metacyclic, supersoluble, monomial, A-group
Aliases: C3⋊F7, D21⋊C3, C21⋊1C6, C7⋊C3⋊S3, C7⋊(C3×S3), (C3×C7⋊C3)⋊1C2, SmallGroup(126,9)
Series: Derived ►Chief ►Lower central ►Upper central
| C21 — C3⋊F7 |
Generators and relations for C3⋊F7
G = < a,b,c | a3=b7=c6=1, ab=ba, cac-1=a-1, cbc-1=b5 >
Character table of C3⋊F7
| class | 1 | 2 | 3A | 3B | 3C | 3D | 3E | 6A | 6B | 7 | 21A | 21B | |
| size | 1 | 21 | 2 | 7 | 7 | 14 | 14 | 21 | 21 | 6 | 6 | 6 | |
| ρ1 | 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 | linear of order 2 |
| ρ3 | 1 | 1 | 1 | ζ3 | ζ32 | ζ3 | ζ32 | ζ3 | ζ32 | 1 | 1 | 1 | linear of order 3 |
| ρ4 | 1 | 1 | 1 | ζ32 | ζ3 | ζ32 | ζ3 | ζ32 | ζ3 | 1 | 1 | 1 | linear of order 3 |
| ρ5 | 1 | -1 | 1 | ζ3 | ζ32 | ζ3 | ζ32 | ζ65 | ζ6 | 1 | 1 | 1 | linear of order 6 |
| ρ6 | 1 | -1 | 1 | ζ32 | ζ3 | ζ32 | ζ3 | ζ6 | ζ65 | 1 | 1 | 1 | linear of order 6 |
| ρ7 | 2 | 0 | -1 | 2 | 2 | -1 | -1 | 0 | 0 | 2 | -1 | -1 | orthogonal lifted from S3 |
| ρ8 | 2 | 0 | -1 | -1-√-3 | -1+√-3 | ζ6 | ζ65 | 0 | 0 | 2 | -1 | -1 | complex lifted from C3×S3 |
| ρ9 | 2 | 0 | -1 | -1+√-3 | -1-√-3 | ζ65 | ζ6 | 0 | 0 | 2 | -1 | -1 | complex lifted from C3×S3 |
| ρ10 | 6 | 0 | 6 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | orthogonal lifted from F7 |
| ρ11 | 6 | 0 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 1+√21/2 | 1-√21/2 | orthogonal faithful |
| ρ12 | 6 | 0 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 1-√21/2 | 1+√21/2 | orthogonal faithful |
►On 21 points - transitive group 21T10
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)
(2 4 3 7 5 6)(8 15)(9 18 10 21 12 20)(11 17 14 19 13 16)
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), (2,4,3,7,5,6)(8,15)(9,18,10,21,12,20)(11,17,14,19,13,16)>;
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), (2,4,3,7,5,6)(8,15)(9,18,10,21,12,20)(11,17,14,19,13,16) );
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)], [(2,4,3,7,5,6),(8,15),(9,18,10,21,12,20),(11,17,14,19,13,16)]])
G:=TransitiveGroup(21,10);
C3⋊F7 is a maximal subgroup of
S3×F7 C9⋊F7 C9⋊2F7 C9⋊5F7 C32⋊F7 C32⋊4F7
C3⋊F7 is a maximal quotient of C6.F7 C9⋊F7 C9⋊2F7 C9⋊5F7 D21⋊C9 C32⋊F7 C32⋊4F7
Matrix representation of C3⋊F7 ►in GL6(𝔽43)
| 40 | 38 | 38 | 0 | 38 | 0 |
| 0 | 40 | 38 | 38 | 0 | 38 |
| 5 | 5 | 2 | 0 | 0 | 5 |
| 38 | 0 | 0 | 40 | 38 | 38 |
| 5 | 0 | 5 | 5 | 2 | 0 |
| 0 | 5 | 0 | 5 | 5 | 2 |
| 42 | 42 | 42 | 42 | 42 | 42 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 42 | 42 | 42 | 42 | 42 | 42 |
| 0 | 0 | 0 | 0 | 1 | 0 |
G:=sub<GL(6,GF(43))| [40,0,5,38,5,0,38,40,5,0,0,5,38,38,2,0,5,0,0,38,0,40,5,5,38,0,0,38,2,5,0,38,5,38,0,2],[42,1,0,0,0,0,42,0,1,0,0,0,42,0,0,1,0,0,42,0,0,0,1,0,42,0,0,0,0,1,42,0,0,0,0,0],[1,0,0,0,42,0,0,0,0,1,42,0,0,0,0,0,42,0,0,0,1,0,42,0,0,0,0,0,42,1,0,1,0,0,42,0] >;
C3⋊F7 in GAP, Magma, Sage, TeX
C_3\rtimes F_7
% in TeX
G:=Group("C3:F7"); // GroupNames label
G:=SmallGroup(126,9);
// by ID
G=gap.SmallGroup(126,9);
# by ID
G:=PCGroup([4,-2,-3,-3,-7,146,1731,295]);
// Polycyclic
G:=Group<a,b,c|a^3=b^7=c^6=1,a*b=b*a,c*a*c^-1=a^-1,c*b*c^-1=b^5>;
// generators/relations
Export
Subgroup lattice of C3⋊F7 in TeX
Character table of C3⋊F7 in TeX