direct product, metabelian, supersoluble, monomial, A-group
Aliases: D7×C7⋊C3, C72⋊2C6, C7⋊2(C3×D7), (C7×D7)⋊2C3, (C7×C7⋊C3)⋊2C2, C7⋊5(C2×C7⋊C3), SmallGroup(294,9)
Series: Derived ►Chief ►Lower central ►Upper central
| C72 — D7×C7⋊C3 |
Generators and relations for D7×C7⋊C3
G = < a,b,c,d | a7=b2=c7=d3=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c4 >
Character table of D7×C7⋊C3
| class | 1 | 2 | 3A | 3B | 6A | 6B | 7A | 7B | 7C | 7D | 7E | 7F | 7G | 7H | 7I | 7J | 7K | 14A | 14B | 21A | 21B | 21C | 21D | 21E | 21F | |
| size | 1 | 7 | 7 | 7 | 49 | 49 | 2 | 2 | 2 | 3 | 3 | 6 | 6 | 6 | 6 | 6 | 6 | 21 | 21 | 14 | 14 | 14 | 14 | 14 | 14 | |
| 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 | 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 | linear of order 2 |
| ρ3 | 1 | 1 | ζ3 | ζ32 | ζ32 | ζ3 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | ζ32 | ζ32 | ζ3 | ζ3 | linear of order 3 |
| ρ4 | 1 | 1 | ζ32 | ζ3 | ζ3 | ζ32 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | ζ3 | ζ3 | ζ32 | ζ32 | linear of order 3 |
| ρ5 | 1 | -1 | ζ3 | ζ32 | ζ6 | ζ65 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | ζ32 | ζ3 | ζ32 | ζ32 | ζ3 | ζ3 | linear of order 6 |
| ρ6 | 1 | -1 | ζ32 | ζ3 | ζ65 | ζ6 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | ζ3 | ζ32 | ζ3 | ζ3 | ζ32 | ζ32 | linear of order 6 |
| ρ7 | 2 | 0 | 2 | 2 | 0 | 0 | ζ74+ζ73 | ζ76+ζ7 | ζ75+ζ72 | 2 | 2 | ζ76+ζ7 | ζ75+ζ72 | ζ75+ζ72 | ζ76+ζ7 | ζ74+ζ73 | ζ74+ζ73 | 0 | 0 | ζ76+ζ7 | ζ75+ζ72 | ζ75+ζ72 | ζ74+ζ73 | ζ74+ζ73 | ζ76+ζ7 | orthogonal lifted from D7 |
| ρ8 | 2 | 0 | 2 | 2 | 0 | 0 | ζ75+ζ72 | ζ74+ζ73 | ζ76+ζ7 | 2 | 2 | ζ74+ζ73 | ζ76+ζ7 | ζ76+ζ7 | ζ74+ζ73 | ζ75+ζ72 | ζ75+ζ72 | 0 | 0 | ζ74+ζ73 | ζ76+ζ7 | ζ76+ζ7 | ζ75+ζ72 | ζ75+ζ72 | ζ74+ζ73 | orthogonal lifted from D7 |
| ρ9 | 2 | 0 | 2 | 2 | 0 | 0 | ζ76+ζ7 | ζ75+ζ72 | ζ74+ζ73 | 2 | 2 | ζ75+ζ72 | ζ74+ζ73 | ζ74+ζ73 | ζ75+ζ72 | ζ76+ζ7 | ζ76+ζ7 | 0 | 0 | ζ75+ζ72 | ζ74+ζ73 | ζ74+ζ73 | ζ76+ζ7 | ζ76+ζ7 | ζ75+ζ72 | orthogonal lifted from D7 |
| ρ10 | 2 | 0 | -1-√-3 | -1+√-3 | 0 | 0 | ζ74+ζ73 | ζ76+ζ7 | ζ75+ζ72 | 2 | 2 | ζ76+ζ7 | ζ75+ζ72 | ζ75+ζ72 | ζ76+ζ7 | ζ74+ζ73 | ζ74+ζ73 | 0 | 0 | ζ3ζ76+ζ3ζ7 | ζ32ζ75+ζ32ζ72 | ζ3ζ75+ζ3ζ72 | ζ3ζ74+ζ3ζ73 | ζ32ζ74+ζ32ζ73 | ζ32ζ76+ζ32ζ7 | complex lifted from C3×D7 |
| ρ11 | 2 | 0 | -1-√-3 | -1+√-3 | 0 | 0 | ζ76+ζ7 | ζ75+ζ72 | ζ74+ζ73 | 2 | 2 | ζ75+ζ72 | ζ74+ζ73 | ζ74+ζ73 | ζ75+ζ72 | ζ76+ζ7 | ζ76+ζ7 | 0 | 0 | ζ3ζ75+ζ3ζ72 | ζ32ζ74+ζ32ζ73 | ζ3ζ74+ζ3ζ73 | ζ3ζ76+ζ3ζ7 | ζ32ζ76+ζ32ζ7 | ζ32ζ75+ζ32ζ72 | complex lifted from C3×D7 |
| ρ12 | 2 | 0 | -1+√-3 | -1-√-3 | 0 | 0 | ζ76+ζ7 | ζ75+ζ72 | ζ74+ζ73 | 2 | 2 | ζ75+ζ72 | ζ74+ζ73 | ζ74+ζ73 | ζ75+ζ72 | ζ76+ζ7 | ζ76+ζ7 | 0 | 0 | ζ32ζ75+ζ32ζ72 | ζ3ζ74+ζ3ζ73 | ζ32ζ74+ζ32ζ73 | ζ32ζ76+ζ32ζ7 | ζ3ζ76+ζ3ζ7 | ζ3ζ75+ζ3ζ72 | complex lifted from C3×D7 |
| ρ13 | 2 | 0 | -1+√-3 | -1-√-3 | 0 | 0 | ζ74+ζ73 | ζ76+ζ7 | ζ75+ζ72 | 2 | 2 | ζ76+ζ7 | ζ75+ζ72 | ζ75+ζ72 | ζ76+ζ7 | ζ74+ζ73 | ζ74+ζ73 | 0 | 0 | ζ32ζ76+ζ32ζ7 | ζ3ζ75+ζ3ζ72 | ζ32ζ75+ζ32ζ72 | ζ32ζ74+ζ32ζ73 | ζ3ζ74+ζ3ζ73 | ζ3ζ76+ζ3ζ7 | complex lifted from C3×D7 |
| ρ14 | 2 | 0 | -1+√-3 | -1-√-3 | 0 | 0 | ζ75+ζ72 | ζ74+ζ73 | ζ76+ζ7 | 2 | 2 | ζ74+ζ73 | ζ76+ζ7 | ζ76+ζ7 | ζ74+ζ73 | ζ75+ζ72 | ζ75+ζ72 | 0 | 0 | ζ32ζ74+ζ32ζ73 | ζ3ζ76+ζ3ζ7 | ζ32ζ76+ζ32ζ7 | ζ32ζ75+ζ32ζ72 | ζ3ζ75+ζ3ζ72 | ζ3ζ74+ζ3ζ73 | complex lifted from C3×D7 |
| ρ15 | 2 | 0 | -1-√-3 | -1+√-3 | 0 | 0 | ζ75+ζ72 | ζ74+ζ73 | ζ76+ζ7 | 2 | 2 | ζ74+ζ73 | ζ76+ζ7 | ζ76+ζ7 | ζ74+ζ73 | ζ75+ζ72 | ζ75+ζ72 | 0 | 0 | ζ3ζ74+ζ3ζ73 | ζ32ζ76+ζ32ζ7 | ζ3ζ76+ζ3ζ7 | ζ3ζ75+ζ3ζ72 | ζ32ζ75+ζ32ζ72 | ζ32ζ74+ζ32ζ73 | complex lifted from C3×D7 |
| ρ16 | 3 | -3 | 0 | 0 | 0 | 0 | 3 | 3 | 3 | -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 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C2×C7⋊C3 |
| ρ17 | 3 | 3 | 0 | 0 | 0 | 0 | 3 | 3 | 3 | -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 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C7⋊C3 |
| ρ18 | 3 | -3 | 0 | 0 | 0 | 0 | 3 | 3 | 3 | -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 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C2×C7⋊C3 |
| ρ19 | 3 | 3 | 0 | 0 | 0 | 0 | 3 | 3 | 3 | -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 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C7⋊C3 |
| ρ20 | 6 | 0 | 0 | 0 | 0 | 0 | 3ζ76+3ζ7 | 3ζ75+3ζ72 | 3ζ74+3ζ73 | -1-√-7 | -1+√-7 | -ζ76-ζ72+ζ7 | -ζ75-ζ74+ζ72 | ζ75-ζ73-ζ72 | ζ76-ζ75-ζ7 | ζ74-ζ73-ζ7 | -ζ76-ζ74+ζ73 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
| ρ21 | 6 | 0 | 0 | 0 | 0 | 0 | 3ζ75+3ζ72 | 3ζ74+3ζ73 | 3ζ76+3ζ7 | -1-√-7 | -1+√-7 | -ζ75-ζ74+ζ72 | ζ74-ζ73-ζ7 | -ζ76-ζ74+ζ73 | ζ75-ζ73-ζ72 | -ζ76-ζ72+ζ7 | ζ76-ζ75-ζ7 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
| ρ22 | 6 | 0 | 0 | 0 | 0 | 0 | 3ζ74+3ζ73 | 3ζ76+3ζ7 | 3ζ75+3ζ72 | -1+√-7 | -1-√-7 | -ζ76-ζ74+ζ73 | ζ76-ζ75-ζ7 | -ζ76-ζ72+ζ7 | ζ74-ζ73-ζ7 | ζ75-ζ73-ζ72 | -ζ75-ζ74+ζ72 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
| ρ23 | 6 | 0 | 0 | 0 | 0 | 0 | 3ζ74+3ζ73 | 3ζ76+3ζ7 | 3ζ75+3ζ72 | -1-√-7 | -1+√-7 | ζ74-ζ73-ζ7 | -ζ76-ζ72+ζ7 | ζ76-ζ75-ζ7 | -ζ76-ζ74+ζ73 | -ζ75-ζ74+ζ72 | ζ75-ζ73-ζ72 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
| ρ24 | 6 | 0 | 0 | 0 | 0 | 0 | 3ζ75+3ζ72 | 3ζ74+3ζ73 | 3ζ76+3ζ7 | -1+√-7 | -1-√-7 | ζ75-ζ73-ζ72 | -ζ76-ζ74+ζ73 | ζ74-ζ73-ζ7 | -ζ75-ζ74+ζ72 | ζ76-ζ75-ζ7 | -ζ76-ζ72+ζ7 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
| ρ25 | 6 | 0 | 0 | 0 | 0 | 0 | 3ζ76+3ζ7 | 3ζ75+3ζ72 | 3ζ74+3ζ73 | -1+√-7 | -1-√-7 | ζ76-ζ75-ζ7 | ζ75-ζ73-ζ72 | -ζ75-ζ74+ζ72 | -ζ76-ζ72+ζ7 | -ζ76-ζ74+ζ73 | ζ74-ζ73-ζ7 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
►On 42 points
Generators in S42
(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 33)(9 32)(10 31)(11 30)(12 29)(13 35)(14 34)(15 40)(16 39)(17 38)(18 37)(19 36)(20 42)(21 41)
(1 2 3 4 5 6 7)(8 10 12 14 9 11 13)(15 19 16 20 17 21 18)(22 28 27 26 25 24 23)(29 34 32 30 35 33 31)(36 39 42 38 41 37 40)
(1 15 8)(2 16 9)(3 17 10)(4 18 11)(5 19 12)(6 20 13)(7 21 14)(22 36 29)(23 37 30)(24 38 31)(25 39 32)(26 40 33)(27 41 34)(28 42 35)
G:=sub<Sym(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), (1,26)(2,25)(3,24)(4,23)(5,22)(6,28)(7,27)(8,33)(9,32)(10,31)(11,30)(12,29)(13,35)(14,34)(15,40)(16,39)(17,38)(18,37)(19,36)(20,42)(21,41), (1,2,3,4,5,6,7)(8,10,12,14,9,11,13)(15,19,16,20,17,21,18)(22,28,27,26,25,24,23)(29,34,32,30,35,33,31)(36,39,42,38,41,37,40), (1,15,8)(2,16,9)(3,17,10)(4,18,11)(5,19,12)(6,20,13)(7,21,14)(22,36,29)(23,37,30)(24,38,31)(25,39,32)(26,40,33)(27,41,34)(28,42,35)>;
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), (1,26)(2,25)(3,24)(4,23)(5,22)(6,28)(7,27)(8,33)(9,32)(10,31)(11,30)(12,29)(13,35)(14,34)(15,40)(16,39)(17,38)(18,37)(19,36)(20,42)(21,41), (1,2,3,4,5,6,7)(8,10,12,14,9,11,13)(15,19,16,20,17,21,18)(22,28,27,26,25,24,23)(29,34,32,30,35,33,31)(36,39,42,38,41,37,40), (1,15,8)(2,16,9)(3,17,10)(4,18,11)(5,19,12)(6,20,13)(7,21,14)(22,36,29)(23,37,30)(24,38,31)(25,39,32)(26,40,33)(27,41,34)(28,42,35) );
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)], [(1,26),(2,25),(3,24),(4,23),(5,22),(6,28),(7,27),(8,33),(9,32),(10,31),(11,30),(12,29),(13,35),(14,34),(15,40),(16,39),(17,38),(18,37),(19,36),(20,42),(21,41)], [(1,2,3,4,5,6,7),(8,10,12,14,9,11,13),(15,19,16,20,17,21,18),(22,28,27,26,25,24,23),(29,34,32,30,35,33,31),(36,39,42,38,41,37,40)], [(1,15,8),(2,16,9),(3,17,10),(4,18,11),(5,19,12),(6,20,13),(7,21,14),(22,36,29),(23,37,30),(24,38,31),(25,39,32),(26,40,33),(27,41,34),(28,42,35)]])
Matrix representation of D7×C7⋊C3 ►in GL5(𝔽43)
| 8 | 42 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 8 | 42 | 0 | 0 | 0 |
| 20 | 35 | 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 | 9 | 10 | 19 |
| 0 | 0 | 33 | 33 | 42 |
| 0 | 0 | 34 | 34 | 25 |
| 6 | 0 | 0 | 0 | 0 |
| 0 | 6 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
G:=sub<GL(5,GF(43))| [8,1,0,0,0,42,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[8,20,0,0,0,42,35,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,9,33,34,0,0,10,33,34,0,0,19,42,25],[6,0,0,0,0,0,6,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0] >;
D7×C7⋊C3 in GAP, Magma, Sage, TeX
D_7\times C_7\rtimes C_3
% in TeX
G:=Group("D7xC7:C3"); // GroupNames label
G:=SmallGroup(294,9);
// by ID
G=gap.SmallGroup(294,9);
# by ID
G:=PCGroup([4,-2,-3,-7,-7,434,679]);
// Polycyclic
G:=Group<a,b,c,d|a^7=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 D7×C7⋊C3 in TeX
Character table of D7×C7⋊C3 in TeX