direct product, metacyclic, supersoluble, monomial, A-group
Aliases: D9×C7⋊C3, C63⋊8C6, C7⋊2(C3×D9), (C7×D9)⋊4C3, C21.8(C3×S3), (C9×C7⋊C3)⋊2C2, C9⋊5(C2×C7⋊C3), C3.3(S3×C7⋊C3), (C3×C7⋊C3).6S3, SmallGroup(378,15)
Series: Derived ►Chief ►Lower central ►Upper central
| C63 — D9×C7⋊C3 |
Generators and relations for D9×C7⋊C3
G = < a,b,c,d | a9=b2=c7=d3=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c4 >
Character table of D9×C7⋊C3
| class | 1 | 2 | 3A | 3B | 3C | 3D | 3E | 6A | 6B | 7A | 7B | 9A | 9B | 9C | 9D | 9E | 9F | 9G | 9H | 9I | 14A | 14B | 21A | 21B | 63A | 63B | 63C | 63D | 63E | 63F | |
| size | 1 | 9 | 2 | 7 | 7 | 14 | 14 | 63 | 63 | 3 | 3 | 2 | 2 | 2 | 14 | 14 | 14 | 14 | 14 | 14 | 27 | 27 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | |
| 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 | ζ32 | ζ3 | ζ3 | ζ32 | ζ65 | ζ6 | 1 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | ζ3 | ζ32 | ζ32 | ζ3 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 6 |
| ρ4 | 1 | 1 | 1 | ζ3 | ζ32 | ζ32 | ζ3 | ζ32 | ζ3 | 1 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | ζ32 | ζ3 | ζ3 | ζ32 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 3 |
| ρ5 | 1 | 1 | 1 | ζ32 | ζ3 | ζ3 | ζ32 | ζ3 | ζ32 | 1 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | ζ3 | ζ32 | ζ32 | ζ3 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 3 |
| ρ6 | 1 | -1 | 1 | ζ3 | ζ32 | ζ32 | ζ3 | ζ6 | ζ65 | 1 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | ζ32 | ζ3 | ζ3 | ζ32 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 6 |
| ρ7 | 2 | 0 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | 2 | 2 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | 0 | 0 | 2 | 2 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ8 | 2 | 0 | -1 | 2 | 2 | -1 | -1 | 0 | 0 | 2 | 2 | ζ98+ζ9 | ζ95+ζ94 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | ζ97+ζ92 | ζ98+ζ9 | ζ95+ζ94 | 0 | 0 | -1 | -1 | ζ98+ζ9 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | ζ97+ζ92 | ζ95+ζ94 | orthogonal lifted from D9 |
| ρ9 | 2 | 0 | -1 | 2 | 2 | -1 | -1 | 0 | 0 | 2 | 2 | ζ95+ζ94 | ζ97+ζ92 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | ζ98+ζ9 | ζ95+ζ94 | ζ97+ζ92 | 0 | 0 | -1 | -1 | ζ95+ζ94 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | ζ98+ζ9 | ζ97+ζ92 | orthogonal lifted from D9 |
| ρ10 | 2 | 0 | -1 | 2 | 2 | -1 | -1 | 0 | 0 | 2 | 2 | ζ97+ζ92 | ζ98+ζ9 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | ζ95+ζ94 | ζ97+ζ92 | ζ98+ζ9 | 0 | 0 | -1 | -1 | ζ97+ζ92 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | ζ95+ζ94 | ζ98+ζ9 | orthogonal lifted from D9 |
| ρ11 | 2 | 0 | 2 | -1+√-3 | -1-√-3 | -1-√-3 | -1+√-3 | 0 | 0 | 2 | 2 | -1 | -1 | -1 | ζ65 | ζ6 | ζ6 | ζ65 | ζ65 | ζ6 | 0 | 0 | 2 | 2 | -1 | -1 | -1 | -1 | -1 | -1 | complex lifted from C3×S3 |
| ρ12 | 2 | 0 | -1 | -1+√-3 | -1-√-3 | ζ6 | ζ65 | 0 | 0 | 2 | 2 | ζ97+ζ92 | ζ98+ζ9 | ζ95+ζ94 | ζ94+ζ92 | ζ98+ζ94 | ζ92+ζ9 | ζ98+ζ97 | ζ95+ζ9 | ζ97+ζ95 | 0 | 0 | -1 | -1 | ζ97+ζ92 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | ζ95+ζ94 | ζ98+ζ9 | complex lifted from C3×D9 |
| ρ13 | 2 | 0 | 2 | -1-√-3 | -1+√-3 | -1+√-3 | -1-√-3 | 0 | 0 | 2 | 2 | -1 | -1 | -1 | ζ6 | ζ65 | ζ65 | ζ6 | ζ6 | ζ65 | 0 | 0 | 2 | 2 | -1 | -1 | -1 | -1 | -1 | -1 | complex lifted from C3×S3 |
| ρ14 | 2 | 0 | -1 | -1-√-3 | -1+√-3 | ζ65 | ζ6 | 0 | 0 | 2 | 2 | ζ95+ζ94 | ζ97+ζ92 | ζ98+ζ9 | ζ98+ζ94 | ζ98+ζ97 | ζ94+ζ92 | ζ97+ζ95 | ζ92+ζ9 | ζ95+ζ9 | 0 | 0 | -1 | -1 | ζ95+ζ94 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | ζ98+ζ9 | ζ97+ζ92 | complex lifted from C3×D9 |
| ρ15 | 2 | 0 | -1 | -1-√-3 | -1+√-3 | ζ65 | ζ6 | 0 | 0 | 2 | 2 | ζ98+ζ9 | ζ95+ζ94 | ζ97+ζ92 | ζ92+ζ9 | ζ94+ζ92 | ζ95+ζ9 | ζ98+ζ94 | ζ97+ζ95 | ζ98+ζ97 | 0 | 0 | -1 | -1 | ζ98+ζ9 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | ζ97+ζ92 | ζ95+ζ94 | complex lifted from C3×D9 |
| ρ16 | 2 | 0 | -1 | -1+√-3 | -1-√-3 | ζ6 | ζ65 | 0 | 0 | 2 | 2 | ζ95+ζ94 | ζ97+ζ92 | ζ98+ζ9 | ζ95+ζ9 | ζ92+ζ9 | ζ97+ζ95 | ζ94+ζ92 | ζ98+ζ97 | ζ98+ζ94 | 0 | 0 | -1 | -1 | ζ95+ζ94 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | ζ98+ζ9 | ζ97+ζ92 | complex lifted from C3×D9 |
| ρ17 | 2 | 0 | -1 | -1-√-3 | -1+√-3 | ζ65 | ζ6 | 0 | 0 | 2 | 2 | ζ97+ζ92 | ζ98+ζ9 | ζ95+ζ94 | ζ97+ζ95 | ζ95+ζ9 | ζ98+ζ97 | ζ92+ζ9 | ζ98+ζ94 | ζ94+ζ92 | 0 | 0 | -1 | -1 | ζ97+ζ92 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | ζ95+ζ94 | ζ98+ζ9 | complex lifted from C3×D9 |
| ρ18 | 2 | 0 | -1 | -1+√-3 | -1-√-3 | ζ6 | ζ65 | 0 | 0 | 2 | 2 | ζ98+ζ9 | ζ95+ζ94 | ζ97+ζ92 | ζ98+ζ97 | ζ97+ζ95 | ζ98+ζ94 | ζ95+ζ9 | ζ94+ζ92 | ζ92+ζ9 | 0 | 0 | -1 | -1 | ζ98+ζ9 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | ζ97+ζ92 | ζ95+ζ94 | complex lifted from C3×D9 |
| ρ19 | 3 | -3 | 3 | 0 | 0 | 0 | 0 | 0 | 0 | -1+√-7/2 | -1-√-7/2 | 3 | 3 | 3 | 0 | 0 | 0 | 0 | 0 | 0 | 1+√-7/2 | 1-√-7/2 | -1+√-7/2 | -1-√-7/2 | -1+√-7/2 | -1-√-7/2 | -1-√-7/2 | -1+√-7/2 | -1+√-7/2 | -1-√-7/2 | complex lifted from C2×C7⋊C3 |
| ρ20 | 3 | 3 | 3 | 0 | 0 | 0 | 0 | 0 | 0 | -1+√-7/2 | -1-√-7/2 | 3 | 3 | 3 | 0 | 0 | 0 | 0 | 0 | 0 | -1-√-7/2 | -1+√-7/2 | -1+√-7/2 | -1-√-7/2 | -1+√-7/2 | -1-√-7/2 | -1-√-7/2 | -1+√-7/2 | -1+√-7/2 | -1-√-7/2 | complex lifted from C7⋊C3 |
| ρ21 | 3 | 3 | 3 | 0 | 0 | 0 | 0 | 0 | 0 | -1-√-7/2 | -1+√-7/2 | 3 | 3 | 3 | 0 | 0 | 0 | 0 | 0 | 0 | -1+√-7/2 | -1-√-7/2 | -1-√-7/2 | -1+√-7/2 | -1-√-7/2 | -1+√-7/2 | -1+√-7/2 | -1-√-7/2 | -1-√-7/2 | -1+√-7/2 | complex lifted from C7⋊C3 |
| ρ22 | 3 | -3 | 3 | 0 | 0 | 0 | 0 | 0 | 0 | -1-√-7/2 | -1+√-7/2 | 3 | 3 | 3 | 0 | 0 | 0 | 0 | 0 | 0 | 1-√-7/2 | 1+√-7/2 | -1-√-7/2 | -1+√-7/2 | -1-√-7/2 | -1+√-7/2 | -1+√-7/2 | -1-√-7/2 | -1-√-7/2 | -1+√-7/2 | complex lifted from C2×C7⋊C3 |
| ρ23 | 6 | 0 | 6 | 0 | 0 | 0 | 0 | 0 | 0 | -1+√-7 | -1-√-7 | -3 | -3 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1+√-7 | -1-√-7 | 1-√-7/2 | 1+√-7/2 | 1+√-7/2 | 1-√-7/2 | 1-√-7/2 | 1+√-7/2 | complex lifted from S3×C7⋊C3 |
| ρ24 | 6 | 0 | 6 | 0 | 0 | 0 | 0 | 0 | 0 | -1-√-7 | -1+√-7 | -3 | -3 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1-√-7 | -1+√-7 | 1+√-7/2 | 1-√-7/2 | 1-√-7/2 | 1+√-7/2 | 1+√-7/2 | 1-√-7/2 | complex lifted from S3×C7⋊C3 |
| ρ25 | 6 | 0 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | -1+√-7 | -1-√-7 | 3ζ97+3ζ92 | 3ζ98+3ζ9 | 3ζ95+3ζ94 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1-√-7/2 | 1+√-7/2 | ζ97ζ74+ζ97ζ72+ζ97ζ7+ζ92ζ74+ζ92ζ72+ζ92ζ7 | ζ97ζ76+ζ97ζ75+ζ97ζ73+ζ92ζ76+ζ92ζ75+ζ92ζ73 | ζ95ζ76+ζ95ζ75+ζ95ζ73+ζ94ζ76+ζ94ζ75+ζ94ζ73 | ζ98ζ74+ζ98ζ72+ζ98ζ7+ζ9ζ74+ζ9ζ72+ζ9ζ7 | ζ95ζ74+ζ95ζ72+ζ95ζ7+ζ94ζ74+ζ94ζ72+ζ94ζ7 | ζ98ζ76+ζ98ζ75+ζ98ζ73+ζ9ζ76+ζ9ζ75+ζ9ζ73 | complex faithful |
| ρ26 | 6 | 0 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | -1+√-7 | -1-√-7 | 3ζ95+3ζ94 | 3ζ97+3ζ92 | 3ζ98+3ζ9 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1-√-7/2 | 1+√-7/2 | ζ95ζ74+ζ95ζ72+ζ95ζ7+ζ94ζ74+ζ94ζ72+ζ94ζ7 | ζ95ζ76+ζ95ζ75+ζ95ζ73+ζ94ζ76+ζ94ζ75+ζ94ζ73 | ζ98ζ76+ζ98ζ75+ζ98ζ73+ζ9ζ76+ζ9ζ75+ζ9ζ73 | ζ97ζ74+ζ97ζ72+ζ97ζ7+ζ92ζ74+ζ92ζ72+ζ92ζ7 | ζ98ζ74+ζ98ζ72+ζ98ζ7+ζ9ζ74+ζ9ζ72+ζ9ζ7 | ζ97ζ76+ζ97ζ75+ζ97ζ73+ζ92ζ76+ζ92ζ75+ζ92ζ73 | complex faithful |
| ρ27 | 6 | 0 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | -1+√-7 | -1-√-7 | 3ζ98+3ζ9 | 3ζ95+3ζ94 | 3ζ97+3ζ92 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1-√-7/2 | 1+√-7/2 | ζ98ζ74+ζ98ζ72+ζ98ζ7+ζ9ζ74+ζ9ζ72+ζ9ζ7 | ζ98ζ76+ζ98ζ75+ζ98ζ73+ζ9ζ76+ζ9ζ75+ζ9ζ73 | ζ97ζ76+ζ97ζ75+ζ97ζ73+ζ92ζ76+ζ92ζ75+ζ92ζ73 | ζ95ζ74+ζ95ζ72+ζ95ζ7+ζ94ζ74+ζ94ζ72+ζ94ζ7 | ζ97ζ74+ζ97ζ72+ζ97ζ7+ζ92ζ74+ζ92ζ72+ζ92ζ7 | ζ95ζ76+ζ95ζ75+ζ95ζ73+ζ94ζ76+ζ94ζ75+ζ94ζ73 | complex faithful |
| ρ28 | 6 | 0 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | -1-√-7 | -1+√-7 | 3ζ95+3ζ94 | 3ζ97+3ζ92 | 3ζ98+3ζ9 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1+√-7/2 | 1-√-7/2 | ζ95ζ76+ζ95ζ75+ζ95ζ73+ζ94ζ76+ζ94ζ75+ζ94ζ73 | ζ95ζ74+ζ95ζ72+ζ95ζ7+ζ94ζ74+ζ94ζ72+ζ94ζ7 | ζ98ζ74+ζ98ζ72+ζ98ζ7+ζ9ζ74+ζ9ζ72+ζ9ζ7 | ζ97ζ76+ζ97ζ75+ζ97ζ73+ζ92ζ76+ζ92ζ75+ζ92ζ73 | ζ98ζ76+ζ98ζ75+ζ98ζ73+ζ9ζ76+ζ9ζ75+ζ9ζ73 | ζ97ζ74+ζ97ζ72+ζ97ζ7+ζ92ζ74+ζ92ζ72+ζ92ζ7 | complex faithful |
| ρ29 | 6 | 0 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | -1-√-7 | -1+√-7 | 3ζ98+3ζ9 | 3ζ95+3ζ94 | 3ζ97+3ζ92 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1+√-7/2 | 1-√-7/2 | ζ98ζ76+ζ98ζ75+ζ98ζ73+ζ9ζ76+ζ9ζ75+ζ9ζ73 | ζ98ζ74+ζ98ζ72+ζ98ζ7+ζ9ζ74+ζ9ζ72+ζ9ζ7 | ζ97ζ74+ζ97ζ72+ζ97ζ7+ζ92ζ74+ζ92ζ72+ζ92ζ7 | ζ95ζ76+ζ95ζ75+ζ95ζ73+ζ94ζ76+ζ94ζ75+ζ94ζ73 | ζ97ζ76+ζ97ζ75+ζ97ζ73+ζ92ζ76+ζ92ζ75+ζ92ζ73 | ζ95ζ74+ζ95ζ72+ζ95ζ7+ζ94ζ74+ζ94ζ72+ζ94ζ7 | complex faithful |
| ρ30 | 6 | 0 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | -1-√-7 | -1+√-7 | 3ζ97+3ζ92 | 3ζ98+3ζ9 | 3ζ95+3ζ94 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1+√-7/2 | 1-√-7/2 | ζ97ζ76+ζ97ζ75+ζ97ζ73+ζ92ζ76+ζ92ζ75+ζ92ζ73 | ζ97ζ74+ζ97ζ72+ζ97ζ7+ζ92ζ74+ζ92ζ72+ζ92ζ7 | ζ95ζ74+ζ95ζ72+ζ95ζ7+ζ94ζ74+ζ94ζ72+ζ94ζ7 | ζ98ζ76+ζ98ζ75+ζ98ζ73+ζ9ζ76+ζ9ζ75+ζ9ζ73 | ζ95ζ76+ζ95ζ75+ζ95ζ73+ζ94ζ76+ζ94ζ75+ζ94ζ73 | ζ98ζ74+ζ98ζ72+ζ98ζ7+ζ9ζ74+ζ9ζ72+ζ9ζ7 | complex faithful |
►On 63 points
Generators in S63
(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 43 44 45)(46 47 48 49 50 51 52 53 54)(55 56 57 58 59 60 61 62 63)
(1 9)(2 8)(3 7)(4 6)(10 17)(11 16)(12 15)(13 14)(19 25)(20 24)(21 23)(26 27)(28 36)(29 35)(30 34)(31 33)(37 42)(38 41)(39 40)(43 45)(46 54)(47 53)(48 52)(49 51)(55 56)(57 63)(58 62)(59 61)
(1 28 46 27 56 14 40)(2 29 47 19 57 15 41)(3 30 48 20 58 16 42)(4 31 49 21 59 17 43)(5 32 50 22 60 18 44)(6 33 51 23 61 10 45)(7 34 52 24 62 11 37)(8 35 53 25 63 12 38)(9 36 54 26 55 13 39)
(10 23 45)(11 24 37)(12 25 38)(13 26 39)(14 27 40)(15 19 41)(16 20 42)(17 21 43)(18 22 44)(28 46 56)(29 47 57)(30 48 58)(31 49 59)(32 50 60)(33 51 61)(34 52 62)(35 53 63)(36 54 55)
G:=sub<Sym(63)| (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)(37,38,39,40,41,42,43,44,45)(46,47,48,49,50,51,52,53,54)(55,56,57,58,59,60,61,62,63), (1,9)(2,8)(3,7)(4,6)(10,17)(11,16)(12,15)(13,14)(19,25)(20,24)(21,23)(26,27)(28,36)(29,35)(30,34)(31,33)(37,42)(38,41)(39,40)(43,45)(46,54)(47,53)(48,52)(49,51)(55,56)(57,63)(58,62)(59,61), (1,28,46,27,56,14,40)(2,29,47,19,57,15,41)(3,30,48,20,58,16,42)(4,31,49,21,59,17,43)(5,32,50,22,60,18,44)(6,33,51,23,61,10,45)(7,34,52,24,62,11,37)(8,35,53,25,63,12,38)(9,36,54,26,55,13,39), (10,23,45)(11,24,37)(12,25,38)(13,26,39)(14,27,40)(15,19,41)(16,20,42)(17,21,43)(18,22,44)(28,46,56)(29,47,57)(30,48,58)(31,49,59)(32,50,60)(33,51,61)(34,52,62)(35,53,63)(36,54,55)>;
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)(28,29,30,31,32,33,34,35,36)(37,38,39,40,41,42,43,44,45)(46,47,48,49,50,51,52,53,54)(55,56,57,58,59,60,61,62,63), (1,9)(2,8)(3,7)(4,6)(10,17)(11,16)(12,15)(13,14)(19,25)(20,24)(21,23)(26,27)(28,36)(29,35)(30,34)(31,33)(37,42)(38,41)(39,40)(43,45)(46,54)(47,53)(48,52)(49,51)(55,56)(57,63)(58,62)(59,61), (1,28,46,27,56,14,40)(2,29,47,19,57,15,41)(3,30,48,20,58,16,42)(4,31,49,21,59,17,43)(5,32,50,22,60,18,44)(6,33,51,23,61,10,45)(7,34,52,24,62,11,37)(8,35,53,25,63,12,38)(9,36,54,26,55,13,39), (10,23,45)(11,24,37)(12,25,38)(13,26,39)(14,27,40)(15,19,41)(16,20,42)(17,21,43)(18,22,44)(28,46,56)(29,47,57)(30,48,58)(31,49,59)(32,50,60)(33,51,61)(34,52,62)(35,53,63)(36,54,55) );
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),(28,29,30,31,32,33,34,35,36),(37,38,39,40,41,42,43,44,45),(46,47,48,49,50,51,52,53,54),(55,56,57,58,59,60,61,62,63)], [(1,9),(2,8),(3,7),(4,6),(10,17),(11,16),(12,15),(13,14),(19,25),(20,24),(21,23),(26,27),(28,36),(29,35),(30,34),(31,33),(37,42),(38,41),(39,40),(43,45),(46,54),(47,53),(48,52),(49,51),(55,56),(57,63),(58,62),(59,61)], [(1,28,46,27,56,14,40),(2,29,47,19,57,15,41),(3,30,48,20,58,16,42),(4,31,49,21,59,17,43),(5,32,50,22,60,18,44),(6,33,51,23,61,10,45),(7,34,52,24,62,11,37),(8,35,53,25,63,12,38),(9,36,54,26,55,13,39)], [(10,23,45),(11,24,37),(12,25,38),(13,26,39),(14,27,40),(15,19,41),(16,20,42),(17,21,43),(18,22,44),(28,46,56),(29,47,57),(30,48,58),(31,49,59),(32,50,60),(33,51,61),(34,52,62),(35,53,63),(36,54,55)]])
Matrix representation of D9×C7⋊C3 ►in GL5(𝔽127)
| 96 | 9 | 0 | 0 | 0 |
| 118 | 105 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 105 | 31 | 0 | 0 | 0 |
| 9 | 22 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 126 | 104 | 1 |
| 0 | 0 | 0 | 104 | 1 |
| 0 | 0 | 126 | 105 | 1 |
| 107 | 0 | 0 | 0 | 0 |
| 0 | 107 | 0 | 0 | 0 |
| 0 | 0 | 105 | 1 | 23 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 1 | 22 |
G:=sub<GL(5,GF(127))| [96,118,0,0,0,9,105,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[105,9,0,0,0,31,22,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,126,0,126,0,0,104,104,105,0,0,1,1,1],[107,0,0,0,0,0,107,0,0,0,0,0,105,1,1,0,0,1,0,1,0,0,23,0,22] >;
D9×C7⋊C3 in GAP, Magma, Sage, TeX
D_9\times C_7\rtimes C_3
% in TeX
G:=Group("D9xC7:C3"); // GroupNames label
G:=SmallGroup(378,15);
// by ID
G=gap.SmallGroup(378,15);
# by ID
G:=PCGroup([5,-2,-3,-3,-7,-3,2072,642,368,6304]);
// Polycyclic
G:=Group<a,b,c,d|a^9=b^2=c^7=d^3=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^4>;
// generators/relations
Export
Subgroup lattice of D9×C7⋊C3 in TeX
Character table of D9×C7⋊C3 in TeX