metacyclic, supersoluble, monomial, Z-group
Aliases: C5⋊F7, D35⋊C3, C35⋊1C6, C7⋊C3⋊D5, C7⋊(C3×D5), (C5×C7⋊C3)⋊1C2, SmallGroup(210,3)
Series: Derived ►Chief ►Lower central ►Upper central
| C35 — C5⋊F7 |
Generators and relations for C5⋊F7
G = < a,b,c | a5=b7=c6=1, ab=ba, cac-1=a-1, cbc-1=b5 >
Character table of C5⋊F7
| class | 1 | 2 | 3A | 3B | 5A | 5B | 6A | 6B | 7 | 15A | 15B | 15C | 15D | 35A | 35B | 35C | 35D | |
| size | 1 | 35 | 7 | 7 | 2 | 2 | 35 | 35 | 6 | 14 | 14 | 14 | 14 | 6 | 6 | 6 | 6 | |
| 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 | 1 | 1 | ζ3 | ζ32 | 1 | ζ32 | ζ3 | ζ32 | ζ3 | 1 | 1 | 1 | 1 | linear of order 3 |
| ρ4 | 1 | -1 | ζ32 | ζ3 | 1 | 1 | ζ65 | ζ6 | 1 | ζ32 | ζ3 | ζ32 | ζ3 | 1 | 1 | 1 | 1 | linear of order 6 |
| ρ5 | 1 | 1 | ζ3 | ζ32 | 1 | 1 | ζ32 | ζ3 | 1 | ζ3 | ζ32 | ζ3 | ζ32 | 1 | 1 | 1 | 1 | linear of order 3 |
| ρ6 | 1 | -1 | ζ3 | ζ32 | 1 | 1 | ζ6 | ζ65 | 1 | ζ3 | ζ32 | ζ3 | ζ32 | 1 | 1 | 1 | 1 | linear of order 6 |
| ρ7 | 2 | 0 | 2 | 2 | -1-√5/2 | -1+√5/2 | 0 | 0 | 2 | -1-√5/2 | -1-√5/2 | -1+√5/2 | -1+√5/2 | -1+√5/2 | -1+√5/2 | -1-√5/2 | -1-√5/2 | orthogonal lifted from D5 |
| ρ8 | 2 | 0 | 2 | 2 | -1+√5/2 | -1-√5/2 | 0 | 0 | 2 | -1+√5/2 | -1+√5/2 | -1-√5/2 | -1-√5/2 | -1-√5/2 | -1-√5/2 | -1+√5/2 | -1+√5/2 | orthogonal lifted from D5 |
| ρ9 | 2 | 0 | -1-√-3 | -1+√-3 | -1-√5/2 | -1+√5/2 | 0 | 0 | 2 | ζ32ζ53+ζ32ζ52 | ζ3ζ53+ζ3ζ52 | ζ32ζ54+ζ32ζ5 | ζ3ζ54+ζ3ζ5 | -1+√5/2 | -1+√5/2 | -1-√5/2 | -1-√5/2 | complex lifted from C3×D5 |
| ρ10 | 2 | 0 | -1+√-3 | -1-√-3 | -1-√5/2 | -1+√5/2 | 0 | 0 | 2 | ζ3ζ53+ζ3ζ52 | ζ32ζ53+ζ32ζ52 | ζ3ζ54+ζ3ζ5 | ζ32ζ54+ζ32ζ5 | -1+√5/2 | -1+√5/2 | -1-√5/2 | -1-√5/2 | complex lifted from C3×D5 |
| ρ11 | 2 | 0 | -1-√-3 | -1+√-3 | -1+√5/2 | -1-√5/2 | 0 | 0 | 2 | ζ32ζ54+ζ32ζ5 | ζ3ζ54+ζ3ζ5 | ζ32ζ53+ζ32ζ52 | ζ3ζ53+ζ3ζ52 | -1-√5/2 | -1-√5/2 | -1+√5/2 | -1+√5/2 | complex lifted from C3×D5 |
| ρ12 | 2 | 0 | -1+√-3 | -1-√-3 | -1+√5/2 | -1-√5/2 | 0 | 0 | 2 | ζ3ζ54+ζ3ζ5 | ζ32ζ54+ζ32ζ5 | ζ3ζ53+ζ3ζ52 | ζ32ζ53+ζ32ζ52 | -1-√5/2 | -1-√5/2 | -1+√5/2 | -1+√5/2 | complex lifted from C3×D5 |
| ρ13 | 6 | 0 | 0 | 0 | 6 | 6 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | orthogonal lifted from F7 |
| ρ14 | 6 | 0 | 0 | 0 | -3+3√5/2 | -3-3√5/2 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | -ζ53ζ74-ζ53ζ72-ζ53ζ7-ζ53+ζ52ζ74+ζ52ζ72+ζ52ζ7 | ζ53ζ74+ζ53ζ72+ζ53ζ7-ζ52ζ74-ζ52ζ72-ζ52ζ7-ζ52 | -ζ54ζ74-ζ54ζ72-ζ54ζ7-ζ54+ζ5ζ74+ζ5ζ72+ζ5ζ7 | ζ54ζ74+ζ54ζ72+ζ54ζ7-ζ5ζ74-ζ5ζ72-ζ5ζ7-ζ5 | orthogonal faithful |
| ρ15 | 6 | 0 | 0 | 0 | -3+3√5/2 | -3-3√5/2 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | ζ53ζ74+ζ53ζ72+ζ53ζ7-ζ52ζ74-ζ52ζ72-ζ52ζ7-ζ52 | -ζ53ζ74-ζ53ζ72-ζ53ζ7-ζ53+ζ52ζ74+ζ52ζ72+ζ52ζ7 | ζ54ζ74+ζ54ζ72+ζ54ζ7-ζ5ζ74-ζ5ζ72-ζ5ζ7-ζ5 | -ζ54ζ74-ζ54ζ72-ζ54ζ7-ζ54+ζ5ζ74+ζ5ζ72+ζ5ζ7 | orthogonal faithful |
| ρ16 | 6 | 0 | 0 | 0 | -3-3√5/2 | -3+3√5/2 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | ζ54ζ74+ζ54ζ72+ζ54ζ7-ζ5ζ74-ζ5ζ72-ζ5ζ7-ζ5 | -ζ54ζ74-ζ54ζ72-ζ54ζ7-ζ54+ζ5ζ74+ζ5ζ72+ζ5ζ7 | -ζ53ζ74-ζ53ζ72-ζ53ζ7-ζ53+ζ52ζ74+ζ52ζ72+ζ52ζ7 | ζ53ζ74+ζ53ζ72+ζ53ζ7-ζ52ζ74-ζ52ζ72-ζ52ζ7-ζ52 | orthogonal faithful |
| ρ17 | 6 | 0 | 0 | 0 | -3-3√5/2 | -3+3√5/2 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | -ζ54ζ74-ζ54ζ72-ζ54ζ7-ζ54+ζ5ζ74+ζ5ζ72+ζ5ζ7 | ζ54ζ74+ζ54ζ72+ζ54ζ7-ζ5ζ74-ζ5ζ72-ζ5ζ7-ζ5 | ζ53ζ74+ζ53ζ72+ζ53ζ7-ζ52ζ74-ζ52ζ72-ζ52ζ7-ζ52 | -ζ53ζ74-ζ53ζ72-ζ53ζ7-ζ53+ζ52ζ74+ζ52ζ72+ζ52ζ7 | orthogonal faithful |
►On 35 points
Generators in S35
(1 29 22 15 8)(2 30 23 16 9)(3 31 24 17 10)(4 32 25 18 11)(5 33 26 19 12)(6 34 27 20 13)(7 35 28 21 14)
(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)
(2 4 3 7 5 6)(8 29)(9 32 10 35 12 34)(11 31 14 33 13 30)(15 22)(16 25 17 28 19 27)(18 24 21 26 20 23)
G:=sub<Sym(35)| (1,29,22,15,8)(2,30,23,16,9)(3,31,24,17,10)(4,32,25,18,11)(5,33,26,19,12)(6,34,27,20,13)(7,35,28,21,14), (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), (2,4,3,7,5,6)(8,29)(9,32,10,35,12,34)(11,31,14,33,13,30)(15,22)(16,25,17,28,19,27)(18,24,21,26,20,23)>;
G:=Group( (1,29,22,15,8)(2,30,23,16,9)(3,31,24,17,10)(4,32,25,18,11)(5,33,26,19,12)(6,34,27,20,13)(7,35,28,21,14), (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), (2,4,3,7,5,6)(8,29)(9,32,10,35,12,34)(11,31,14,33,13,30)(15,22)(16,25,17,28,19,27)(18,24,21,26,20,23) );
G=PermutationGroup([[(1,29,22,15,8),(2,30,23,16,9),(3,31,24,17,10),(4,32,25,18,11),(5,33,26,19,12),(6,34,27,20,13),(7,35,28,21,14)], [(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)], [(2,4,3,7,5,6),(8,29),(9,32,10,35,12,34),(11,31,14,33,13,30),(15,22),(16,25,17,28,19,27),(18,24,21,26,20,23)]])
C5⋊F7 is a maximal subgroup of
D5×F7
C5⋊F7 is a maximal quotient of C35⋊3C12
Matrix representation of C5⋊F7 ►in GL6(𝔽211)
| 104 | 0 | 181 | 30 | 181 | 0 |
| 30 | 104 | 181 | 0 | 0 | 181 |
| 30 | 30 | 74 | 0 | 181 | 0 |
| 0 | 30 | 0 | 104 | 181 | 181 |
| 30 | 0 | 0 | 30 | 74 | 181 |
| 0 | 30 | 181 | 30 | 0 | 74 |
| 0 | 0 | 0 | 0 | 0 | 210 |
| 1 | 0 | 0 | 0 | 0 | 210 |
| 0 | 1 | 0 | 0 | 0 | 210 |
| 0 | 0 | 1 | 0 | 0 | 210 |
| 0 | 0 | 0 | 1 | 0 | 210 |
| 0 | 0 | 0 | 0 | 1 | 210 |
| 0 | 0 | 0 | 0 | 1 | 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 | 1 | 0 | 0 | 0 | 0 |
G:=sub<GL(6,GF(211))| [104,30,30,0,30,0,0,104,30,30,0,30,181,181,74,0,0,181,30,0,0,104,30,30,181,0,181,181,74,0,0,181,0,181,181,74],[0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,210,210,210,210,210,210],[0,0,1,0,0,0,0,0,0,0,0,1,0,1,0,0,0,0,0,0,0,0,1,0,1,0,0,0,0,0,0,0,0,1,0,0] >;
C5⋊F7 in GAP, Magma, Sage, TeX
C_5\rtimes F_7
% in TeX
G:=Group("C5:F7"); // GroupNames label
G:=SmallGroup(210,3);
// by ID
G=gap.SmallGroup(210,3);
# by ID
G:=PCGroup([4,-2,-3,-5,-7,290,2883,487]);
// Polycyclic
G:=Group<a,b,c|a^5=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 C5⋊F7 in TeX
Character table of C5⋊F7 in TeX