direct product, metacyclic, supersoluble, monomial, A-group
Aliases: C2×C3⋊F7, C6⋊F7, D42⋊C3, C42⋊1C6, D21⋊2C6, C14⋊(C3×S3), C7⋊2(S3×C6), C7⋊C3⋊2D6, C3⋊2(C2×F7), C21⋊2(C2×C6), (C2×C7⋊C3)⋊S3, (C6×C7⋊C3)⋊1C2, (C3×C7⋊C3)⋊2C22, SmallGroup(252,30)
Series: Derived ►Chief ►Lower central ►Upper central
| C21 — C2×C3⋊F7 |
Generators and relations for C2×C3⋊F7
G = < a,b,c,d | a2=b3=c7=d6=1, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=c5 >
Character table of C2×C3⋊F7
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 3D | 3E | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 6H | 6I | 7 | 14 | 21A | 21B | 42A | 42B | |
| size | 1 | 1 | 21 | 21 | 2 | 7 | 7 | 14 | 14 | 2 | 7 | 7 | 14 | 14 | 21 | 21 | 21 | 21 | 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 | 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 | 1 | ζ32 | ζ3 | ζ3 | ζ32 | -1 | ζ6 | ζ65 | ζ65 | ζ6 | ζ6 | ζ3 | ζ65 | ζ32 | 1 | -1 | 1 | 1 | -1 | -1 | linear of order 6 |
| ρ6 | 1 | 1 | -1 | -1 | 1 | ζ3 | ζ32 | ζ32 | ζ3 | 1 | ζ3 | ζ32 | ζ32 | ζ3 | ζ65 | ζ6 | ζ6 | ζ65 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 6 |
| ρ7 | 1 | -1 | 1 | -1 | 1 | ζ3 | ζ32 | ζ32 | ζ3 | -1 | ζ65 | ζ6 | ζ6 | ζ65 | ζ3 | ζ6 | ζ32 | ζ65 | 1 | -1 | 1 | 1 | -1 | -1 | linear of order 6 |
| ρ8 | 1 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | ζ3 | ζ32 | 1 | ζ32 | ζ3 | ζ3 | ζ32 | ζ32 | ζ3 | ζ3 | ζ32 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 3 |
| ρ9 | 1 | -1 | 1 | -1 | 1 | ζ32 | ζ3 | ζ3 | ζ32 | -1 | ζ6 | ζ65 | ζ65 | ζ6 | ζ32 | ζ65 | ζ3 | ζ6 | 1 | -1 | 1 | 1 | -1 | -1 | linear of order 6 |
| ρ10 | 1 | -1 | -1 | 1 | 1 | ζ3 | ζ32 | ζ32 | ζ3 | -1 | ζ65 | ζ6 | ζ6 | ζ65 | ζ65 | ζ32 | ζ6 | ζ3 | 1 | -1 | 1 | 1 | -1 | -1 | linear of order 6 |
| ρ11 | 1 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | ζ32 | ζ3 | 1 | ζ3 | ζ32 | ζ32 | ζ3 | ζ3 | ζ32 | ζ32 | ζ3 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 3 |
| ρ12 | 1 | 1 | -1 | -1 | 1 | ζ32 | ζ3 | ζ3 | ζ32 | 1 | ζ32 | ζ3 | ζ3 | ζ32 | ζ6 | ζ65 | ζ65 | ζ6 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 6 |
| ρ13 | 2 | 2 | 0 | 0 | -1 | 2 | 2 | -1 | -1 | -1 | 2 | 2 | -1 | -1 | 0 | 0 | 0 | 0 | 2 | 2 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ14 | 2 | -2 | 0 | 0 | -1 | 2 | 2 | -1 | -1 | 1 | -2 | -2 | 1 | 1 | 0 | 0 | 0 | 0 | 2 | -2 | -1 | -1 | 1 | 1 | orthogonal lifted from D6 |
| ρ15 | 2 | 2 | 0 | 0 | -1 | -1+√-3 | -1-√-3 | ζ6 | ζ65 | -1 | -1+√-3 | -1-√-3 | ζ6 | ζ65 | 0 | 0 | 0 | 0 | 2 | 2 | -1 | -1 | -1 | -1 | complex lifted from C3×S3 |
| ρ16 | 2 | 2 | 0 | 0 | -1 | -1-√-3 | -1+√-3 | ζ65 | ζ6 | -1 | -1-√-3 | -1+√-3 | ζ65 | ζ6 | 0 | 0 | 0 | 0 | 2 | 2 | -1 | -1 | -1 | -1 | complex lifted from C3×S3 |
| ρ17 | 2 | -2 | 0 | 0 | -1 | -1-√-3 | -1+√-3 | ζ65 | ζ6 | 1 | 1+√-3 | 1-√-3 | ζ3 | ζ32 | 0 | 0 | 0 | 0 | 2 | -2 | -1 | -1 | 1 | 1 | complex lifted from S3×C6 |
| ρ18 | 2 | -2 | 0 | 0 | -1 | -1+√-3 | -1-√-3 | ζ6 | ζ65 | 1 | 1-√-3 | 1+√-3 | ζ32 | ζ3 | 0 | 0 | 0 | 0 | 2 | -2 | -1 | -1 | 1 | 1 | complex lifted from S3×C6 |
| ρ19 | 6 | -6 | 0 | 0 | 6 | 0 | 0 | 0 | 0 | -6 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 1 | -1 | -1 | 1 | 1 | orthogonal lifted from C2×F7 |
| ρ20 | 6 | 6 | 0 | 0 | 6 | 0 | 0 | 0 | 0 | 6 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from F7 |
| ρ21 | 6 | 6 | 0 | 0 | -3 | 0 | 0 | 0 | 0 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | 1+√21/2 | 1-√21/2 | 1+√21/2 | 1-√21/2 | orthogonal lifted from C3⋊F7 |
| ρ22 | 6 | -6 | 0 | 0 | -3 | 0 | 0 | 0 | 0 | 3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 1 | 1-√21/2 | 1+√21/2 | -1+√21/2 | -1-√21/2 | orthogonal faithful |
| ρ23 | 6 | 6 | 0 | 0 | -3 | 0 | 0 | 0 | 0 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | 1-√21/2 | 1+√21/2 | 1-√21/2 | 1+√21/2 | orthogonal lifted from C3⋊F7 |
| ρ24 | 6 | -6 | 0 | 0 | -3 | 0 | 0 | 0 | 0 | 3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 1 | 1+√21/2 | 1-√21/2 | -1-√21/2 | -1+√21/2 | orthogonal faithful |
►On 42 points
Generators in S42
(1 27)(2 28)(3 22)(4 23)(5 24)(6 25)(7 26)(8 29)(9 30)(10 31)(11 32)(12 33)(13 34)(14 35)(15 36)(16 37)(17 38)(18 39)(19 40)(20 41)(21 42)
(1 20 13)(2 21 14)(3 15 8)(4 16 9)(5 17 10)(6 18 11)(7 19 12)(22 36 29)(23 37 30)(24 38 31)(25 39 32)(26 40 33)(27 41 34)(28 42 35)
(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 27)(2 23 3 26 5 25)(4 22 7 24 6 28)(8 40 10 39 14 37)(9 36 12 38 11 42)(13 41)(15 33 17 32 21 30)(16 29 19 31 18 35)(20 34)
G:=sub<Sym(42)| (1,27)(2,28)(3,22)(4,23)(5,24)(6,25)(7,26)(8,29)(9,30)(10,31)(11,32)(12,33)(13,34)(14,35)(15,36)(16,37)(17,38)(18,39)(19,40)(20,41)(21,42), (1,20,13)(2,21,14)(3,15,8)(4,16,9)(5,17,10)(6,18,11)(7,19,12)(22,36,29)(23,37,30)(24,38,31)(25,39,32)(26,40,33)(27,41,34)(28,42,35), (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), (1,27)(2,23,3,26,5,25)(4,22,7,24,6,28)(8,40,10,39,14,37)(9,36,12,38,11,42)(13,41)(15,33,17,32,21,30)(16,29,19,31,18,35)(20,34)>;
G:=Group( (1,27)(2,28)(3,22)(4,23)(5,24)(6,25)(7,26)(8,29)(9,30)(10,31)(11,32)(12,33)(13,34)(14,35)(15,36)(16,37)(17,38)(18,39)(19,40)(20,41)(21,42), (1,20,13)(2,21,14)(3,15,8)(4,16,9)(5,17,10)(6,18,11)(7,19,12)(22,36,29)(23,37,30)(24,38,31)(25,39,32)(26,40,33)(27,41,34)(28,42,35), (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), (1,27)(2,23,3,26,5,25)(4,22,7,24,6,28)(8,40,10,39,14,37)(9,36,12,38,11,42)(13,41)(15,33,17,32,21,30)(16,29,19,31,18,35)(20,34) );
G=PermutationGroup([[(1,27),(2,28),(3,22),(4,23),(5,24),(6,25),(7,26),(8,29),(9,30),(10,31),(11,32),(12,33),(13,34),(14,35),(15,36),(16,37),(17,38),(18,39),(19,40),(20,41),(21,42)], [(1,20,13),(2,21,14),(3,15,8),(4,16,9),(5,17,10),(6,18,11),(7,19,12),(22,36,29),(23,37,30),(24,38,31),(25,39,32),(26,40,33),(27,41,34),(28,42,35)], [(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)], [(1,27),(2,23,3,26,5,25),(4,22,7,24,6,28),(8,40,10,39,14,37),(9,36,12,38,11,42),(13,41),(15,33,17,32,21,30),(16,29,19,31,18,35),(20,34)]])
Matrix representation of C2×C3⋊F7 ►in GL8(𝔽43)
| 42 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 42 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 42 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 42 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 5 | 38 | 5 | 0 |
| 0 | 0 | 38 | 40 | 5 | 0 | 0 | 5 |
| 0 | 0 | 38 | 38 | 2 | 0 | 5 | 0 |
| 0 | 0 | 0 | 38 | 0 | 40 | 5 | 5 |
| 0 | 0 | 38 | 0 | 0 | 38 | 2 | 5 |
| 0 | 0 | 0 | 38 | 5 | 38 | 0 | 2 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 42 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 42 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 42 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 42 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 42 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 42 | 0 | 0 | 0 | 0 | 0 |
| 0 | 42 | 0 | 0 | 0 | 0 | 0 | 0 |
| 42 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
G:=sub<GL(8,GF(43))| [42,0,0,0,0,0,0,0,0,42,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[42,42,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,40,38,38,0,38,0,0,0,0,40,38,38,0,38,0,0,5,5,2,0,0,5,0,0,38,0,0,40,38,38,0,0,5,0,5,5,2,0,0,0,0,5,0,5,5,2],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,42,42,42,42,42,42,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0],[0,42,0,0,0,0,0,0,42,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0] >;
C2×C3⋊F7 in GAP, Magma, Sage, TeX
C_2\times C_3\rtimes F_7
% in TeX
G:=Group("C2xC3:F7"); // GroupNames label
G:=SmallGroup(252,30);
// by ID
G=gap.SmallGroup(252,30);
# by ID
G:=PCGroup([5,-2,-2,-3,-3,-7,483,5404,464]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^3=c^7=d^6=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=c^5>;
// generators/relations
Export
Subgroup lattice of C2×C3⋊F7 in TeX
Character table of C2×C3⋊F7 in TeX