direct product, metacyclic, supersoluble, monomial, A-group, 3-hyperelementary
Aliases: C6×C7⋊C3, C42⋊C3, C14⋊C32, C21⋊4C6, C7⋊2(C3×C6), SmallGroup(126,10)
Series: Derived ►Chief ►Lower central ►Upper central
| C7 — C6×C7⋊C3 |
Generators and relations for C6×C7⋊C3
G = < a,b,c | a6=b7=c3=1, ab=ba, ac=ca, cbc-1=b4 >
Character table of C6×C7⋊C3
| class | 1 | 2 | 3A | 3B | 3C | 3D | 3E | 3F | 3G | 3H | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 6H | 7A | 7B | 14A | 14B | 21A | 21B | 21C | 21D | 42A | 42B | 42C | 42D | |
| size | 1 | 1 | 1 | 1 | 7 | 7 | 7 | 7 | 7 | 7 | 1 | 1 | 7 | 7 | 7 | 7 | 7 | 7 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | |
| 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 | ζ3 | ζ32 | 1 | ζ3 | 1 | ζ32 | ζ32 | ζ3 | ζ65 | ζ6 | -1 | ζ65 | -1 | ζ6 | ζ6 | ζ65 | 1 | 1 | -1 | -1 | ζ3 | ζ3 | ζ32 | ζ32 | ζ6 | ζ65 | ζ65 | ζ6 | linear of order 6 |
| ρ4 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | ζ32 | ζ32 | ζ3 | ζ3 | 1 | 1 | ζ3 | ζ32 | ζ32 | ζ32 | ζ3 | ζ3 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 3 |
| ρ5 | 1 | -1 | 1 | 1 | ζ3 | ζ32 | ζ32 | ζ32 | ζ3 | ζ3 | -1 | -1 | ζ65 | ζ6 | ζ6 | ζ6 | ζ65 | ζ65 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | linear of order 6 |
| ρ6 | 1 | 1 | ζ32 | ζ3 | ζ32 | 1 | ζ3 | ζ32 | 1 | ζ3 | ζ32 | ζ3 | ζ32 | 1 | ζ3 | ζ32 | 1 | ζ3 | 1 | 1 | 1 | 1 | ζ32 | ζ32 | ζ3 | ζ3 | ζ3 | ζ32 | ζ32 | ζ3 | linear of order 3 |
| ρ7 | 1 | 1 | ζ3 | ζ32 | ζ32 | ζ32 | ζ3 | 1 | ζ3 | 1 | ζ3 | ζ32 | ζ32 | ζ32 | ζ3 | 1 | ζ3 | 1 | 1 | 1 | 1 | 1 | ζ3 | ζ3 | ζ32 | ζ32 | ζ32 | ζ3 | ζ3 | ζ32 | linear of order 3 |
| ρ8 | 1 | -1 | ζ32 | ζ3 | ζ32 | 1 | ζ3 | ζ32 | 1 | ζ3 | ζ6 | ζ65 | ζ6 | -1 | ζ65 | ζ6 | -1 | ζ65 | 1 | 1 | -1 | -1 | ζ32 | ζ32 | ζ3 | ζ3 | ζ65 | ζ6 | ζ6 | ζ65 | linear of order 6 |
| ρ9 | 1 | 1 | ζ3 | ζ32 | ζ3 | 1 | ζ32 | ζ3 | 1 | ζ32 | ζ3 | ζ32 | ζ3 | 1 | ζ32 | ζ3 | 1 | ζ32 | 1 | 1 | 1 | 1 | ζ3 | ζ3 | ζ32 | ζ32 | ζ32 | ζ3 | ζ3 | ζ32 | linear of order 3 |
| ρ10 | 1 | 1 | ζ3 | ζ32 | 1 | ζ3 | 1 | ζ32 | ζ32 | ζ3 | ζ3 | ζ32 | 1 | ζ3 | 1 | ζ32 | ζ32 | ζ3 | 1 | 1 | 1 | 1 | ζ3 | ζ3 | ζ32 | ζ32 | ζ32 | ζ3 | ζ3 | ζ32 | linear of order 3 |
| ρ11 | 1 | -1 | ζ3 | ζ32 | ζ32 | ζ32 | ζ3 | 1 | ζ3 | 1 | ζ65 | ζ6 | ζ6 | ζ6 | ζ65 | -1 | ζ65 | -1 | 1 | 1 | -1 | -1 | ζ3 | ζ3 | ζ32 | ζ32 | ζ6 | ζ65 | ζ65 | ζ6 | linear of order 6 |
| ρ12 | 1 | 1 | ζ32 | ζ3 | ζ3 | ζ3 | ζ32 | 1 | ζ32 | 1 | ζ32 | ζ3 | ζ3 | ζ3 | ζ32 | 1 | ζ32 | 1 | 1 | 1 | 1 | 1 | ζ32 | ζ32 | ζ3 | ζ3 | ζ3 | ζ32 | ζ32 | ζ3 | linear of order 3 |
| ρ13 | 1 | 1 | ζ32 | ζ3 | 1 | ζ32 | 1 | ζ3 | ζ3 | ζ32 | ζ32 | ζ3 | 1 | ζ32 | 1 | ζ3 | ζ3 | ζ32 | 1 | 1 | 1 | 1 | ζ32 | ζ32 | ζ3 | ζ3 | ζ3 | ζ32 | ζ32 | ζ3 | linear of order 3 |
| ρ14 | 1 | -1 | 1 | 1 | ζ32 | ζ3 | ζ3 | ζ3 | ζ32 | ζ32 | -1 | -1 | ζ6 | ζ65 | ζ65 | ζ65 | ζ6 | ζ6 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | linear of order 6 |
| ρ15 | 1 | -1 | ζ32 | ζ3 | ζ3 | ζ3 | ζ32 | 1 | ζ32 | 1 | ζ6 | ζ65 | ζ65 | ζ65 | ζ6 | -1 | ζ6 | -1 | 1 | 1 | -1 | -1 | ζ32 | ζ32 | ζ3 | ζ3 | ζ65 | ζ6 | ζ6 | ζ65 | linear of order 6 |
| ρ16 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | ζ3 | ζ3 | ζ32 | ζ32 | 1 | 1 | ζ32 | ζ3 | ζ3 | ζ3 | ζ32 | ζ32 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 3 |
| ρ17 | 1 | -1 | ζ3 | ζ32 | ζ3 | 1 | ζ32 | ζ3 | 1 | ζ32 | ζ65 | ζ6 | ζ65 | -1 | ζ6 | ζ65 | -1 | ζ6 | 1 | 1 | -1 | -1 | ζ3 | ζ3 | ζ32 | ζ32 | ζ6 | ζ65 | ζ65 | ζ6 | linear of order 6 |
| ρ18 | 1 | -1 | ζ32 | ζ3 | 1 | ζ32 | 1 | ζ3 | ζ3 | ζ32 | ζ6 | ζ65 | -1 | ζ6 | -1 | ζ65 | ζ65 | ζ6 | 1 | 1 | -1 | -1 | ζ32 | ζ32 | ζ3 | ζ3 | ζ65 | ζ6 | ζ6 | ζ65 | linear of order 6 |
| ρ19 | 3 | 3 | 3 | 3 | 0 | 0 | 0 | 0 | 0 | 0 | 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 | -1-√-7/2 | -1-√-7/2 | complex lifted from C7⋊C3 |
| ρ20 | 3 | -3 | 3 | 3 | 0 | 0 | 0 | 0 | 0 | 0 | -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 | 1-√-7/2 | 1-√-7/2 | complex lifted from C2×C7⋊C3 |
| ρ21 | 3 | 3 | 3 | 3 | 0 | 0 | 0 | 0 | 0 | 0 | 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 | -1+√-7/2 | -1+√-7/2 | complex lifted from C7⋊C3 |
| ρ22 | 3 | -3 | 3 | 3 | 0 | 0 | 0 | 0 | 0 | 0 | -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 | 1+√-7/2 | 1+√-7/2 | complex lifted from C2×C7⋊C3 |
| ρ23 | 3 | -3 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | 3+3√-3/2 | 3-3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | -1+√-7/2 | -1-√-7/2 | 1-√-7/2 | 1+√-7/2 | ζ32ζ76+ζ32ζ75+ζ32ζ73 | ζ32ζ74+ζ32ζ72+ζ32ζ7 | ζ3ζ74+ζ3ζ72+ζ3ζ7 | ζ3ζ76+ζ3ζ75+ζ3ζ73 | -ζ3ζ74-ζ3ζ72-ζ3ζ7 | -ζ32ζ74-ζ32ζ72-ζ32ζ7 | -ζ32ζ76-ζ32ζ75-ζ32ζ73 | -ζ3ζ76-ζ3ζ75-ζ3ζ73 | complex faithful |
| ρ24 | 3 | -3 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | 3-3√-3/2 | 3+3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | -1-√-7/2 | -1+√-7/2 | 1+√-7/2 | 1-√-7/2 | ζ3ζ74+ζ3ζ72+ζ3ζ7 | ζ3ζ76+ζ3ζ75+ζ3ζ73 | ζ32ζ76+ζ32ζ75+ζ32ζ73 | ζ32ζ74+ζ32ζ72+ζ32ζ7 | -ζ32ζ76-ζ32ζ75-ζ32ζ73 | -ζ3ζ76-ζ3ζ75-ζ3ζ73 | -ζ3ζ74-ζ3ζ72-ζ3ζ7 | -ζ32ζ74-ζ32ζ72-ζ32ζ7 | complex faithful |
| ρ25 | 3 | 3 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | -1+√-7/2 | -1-√-7/2 | -1+√-7/2 | -1-√-7/2 | ζ3ζ76+ζ3ζ75+ζ3ζ73 | ζ3ζ74+ζ3ζ72+ζ3ζ7 | ζ32ζ74+ζ32ζ72+ζ32ζ7 | ζ32ζ76+ζ32ζ75+ζ32ζ73 | ζ32ζ74+ζ32ζ72+ζ32ζ7 | ζ3ζ74+ζ3ζ72+ζ3ζ7 | ζ3ζ76+ζ3ζ75+ζ3ζ73 | ζ32ζ76+ζ32ζ75+ζ32ζ73 | complex lifted from C3×C7⋊C3 |
| ρ26 | 3 | 3 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | -1-√-7/2 | -1+√-7/2 | -1-√-7/2 | -1+√-7/2 | ζ3ζ74+ζ3ζ72+ζ3ζ7 | ζ3ζ76+ζ3ζ75+ζ3ζ73 | ζ32ζ76+ζ32ζ75+ζ32ζ73 | ζ32ζ74+ζ32ζ72+ζ32ζ7 | ζ32ζ76+ζ32ζ75+ζ32ζ73 | ζ3ζ76+ζ3ζ75+ζ3ζ73 | ζ3ζ74+ζ3ζ72+ζ3ζ7 | ζ32ζ74+ζ32ζ72+ζ32ζ7 | complex lifted from C3×C7⋊C3 |
| ρ27 | 3 | 3 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | -1+√-7/2 | -1-√-7/2 | -1+√-7/2 | -1-√-7/2 | ζ32ζ76+ζ32ζ75+ζ32ζ73 | ζ32ζ74+ζ32ζ72+ζ32ζ7 | ζ3ζ74+ζ3ζ72+ζ3ζ7 | ζ3ζ76+ζ3ζ75+ζ3ζ73 | ζ3ζ74+ζ3ζ72+ζ3ζ7 | ζ32ζ74+ζ32ζ72+ζ32ζ7 | ζ32ζ76+ζ32ζ75+ζ32ζ73 | ζ3ζ76+ζ3ζ75+ζ3ζ73 | complex lifted from C3×C7⋊C3 |
| ρ28 | 3 | -3 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | 3+3√-3/2 | 3-3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | -1-√-7/2 | -1+√-7/2 | 1+√-7/2 | 1-√-7/2 | ζ32ζ74+ζ32ζ72+ζ32ζ7 | ζ32ζ76+ζ32ζ75+ζ32ζ73 | ζ3ζ76+ζ3ζ75+ζ3ζ73 | ζ3ζ74+ζ3ζ72+ζ3ζ7 | -ζ3ζ76-ζ3ζ75-ζ3ζ73 | -ζ32ζ76-ζ32ζ75-ζ32ζ73 | -ζ32ζ74-ζ32ζ72-ζ32ζ7 | -ζ3ζ74-ζ3ζ72-ζ3ζ7 | complex faithful |
| ρ29 | 3 | 3 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | -1-√-7/2 | -1+√-7/2 | -1-√-7/2 | -1+√-7/2 | ζ32ζ74+ζ32ζ72+ζ32ζ7 | ζ32ζ76+ζ32ζ75+ζ32ζ73 | ζ3ζ76+ζ3ζ75+ζ3ζ73 | ζ3ζ74+ζ3ζ72+ζ3ζ7 | ζ3ζ76+ζ3ζ75+ζ3ζ73 | ζ32ζ76+ζ32ζ75+ζ32ζ73 | ζ32ζ74+ζ32ζ72+ζ32ζ7 | ζ3ζ74+ζ3ζ72+ζ3ζ7 | complex lifted from C3×C7⋊C3 |
| ρ30 | 3 | -3 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | 3-3√-3/2 | 3+3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | -1+√-7/2 | -1-√-7/2 | 1-√-7/2 | 1+√-7/2 | ζ3ζ76+ζ3ζ75+ζ3ζ73 | ζ3ζ74+ζ3ζ72+ζ3ζ7 | ζ32ζ74+ζ32ζ72+ζ32ζ7 | ζ32ζ76+ζ32ζ75+ζ32ζ73 | -ζ32ζ74-ζ32ζ72-ζ32ζ7 | -ζ3ζ74-ζ3ζ72-ζ3ζ7 | -ζ3ζ76-ζ3ζ75-ζ3ζ73 | -ζ32ζ76-ζ32ζ75-ζ32ζ73 | complex faithful |
►On 42 points
Generators in S42
(1 29 15 22 8 36)(2 30 16 23 9 37)(3 31 17 24 10 38)(4 32 18 25 11 39)(5 33 19 26 12 40)(6 34 20 27 13 41)(7 35 21 28 14 42)
(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 15 8)(2 17 12)(3 19 9)(4 21 13)(5 16 10)(6 18 14)(7 20 11)(22 36 29)(23 38 33)(24 40 30)(25 42 34)(26 37 31)(27 39 35)(28 41 32)
G:=sub<Sym(42)| (1,29,15,22,8,36)(2,30,16,23,9,37)(3,31,17,24,10,38)(4,32,18,25,11,39)(5,33,19,26,12,40)(6,34,20,27,13,41)(7,35,21,28,14,42), (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), (1,15,8)(2,17,12)(3,19,9)(4,21,13)(5,16,10)(6,18,14)(7,20,11)(22,36,29)(23,38,33)(24,40,30)(25,42,34)(26,37,31)(27,39,35)(28,41,32)>;
G:=Group( (1,29,15,22,8,36)(2,30,16,23,9,37)(3,31,17,24,10,38)(4,32,18,25,11,39)(5,33,19,26,12,40)(6,34,20,27,13,41)(7,35,21,28,14,42), (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), (1,15,8)(2,17,12)(3,19,9)(4,21,13)(5,16,10)(6,18,14)(7,20,11)(22,36,29)(23,38,33)(24,40,30)(25,42,34)(26,37,31)(27,39,35)(28,41,32) );
G=PermutationGroup([[(1,29,15,22,8,36),(2,30,16,23,9,37),(3,31,17,24,10,38),(4,32,18,25,11,39),(5,33,19,26,12,40),(6,34,20,27,13,41),(7,35,21,28,14,42)], [(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)], [(1,15,8),(2,17,12),(3,19,9),(4,21,13),(5,16,10),(6,18,14),(7,20,11),(22,36,29),(23,38,33),(24,40,30),(25,42,34),(26,37,31),(27,39,35),(28,41,32)]])
C6×C7⋊C3 is a maximal subgroup of
C6.F7
Matrix representation of C6×C7⋊C3 ►in GL3(𝔽43) generated by
| 37 | 0 | 0 |
| 0 | 37 | 0 |
| 0 | 0 | 37 |
| 24 | 25 | 1 |
| 1 | 0 | 0 |
| 0 | 1 | 0 |
| 36 | 0 | 0 |
| 3 | 7 | 7 |
| 0 | 36 | 0 |
G:=sub<GL(3,GF(43))| [37,0,0,0,37,0,0,0,37],[24,1,0,25,0,1,1,0,0],[36,3,0,0,7,36,0,7,0] >;
C6×C7⋊C3 in GAP, Magma, Sage, TeX
C_6\times C_7\rtimes C_3
% in TeX
G:=Group("C6xC7:C3"); // GroupNames label
G:=SmallGroup(126,10);
// by ID
G=gap.SmallGroup(126,10);
# by ID
G:=PCGroup([4,-2,-3,-3,-7,295]);
// Polycyclic
G:=Group<a,b,c|a^6=b^7=c^3=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^4>;
// generators/relations
Export
Subgroup lattice of C6×C7⋊C3 in TeX
Character table of C6×C7⋊C3 in TeX