metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C7⋊F5, C5⋊Dic7, C35⋊1C4, D5.D7, (C7×D5).1C2, SmallGroup(140,6)
Series: Derived ►Chief ►Lower central ►Upper central
| C35 — C7⋊F5 |
Generators and relations for C7⋊F5
G = < a,b,c | a7=b5=c4=1, ab=ba, cac-1=a-1, cbc-1=b3 >
Character table of C7⋊F5
| class | 1 | 2 | 4A | 4B | 5 | 7A | 7B | 7C | 14A | 14B | 14C | 35A | 35B | 35C | 35D | 35E | 35F | |
| size | 1 | 5 | 35 | 35 | 4 | 2 | 2 | 2 | 10 | 10 | 10 | 4 | 4 | 4 | 4 | 4 | 4 | |
| 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 | i | -i | 1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 4 |
| ρ4 | 1 | -1 | -i | i | 1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 4 |
| ρ5 | 2 | 2 | 0 | 0 | 2 | ζ76+ζ7 | ζ74+ζ73 | ζ75+ζ72 | ζ76+ζ7 | ζ75+ζ72 | ζ74+ζ73 | ζ76+ζ7 | ζ74+ζ73 | ζ75+ζ72 | ζ75+ζ72 | ζ74+ζ73 | ζ76+ζ7 | orthogonal lifted from D7 |
| ρ6 | 2 | 2 | 0 | 0 | 2 | ζ75+ζ72 | ζ76+ζ7 | ζ74+ζ73 | ζ75+ζ72 | ζ74+ζ73 | ζ76+ζ7 | ζ75+ζ72 | ζ76+ζ7 | ζ74+ζ73 | ζ74+ζ73 | ζ76+ζ7 | ζ75+ζ72 | orthogonal lifted from D7 |
| ρ7 | 2 | 2 | 0 | 0 | 2 | ζ74+ζ73 | ζ75+ζ72 | ζ76+ζ7 | ζ74+ζ73 | ζ76+ζ7 | ζ75+ζ72 | ζ74+ζ73 | ζ75+ζ72 | ζ76+ζ7 | ζ76+ζ7 | ζ75+ζ72 | ζ74+ζ73 | orthogonal lifted from D7 |
| ρ8 | 2 | -2 | 0 | 0 | 2 | ζ76+ζ7 | ζ74+ζ73 | ζ75+ζ72 | -ζ76-ζ7 | -ζ75-ζ72 | -ζ74-ζ73 | ζ76+ζ7 | ζ74+ζ73 | ζ75+ζ72 | ζ75+ζ72 | ζ74+ζ73 | ζ76+ζ7 | symplectic lifted from Dic7, Schur index 2 |
| ρ9 | 2 | -2 | 0 | 0 | 2 | ζ74+ζ73 | ζ75+ζ72 | ζ76+ζ7 | -ζ74-ζ73 | -ζ76-ζ7 | -ζ75-ζ72 | ζ74+ζ73 | ζ75+ζ72 | ζ76+ζ7 | ζ76+ζ7 | ζ75+ζ72 | ζ74+ζ73 | symplectic lifted from Dic7, Schur index 2 |
| ρ10 | 2 | -2 | 0 | 0 | 2 | ζ75+ζ72 | ζ76+ζ7 | ζ74+ζ73 | -ζ75-ζ72 | -ζ74-ζ73 | -ζ76-ζ7 | ζ75+ζ72 | ζ76+ζ7 | ζ74+ζ73 | ζ74+ζ73 | ζ76+ζ7 | ζ75+ζ72 | symplectic lifted from Dic7, Schur index 2 |
| ρ11 | 4 | 0 | 0 | 0 | -1 | 4 | 4 | 4 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from F5 |
| ρ12 | 4 | 0 | 0 | 0 | -1 | 2ζ76+2ζ7 | 2ζ74+2ζ73 | 2ζ75+2ζ72 | 0 | 0 | 0 | ζ54ζ76-ζ54ζ7+ζ5ζ76-ζ5ζ7-ζ7 | -ζ53ζ74+ζ53ζ73-ζ52ζ74+ζ52ζ73-ζ74 | -ζ54ζ75+ζ54ζ72-ζ5ζ75+ζ5ζ72-ζ75 | ζ54ζ75-ζ54ζ72+ζ5ζ75-ζ5ζ72-ζ72 | ζ53ζ74-ζ53ζ73+ζ52ζ74-ζ52ζ73-ζ73 | ζ53ζ76-ζ53ζ7+ζ52ζ76-ζ52ζ7-ζ7 | complex faithful |
| ρ13 | 4 | 0 | 0 | 0 | -1 | 2ζ75+2ζ72 | 2ζ76+2ζ7 | 2ζ74+2ζ73 | 0 | 0 | 0 | -ζ54ζ75+ζ54ζ72-ζ5ζ75+ζ5ζ72-ζ75 | ζ54ζ76-ζ54ζ7+ζ5ζ76-ζ5ζ7-ζ7 | ζ53ζ74-ζ53ζ73+ζ52ζ74-ζ52ζ73-ζ73 | -ζ53ζ74+ζ53ζ73-ζ52ζ74+ζ52ζ73-ζ74 | ζ53ζ76-ζ53ζ7+ζ52ζ76-ζ52ζ7-ζ7 | ζ54ζ75-ζ54ζ72+ζ5ζ75-ζ5ζ72-ζ72 | complex faithful |
| ρ14 | 4 | 0 | 0 | 0 | -1 | 2ζ74+2ζ73 | 2ζ75+2ζ72 | 2ζ76+2ζ7 | 0 | 0 | 0 | ζ53ζ74-ζ53ζ73+ζ52ζ74-ζ52ζ73-ζ73 | -ζ54ζ75+ζ54ζ72-ζ5ζ75+ζ5ζ72-ζ75 | ζ53ζ76-ζ53ζ7+ζ52ζ76-ζ52ζ7-ζ7 | ζ54ζ76-ζ54ζ7+ζ5ζ76-ζ5ζ7-ζ7 | ζ54ζ75-ζ54ζ72+ζ5ζ75-ζ5ζ72-ζ72 | -ζ53ζ74+ζ53ζ73-ζ52ζ74+ζ52ζ73-ζ74 | complex faithful |
| ρ15 | 4 | 0 | 0 | 0 | -1 | 2ζ74+2ζ73 | 2ζ75+2ζ72 | 2ζ76+2ζ7 | 0 | 0 | 0 | -ζ53ζ74+ζ53ζ73-ζ52ζ74+ζ52ζ73-ζ74 | ζ54ζ75-ζ54ζ72+ζ5ζ75-ζ5ζ72-ζ72 | ζ54ζ76-ζ54ζ7+ζ5ζ76-ζ5ζ7-ζ7 | ζ53ζ76-ζ53ζ7+ζ52ζ76-ζ52ζ7-ζ7 | -ζ54ζ75+ζ54ζ72-ζ5ζ75+ζ5ζ72-ζ75 | ζ53ζ74-ζ53ζ73+ζ52ζ74-ζ52ζ73-ζ73 | complex faithful |
| ρ16 | 4 | 0 | 0 | 0 | -1 | 2ζ76+2ζ7 | 2ζ74+2ζ73 | 2ζ75+2ζ72 | 0 | 0 | 0 | ζ53ζ76-ζ53ζ7+ζ52ζ76-ζ52ζ7-ζ7 | ζ53ζ74-ζ53ζ73+ζ52ζ74-ζ52ζ73-ζ73 | ζ54ζ75-ζ54ζ72+ζ5ζ75-ζ5ζ72-ζ72 | -ζ54ζ75+ζ54ζ72-ζ5ζ75+ζ5ζ72-ζ75 | -ζ53ζ74+ζ53ζ73-ζ52ζ74+ζ52ζ73-ζ74 | ζ54ζ76-ζ54ζ7+ζ5ζ76-ζ5ζ7-ζ7 | complex faithful |
| ρ17 | 4 | 0 | 0 | 0 | -1 | 2ζ75+2ζ72 | 2ζ76+2ζ7 | 2ζ74+2ζ73 | 0 | 0 | 0 | ζ54ζ75-ζ54ζ72+ζ5ζ75-ζ5ζ72-ζ72 | ζ53ζ76-ζ53ζ7+ζ52ζ76-ζ52ζ7-ζ7 | -ζ53ζ74+ζ53ζ73-ζ52ζ74+ζ52ζ73-ζ74 | ζ53ζ74-ζ53ζ73+ζ52ζ74-ζ52ζ73-ζ73 | ζ54ζ76-ζ54ζ7+ζ5ζ76-ζ5ζ7-ζ7 | -ζ54ζ75+ζ54ζ72-ζ5ζ75+ζ5ζ72-ζ75 | complex faithful |
►On 35 points
Generators in S35
(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)
(1 13 20 27 34)(2 14 21 28 35)(3 8 15 22 29)(4 9 16 23 30)(5 10 17 24 31)(6 11 18 25 32)(7 12 19 26 33)
(2 7)(3 6)(4 5)(8 18 29 25)(9 17 30 24)(10 16 31 23)(11 15 32 22)(12 21 33 28)(13 20 34 27)(14 19 35 26)
G:=sub<Sym(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), (1,13,20,27,34)(2,14,21,28,35)(3,8,15,22,29)(4,9,16,23,30)(5,10,17,24,31)(6,11,18,25,32)(7,12,19,26,33), (2,7)(3,6)(4,5)(8,18,29,25)(9,17,30,24)(10,16,31,23)(11,15,32,22)(12,21,33,28)(13,20,34,27)(14,19,35,26)>;
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), (1,13,20,27,34)(2,14,21,28,35)(3,8,15,22,29)(4,9,16,23,30)(5,10,17,24,31)(6,11,18,25,32)(7,12,19,26,33), (2,7)(3,6)(4,5)(8,18,29,25)(9,17,30,24)(10,16,31,23)(11,15,32,22)(12,21,33,28)(13,20,34,27)(14,19,35,26) );
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)], [(1,13,20,27,34),(2,14,21,28,35),(3,8,15,22,29),(4,9,16,23,30),(5,10,17,24,31),(6,11,18,25,32),(7,12,19,26,33)], [(2,7),(3,6),(4,5),(8,18,29,25),(9,17,30,24),(10,16,31,23),(11,15,32,22),(12,21,33,28),(13,20,34,27),(14,19,35,26)]])
C7⋊F5 is a maximal subgroup of
D7×F5 C35⋊C12 C5⋊Dic21
C7⋊F5 is a maximal quotient of C35⋊C8 C5⋊Dic21
Matrix representation of C7⋊F5 ►in GL4(𝔽281) generated by
| 280 | 1 | 0 | 0 |
| 39 | 241 | 0 | 0 |
| 0 | 0 | 280 | 1 |
| 0 | 0 | 39 | 241 |
| 267 | 201 | 280 | 0 |
| 252 | 15 | 0 | 280 |
| 268 | 201 | 280 | 0 |
| 252 | 16 | 0 | 280 |
| 27 | 94 | 254 | 187 |
| 58 | 254 | 223 | 27 |
| 40 | 1 | 0 | 0 |
| 87 | 241 | 0 | 0 |
G:=sub<GL(4,GF(281))| [280,39,0,0,1,241,0,0,0,0,280,39,0,0,1,241],[267,252,268,252,201,15,201,16,280,0,280,0,0,280,0,280],[27,58,40,87,94,254,1,241,254,223,0,0,187,27,0,0] >;
C7⋊F5 in GAP, Magma, Sage, TeX
C_7\rtimes F_5
% in TeX
G:=Group("C7:F5"); // GroupNames label
G:=SmallGroup(140,6);
// by ID
G=gap.SmallGroup(140,6);
# by ID
G:=PCGroup([4,-2,-2,-5,-7,8,146,102,1923]);
// Polycyclic
G:=Group<a,b,c|a^7=b^5=c^4=1,a*b=b*a,c*a*c^-1=a^-1,c*b*c^-1=b^3>;
// generators/relations
Export
Subgroup lattice of C7⋊F5 in TeX
Character table of C7⋊F5 in TeX