direct product, metacyclic, supersoluble, monomial, A-group
Aliases: C3×F7, D7⋊C32, C21⋊2C6, C7⋊C3⋊C6, C7⋊(C3×C6), (C3×D7)⋊C3, (C3×C7⋊C3)⋊2C2, SmallGroup(126,7)
Series: Derived ►Chief ►Lower central ►Upper central
| C7 — C3×F7 |
Generators and relations for C3×F7
G = < a,b,c | a3=b7=c6=1, ab=ba, ac=ca, cbc-1=b5 >
Character table of C3×F7
| class | 1 | 2 | 3A | 3B | 3C | 3D | 3E | 3F | 3G | 3H | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 6H | 7 | 21A | 21B | |
| size | 1 | 7 | 1 | 1 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 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 | 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 | ζ32 | ζ3 | 1 | 1 | ζ32 | ζ3 | ζ3 | ζ32 | ζ65 | ζ65 | ζ6 | ζ65 | ζ6 | -1 | ζ6 | -1 | 1 | ζ32 | ζ3 | linear of order 6 |
| ρ4 | 1 | 1 | ζ32 | ζ3 | 1 | 1 | ζ32 | ζ3 | ζ3 | ζ32 | ζ3 | ζ3 | ζ32 | ζ3 | ζ32 | 1 | ζ32 | 1 | 1 | ζ32 | ζ3 | linear of order 3 |
| ρ5 | 1 | -1 | ζ32 | ζ3 | ζ32 | ζ3 | ζ3 | ζ32 | 1 | 1 | -1 | ζ65 | ζ6 | ζ6 | -1 | ζ65 | ζ65 | ζ6 | 1 | ζ32 | ζ3 | linear of order 6 |
| ρ6 | 1 | -1 | 1 | 1 | ζ3 | ζ32 | ζ3 | ζ32 | ζ3 | ζ32 | ζ65 | -1 | -1 | ζ6 | ζ6 | ζ6 | ζ65 | ζ65 | 1 | 1 | 1 | linear of order 6 |
| ρ7 | 1 | -1 | ζ32 | ζ3 | ζ3 | ζ32 | 1 | 1 | ζ32 | ζ3 | ζ6 | ζ65 | ζ6 | -1 | ζ65 | ζ6 | -1 | ζ65 | 1 | ζ32 | ζ3 | linear of order 6 |
| ρ8 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | ζ3 | ζ32 | ζ3 | ζ32 | ζ3 | 1 | 1 | ζ32 | ζ32 | ζ32 | ζ3 | ζ3 | 1 | 1 | 1 | linear of order 3 |
| ρ9 | 1 | 1 | ζ3 | ζ32 | ζ32 | ζ3 | 1 | 1 | ζ3 | ζ32 | ζ3 | ζ32 | ζ3 | 1 | ζ32 | ζ3 | 1 | ζ32 | 1 | ζ3 | ζ32 | linear of order 3 |
| ρ10 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | ζ32 | ζ3 | ζ32 | ζ3 | ζ32 | 1 | 1 | ζ3 | ζ3 | ζ3 | ζ32 | ζ32 | 1 | 1 | 1 | linear of order 3 |
| ρ11 | 1 | 1 | ζ32 | ζ3 | ζ32 | ζ3 | ζ3 | ζ32 | 1 | 1 | 1 | ζ3 | ζ32 | ζ32 | 1 | ζ3 | ζ3 | ζ32 | 1 | ζ32 | ζ3 | linear of order 3 |
| ρ12 | 1 | 1 | ζ3 | ζ32 | ζ3 | ζ32 | ζ32 | ζ3 | 1 | 1 | 1 | ζ32 | ζ3 | ζ3 | 1 | ζ32 | ζ32 | ζ3 | 1 | ζ3 | ζ32 | linear of order 3 |
| ρ13 | 1 | 1 | ζ3 | ζ32 | 1 | 1 | ζ3 | ζ32 | ζ32 | ζ3 | ζ32 | ζ32 | ζ3 | ζ32 | ζ3 | 1 | ζ3 | 1 | 1 | ζ3 | ζ32 | linear of order 3 |
| ρ14 | 1 | -1 | ζ3 | ζ32 | 1 | 1 | ζ3 | ζ32 | ζ32 | ζ3 | ζ6 | ζ6 | ζ65 | ζ6 | ζ65 | -1 | ζ65 | -1 | 1 | ζ3 | ζ32 | linear of order 6 |
| ρ15 | 1 | -1 | ζ3 | ζ32 | ζ32 | ζ3 | 1 | 1 | ζ3 | ζ32 | ζ65 | ζ6 | ζ65 | -1 | ζ6 | ζ65 | -1 | ζ6 | 1 | ζ3 | ζ32 | linear of order 6 |
| ρ16 | 1 | -1 | ζ3 | ζ32 | ζ3 | ζ32 | ζ32 | ζ3 | 1 | 1 | -1 | ζ6 | ζ65 | ζ65 | -1 | ζ6 | ζ6 | ζ65 | 1 | ζ3 | ζ32 | linear of order 6 |
| ρ17 | 1 | -1 | 1 | 1 | ζ32 | ζ3 | ζ32 | ζ3 | ζ32 | ζ3 | ζ6 | -1 | -1 | ζ65 | ζ65 | ζ65 | ζ6 | ζ6 | 1 | 1 | 1 | linear of order 6 |
| ρ18 | 1 | 1 | ζ32 | ζ3 | ζ3 | ζ32 | 1 | 1 | ζ32 | ζ3 | ζ32 | ζ3 | ζ32 | 1 | ζ3 | ζ32 | 1 | ζ3 | 1 | ζ32 | ζ3 | linear of order 3 |
| ρ19 | 6 | 0 | 6 | 6 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | orthogonal lifted from F7 |
| ρ20 | 6 | 0 | -3+3√-3 | -3-3√-3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | ζ65 | ζ6 | complex faithful |
| ρ21 | 6 | 0 | -3-3√-3 | -3+3√-3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | ζ6 | ζ65 | complex faithful |
►On 21 points - transitive group 21T9
Generators in S21
(1 15 8)(2 16 9)(3 17 10)(4 18 11)(5 19 12)(6 20 13)(7 21 14)
(1 2 3 4 5 6 7)(8 9 10 11 12 13 14)(15 16 17 18 19 20 21)
(1 15 8)(2 18 10 7 19 13)(3 21 12 6 16 11)(4 17 14 5 20 9)
G:=sub<Sym(21)| (1,15,8)(2,16,9)(3,17,10)(4,18,11)(5,19,12)(6,20,13)(7,21,14), (1,2,3,4,5,6,7)(8,9,10,11,12,13,14)(15,16,17,18,19,20,21), (1,15,8)(2,18,10,7,19,13)(3,21,12,6,16,11)(4,17,14,5,20,9)>;
G:=Group( (1,15,8)(2,16,9)(3,17,10)(4,18,11)(5,19,12)(6,20,13)(7,21,14), (1,2,3,4,5,6,7)(8,9,10,11,12,13,14)(15,16,17,18,19,20,21), (1,15,8)(2,18,10,7,19,13)(3,21,12,6,16,11)(4,17,14,5,20,9) );
G=PermutationGroup([[(1,15,8),(2,16,9),(3,17,10),(4,18,11),(5,19,12),(6,20,13),(7,21,14)], [(1,2,3,4,5,6,7),(8,9,10,11,12,13,14),(15,16,17,18,19,20,21)], [(1,15,8),(2,18,10,7,19,13),(3,21,12,6,16,11),(4,17,14,5,20,9)]])
G:=TransitiveGroup(21,9);
C3×F7 is a maximal subgroup of
C9⋊3F7 C9⋊4F7 D7⋊He3
C3×F7 is a maximal quotient of C9⋊3F7 C9⋊4F7 C32.F7 D7⋊He3
Matrix representation of C3×F7 ►in GL7(𝔽43)
| 6 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 42 |
| 0 | 1 | 0 | 0 | 0 | 0 | 42 |
| 0 | 0 | 1 | 0 | 0 | 0 | 42 |
| 0 | 0 | 0 | 1 | 0 | 0 | 42 |
| 0 | 0 | 0 | 0 | 1 | 0 | 42 |
| 0 | 0 | 0 | 0 | 0 | 1 | 42 |
| 42 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 |
G:=sub<GL(7,GF(43))| [6,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,42,42,42,42,42,42],[42,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0] >;
C3×F7 in GAP, Magma, Sage, TeX
C_3\times F_7
% in TeX
G:=Group("C3xF7"); // GroupNames label
G:=SmallGroup(126,7);
// by ID
G=gap.SmallGroup(126,7);
# by ID
G:=PCGroup([4,-2,-3,-3,-7,1731,583]);
// Polycyclic
G:=Group<a,b,c|a^3=b^7=c^6=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^5>;
// generators/relations
Export
Subgroup lattice of C3×F7 in TeX
Character table of C3×F7 in TeX