metabelian, supersoluble, monomial, A-group
Aliases: C7⋊3F7, C72⋊4C6, (C7×D7)⋊3C3, D7⋊1(C7⋊C3), C72⋊3C3⋊2C2, C7⋊1(C2×C7⋊C3), SmallGroup(294,11)
Series: Derived ►Chief ►Lower central ►Upper central
| C72 — C7⋊3F7 |
Generators and relations for C7⋊3F7
G = < a,b,c | a7=b7=c6=1, ab=ba, cac-1=a4, cbc-1=b5 >
Character table of C7⋊3F7
| class | 1 | 2 | 3A | 3B | 6A | 6B | 7A | 7B | 7C | 7D | 7E | 7F | 7G | 7H | 7I | 14A | 14B | |
| size | 1 | 7 | 49 | 49 | 49 | 49 | 3 | 3 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 21 | 21 | |
| ρ1 | 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 | linear of order 2 |
| ρ3 | 1 | 1 | ζ32 | ζ3 | ζ32 | ζ3 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 3 |
| ρ4 | 1 | 1 | ζ3 | ζ32 | ζ3 | ζ32 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 3 |
| ρ5 | 1 | -1 | ζ32 | ζ3 | ζ6 | ζ65 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | linear of order 6 |
| ρ6 | 1 | -1 | ζ3 | ζ32 | ζ65 | ζ6 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | linear of order 6 |
| ρ7 | 3 | 3 | 0 | 0 | 0 | 0 | -1-√-7/2 | -1+√-7/2 | -1-√-7/2 | -1-√-7/2 | 3 | -1-√-7/2 | -1+√-7/2 | -1+√-7/2 | -1+√-7/2 | -1+√-7/2 | -1-√-7/2 | complex lifted from C7⋊C3 |
| ρ8 | 3 | 3 | 0 | 0 | 0 | 0 | -1+√-7/2 | -1-√-7/2 | -1+√-7/2 | -1+√-7/2 | 3 | -1+√-7/2 | -1-√-7/2 | -1-√-7/2 | -1-√-7/2 | -1-√-7/2 | -1+√-7/2 | complex lifted from C7⋊C3 |
| ρ9 | 3 | -3 | 0 | 0 | 0 | 0 | -1-√-7/2 | -1+√-7/2 | -1-√-7/2 | -1-√-7/2 | 3 | -1-√-7/2 | -1+√-7/2 | -1+√-7/2 | -1+√-7/2 | 1-√-7/2 | 1+√-7/2 | complex lifted from C2×C7⋊C3 |
| ρ10 | 3 | -3 | 0 | 0 | 0 | 0 | -1+√-7/2 | -1-√-7/2 | -1+√-7/2 | -1+√-7/2 | 3 | -1+√-7/2 | -1-√-7/2 | -1-√-7/2 | -1-√-7/2 | 1+√-7/2 | 1-√-7/2 | complex lifted from C2×C7⋊C3 |
| ρ11 | 6 | 0 | 0 | 0 | 0 | 0 | 6 | 6 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | 0 | 0 | orthogonal lifted from F7 |
| ρ12 | 6 | 0 | 0 | 0 | 0 | 0 | -1+√-7 | -1-√-7 | 2ζ76+ζ75+2ζ72+1 | ζ76+2ζ73+2ζ7+1 | -1 | 2ζ75+2ζ74+ζ73+1 | 2ζ75+ζ72+2ζ7+1 | 2ζ76+2ζ74+ζ7+1 | ζ74+2ζ73+2ζ72+1 | 0 | 0 | complex faithful |
| ρ13 | 6 | 0 | 0 | 0 | 0 | 0 | -1+√-7 | -1-√-7 | 2ζ75+2ζ74+ζ73+1 | 2ζ76+ζ75+2ζ72+1 | -1 | ζ76+2ζ73+2ζ7+1 | ζ74+2ζ73+2ζ72+1 | 2ζ75+ζ72+2ζ7+1 | 2ζ76+2ζ74+ζ7+1 | 0 | 0 | complex faithful |
| ρ14 | 6 | 0 | 0 | 0 | 0 | 0 | -1-√-7 | -1+√-7 | ζ74+2ζ73+2ζ72+1 | 2ζ75+ζ72+2ζ7+1 | -1 | 2ζ76+2ζ74+ζ7+1 | 2ζ75+2ζ74+ζ73+1 | 2ζ76+ζ75+2ζ72+1 | ζ76+2ζ73+2ζ7+1 | 0 | 0 | complex faithful |
| ρ15 | 6 | 0 | 0 | 0 | 0 | 0 | -1+√-7 | -1-√-7 | ζ76+2ζ73+2ζ7+1 | 2ζ75+2ζ74+ζ73+1 | -1 | 2ζ76+ζ75+2ζ72+1 | 2ζ76+2ζ74+ζ7+1 | ζ74+2ζ73+2ζ72+1 | 2ζ75+ζ72+2ζ7+1 | 0 | 0 | complex faithful |
| ρ16 | 6 | 0 | 0 | 0 | 0 | 0 | -1-√-7 | -1+√-7 | 2ζ76+2ζ74+ζ7+1 | ζ74+2ζ73+2ζ72+1 | -1 | 2ζ75+ζ72+2ζ7+1 | ζ76+2ζ73+2ζ7+1 | 2ζ75+2ζ74+ζ73+1 | 2ζ76+ζ75+2ζ72+1 | 0 | 0 | complex faithful |
| ρ17 | 6 | 0 | 0 | 0 | 0 | 0 | -1-√-7 | -1+√-7 | 2ζ75+ζ72+2ζ7+1 | 2ζ76+2ζ74+ζ7+1 | -1 | ζ74+2ζ73+2ζ72+1 | 2ζ76+ζ75+2ζ72+1 | ζ76+2ζ73+2ζ7+1 | 2ζ75+2ζ74+ζ73+1 | 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 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 37 11 22 16 32)(2 39 8 23 18 29)(3 41 12 24 20 33)(4 36 9 25 15 30)(5 38 13 26 17 34)(6 40 10 27 19 31)(7 42 14 28 21 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), (1,2,3,4,5,6,7)(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), (1,37,11,22,16,32)(2,39,8,23,18,29)(3,41,12,24,20,33)(4,36,9,25,15,30)(5,38,13,26,17,34)(6,40,10,27,19,31)(7,42,14,28,21,35)>;
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), (1,2,3,4,5,6,7)(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), (1,37,11,22,16,32)(2,39,8,23,18,29)(3,41,12,24,20,33)(4,36,9,25,15,30)(5,38,13,26,17,34)(6,40,10,27,19,31)(7,42,14,28,21,35) );
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)], [(1,2,3,4,5,6,7),(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)], [(1,37,11,22,16,32),(2,39,8,23,18,29),(3,41,12,24,20,33),(4,36,9,25,15,30),(5,38,13,26,17,34),(6,40,10,27,19,31),(7,42,14,28,21,35)]])
Matrix representation of C7⋊3F7 ►in GL6(𝔽43)
| 35 | 0 | 0 | 0 | 0 | 0 |
| 0 | 21 | 0 | 0 | 0 | 0 |
| 0 | 0 | 11 | 0 | 0 | 0 |
| 0 | 0 | 0 | 35 | 0 | 0 |
| 0 | 0 | 0 | 0 | 21 | 0 |
| 0 | 0 | 0 | 0 | 0 | 11 |
| 35 | 0 | 0 | 0 | 0 | 0 |
| 0 | 11 | 0 | 0 | 0 | 0 |
| 0 | 0 | 21 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 0 | 41 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
G:=sub<GL(6,GF(43))| [35,0,0,0,0,0,0,21,0,0,0,0,0,0,11,0,0,0,0,0,0,35,0,0,0,0,0,0,21,0,0,0,0,0,0,11],[35,0,0,0,0,0,0,11,0,0,0,0,0,0,21,0,0,0,0,0,0,16,0,0,0,0,0,0,4,0,0,0,0,0,0,41],[0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,1,0,0,0,1,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0] >;
C7⋊3F7 in GAP, Magma, Sage, TeX
C_7\rtimes_3F_7
% in TeX
G:=Group("C7:3F7"); // GroupNames label
G:=SmallGroup(294,11);
// by ID
G=gap.SmallGroup(294,11);
# by ID
G:=PCGroup([4,-2,-3,-7,-7,78,4035,1351]);
// Polycyclic
G:=Group<a,b,c|a^7=b^7=c^6=1,a*b=b*a,c*a*c^-1=a^4,c*b*c^-1=b^5>;
// generators/relations
Export
Subgroup lattice of C7⋊3F7 in TeX
Character table of C7⋊3F7 in TeX