direct product, metabelian, supersoluble, monomial, A-group
Aliases: S3×F7, D21⋊C6, C3⋊F7⋊C2, (S3×C7)⋊C6, C21⋊(C2×C6), C7⋊1(S3×C6), (C3×D7)⋊C6, (S3×D7)⋊C3, D7⋊(C3×S3), C7⋊C3⋊1D6, (C3×F7)⋊C2, C3⋊1(C2×F7), (S3×C7⋊C3)⋊C2, (C3×C7⋊C3)⋊C22, Aut(D21), Hol(C21), SmallGroup(252,26)
Series: Derived ►Chief ►Lower central ►Upper central
| C21 — S3×F7 |
Generators and relations for S3×F7
G = < a,b,c,d | a3=b2=c7=d6=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c5 >
Character table of S3×F7
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 3D | 3E | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 6H | 6I | 7 | 14 | 21 | |
| size | 1 | 3 | 7 | 21 | 2 | 7 | 7 | 14 | 14 | 7 | 7 | 14 | 14 | 14 | 21 | 21 | 21 | 21 | 6 | 18 | 12 | |
| ρ1 | 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 | 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 | 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 | linear of order 2 |
| ρ5 | 1 | -1 | 1 | -1 | 1 | ζ3 | ζ32 | ζ32 | ζ3 | ζ32 | ζ3 | ζ32 | ζ3 | 1 | ζ65 | ζ6 | ζ6 | ζ65 | 1 | -1 | 1 | linear of order 6 |
| ρ6 | 1 | -1 | -1 | 1 | 1 | ζ32 | ζ3 | ζ3 | ζ32 | ζ65 | ζ6 | ζ65 | ζ6 | -1 | ζ32 | ζ65 | ζ3 | ζ6 | 1 | -1 | 1 | linear of order 6 |
| ρ7 | 1 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | ζ32 | ζ3 | ζ32 | ζ3 | ζ32 | ζ3 | 1 | ζ3 | ζ32 | ζ32 | ζ3 | 1 | 1 | 1 | linear of order 3 |
| ρ8 | 1 | -1 | 1 | -1 | 1 | ζ32 | ζ3 | ζ3 | ζ32 | ζ3 | ζ32 | ζ3 | ζ32 | 1 | ζ6 | ζ65 | ζ65 | ζ6 | 1 | -1 | 1 | linear of order 6 |
| ρ9 | 1 | 1 | -1 | -1 | 1 | ζ3 | ζ32 | ζ32 | ζ3 | ζ6 | ζ65 | ζ6 | ζ65 | -1 | ζ65 | ζ32 | ζ6 | ζ3 | 1 | 1 | 1 | linear of order 6 |
| ρ10 | 1 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | ζ3 | ζ32 | ζ3 | ζ32 | ζ3 | ζ32 | 1 | ζ32 | ζ3 | ζ3 | ζ32 | 1 | 1 | 1 | linear of order 3 |
| ρ11 | 1 | -1 | -1 | 1 | 1 | ζ3 | ζ32 | ζ32 | ζ3 | ζ6 | ζ65 | ζ6 | ζ65 | -1 | ζ3 | ζ6 | ζ32 | ζ65 | 1 | -1 | 1 | linear of order 6 |
| ρ12 | 1 | 1 | -1 | -1 | 1 | ζ32 | ζ3 | ζ3 | ζ32 | ζ65 | ζ6 | ζ65 | ζ6 | -1 | ζ6 | ζ3 | ζ65 | ζ32 | 1 | 1 | 1 | linear of order 6 |
| ρ13 | 2 | 0 | -2 | 0 | -1 | 2 | 2 | -1 | -1 | -2 | -2 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 2 | 0 | -1 | orthogonal lifted from D6 |
| ρ14 | 2 | 0 | 2 | 0 | -1 | 2 | 2 | -1 | -1 | 2 | 2 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 2 | 0 | -1 | orthogonal lifted from S3 |
| ρ15 | 2 | 0 | -2 | 0 | -1 | -1+√-3 | -1-√-3 | ζ6 | ζ65 | 1+√-3 | 1-√-3 | ζ32 | ζ3 | 1 | 0 | 0 | 0 | 0 | 2 | 0 | -1 | complex lifted from S3×C6 |
| ρ16 | 2 | 0 | 2 | 0 | -1 | -1+√-3 | -1-√-3 | ζ6 | ζ65 | -1-√-3 | -1+√-3 | ζ6 | ζ65 | -1 | 0 | 0 | 0 | 0 | 2 | 0 | -1 | complex lifted from C3×S3 |
| ρ17 | 2 | 0 | 2 | 0 | -1 | -1-√-3 | -1+√-3 | ζ65 | ζ6 | -1+√-3 | -1-√-3 | ζ65 | ζ6 | -1 | 0 | 0 | 0 | 0 | 2 | 0 | -1 | complex lifted from C3×S3 |
| ρ18 | 2 | 0 | -2 | 0 | -1 | -1-√-3 | -1+√-3 | ζ65 | ζ6 | 1-√-3 | 1+√-3 | ζ3 | ζ32 | 1 | 0 | 0 | 0 | 0 | 2 | 0 | -1 | complex lifted from S3×C6 |
| ρ19 | 6 | -6 | 0 | 0 | 6 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 1 | -1 | orthogonal lifted from C2×F7 |
| ρ20 | 6 | 6 | 0 | 0 | 6 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | orthogonal lifted from F7 |
| ρ21 | 12 | 0 | 0 | 0 | -6 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 0 | 1 | orthogonal faithful |
►On 21 points - transitive group 21T15
Generators in S21
(1 8 15)(2 9 16)(3 10 17)(4 11 18)(5 12 19)(6 13 20)(7 14 21)
(8 15)(9 16)(10 17)(11 18)(12 19)(13 20)(14 21)
(1 2 3 4 5 6 7)(8 9 10 11 12 13 14)(15 16 17 18 19 20 21)
(2 4 3 7 5 6)(9 11 10 14 12 13)(16 18 17 21 19 20)
G:=sub<Sym(21)| (1,8,15)(2,9,16)(3,10,17)(4,11,18)(5,12,19)(6,13,20)(7,14,21), (8,15)(9,16)(10,17)(11,18)(12,19)(13,20)(14,21), (1,2,3,4,5,6,7)(8,9,10,11,12,13,14)(15,16,17,18,19,20,21), (2,4,3,7,5,6)(9,11,10,14,12,13)(16,18,17,21,19,20)>;
G:=Group( (1,8,15)(2,9,16)(3,10,17)(4,11,18)(5,12,19)(6,13,20)(7,14,21), (8,15)(9,16)(10,17)(11,18)(12,19)(13,20)(14,21), (1,2,3,4,5,6,7)(8,9,10,11,12,13,14)(15,16,17,18,19,20,21), (2,4,3,7,5,6)(9,11,10,14,12,13)(16,18,17,21,19,20) );
G=PermutationGroup([[(1,8,15),(2,9,16),(3,10,17),(4,11,18),(5,12,19),(6,13,20),(7,14,21)], [(8,15),(9,16),(10,17),(11,18),(12,19),(13,20),(14,21)], [(1,2,3,4,5,6,7),(8,9,10,11,12,13,14),(15,16,17,18,19,20,21)], [(2,4,3,7,5,6),(9,11,10,14,12,13),(16,18,17,21,19,20)]])
G:=TransitiveGroup(21,15);
Matrix representation of S3×F7 ►in GL8(𝔽43)
| 0 | 42 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 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 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 0 | 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 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 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 | 1 | 0 | 42 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 42 |
| 6 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 6 | 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))| [0,1,0,0,0,0,0,0,42,42,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,42,42,42,42,42,42],[6,0,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0] >;
S3×F7 in GAP, Magma, Sage, TeX
S_3\times F_7
% in TeX
G:=Group("S3xF7"); // GroupNames label
G:=SmallGroup(252,26);
// by ID
G=gap.SmallGroup(252,26);
# by ID
G:=PCGroup([5,-2,-2,-3,-3,-7,248,5404,914]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^2=c^7=d^6=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^5>;
// generators/relations
Export
Subgroup lattice of S3×F7 in TeX
Character table of S3×F7 in TeX