:=sub<Sym(42)| (1,34,20,27,13,41)(2,35,21,28,14,42)(3,29,15,22,8,36)(4,30,16,23,9,37)(5,31,17,24,10,38)(6,32,18,25,11,39)(7,33,19,26,12,40), (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)(29,30,31,32,33,34,35)(36,37,38,39,40,41,42), (1,26)(2,25)(3,24)(4,23)(5,22)(6,28)(7,27)(8,31)(9,30)(10,29)(11,35)(12,34)(13,33)(14,32)(15,38)(16,37)(17,36)(18,42)(19,41)(20,40)(21,39)>;
G:=Group( (1,34,20,27,13,41)(2,35,21,28,14,42)(3,29,15,22,8,36)(4,30,16,23,9,37)(5,31,17,24,10,38)(6,32,18,25,11,39)(7,33,19,26,12,40), (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)(29,30,31,32,33,34,35)(36,37,38,39,40,41,42), (1,26)(2,25)(3,24)(4,23)(5,22)(6,28)(7,27)(8,31)(9,30)(10,29)(11,35)(12,34)(13,33)(14,32)(15,38)(16,37)(17,36)(18,42)(19,41)(20,40)(21,39) );
G=PermutationGroup([[(1,34,20,27,13,41),(2,35,21,28,14,42),(3,29,15,22,8,36),(4,30,16,23,9,37),(5,31,17,24,10,38),(6,32,18,25,11,39),(7,33,19,26,12,40)], [(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),(29,30,31,32,33,34,35),(36,37,38,39,40,41,42)], [(1,26),(2,25),(3,24),(4,23),(5,22),(6,28),(7,27),(8,31),(9,30),(10,29),(11,35),(12,34),(13,33),(14,32),(15,38),(16,37),(17,36),(18,42),(19,41),(20,40),(21,39)]])
C6×D7 is a maximal subgroup of
C21⋊D4 C3⋊D28
Matrix representation of C6×D7 ►in GL2(𝔽13) generated by
| 10 | 0 |
| 0 | 10 |
| 5 | 8 |
| 6 | 2 |
| 2 | 7 |
| 7 | 11 |
G:=sub<GL(2,GF(13))| [10,0,0,10],[5,6,8,2],[2,7,7,11] >;
C6×D7 in GAP, Magma, Sage, TeX
C_6\times D_7
% in TeX
G:=Group("C6xD7"); // GroupNames label
G:=SmallGroup(84,12);
// by ID
G=gap.SmallGroup(84,12);
# by ID
G:=PCGroup([4,-2,-2,-3,-7,1155]);
// Polycyclic
G:=Group<a,b,c|a^6=b^7=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export
Subgroup lattice of C6×D7 in TeX
Character table of C6×D7 in TeX