direct product, metacyclic, supersoluble, monomial, Z-group
Aliases: F5×C7⋊C3, C35⋊2C12, (C7×F5)⋊C3, C7⋊2(C3×F5), (C7×D5).2C6, C5⋊(C4×C7⋊C3), (C5×C7⋊C3)⋊2C4, D5.(C2×C7⋊C3), (D5×C7⋊C3).2C2, SmallGroup(420,14)
Series: Derived ►Chief ►Lower central ►Upper central
| C35 — F5×C7⋊C3 |
Generators and relations for F5×C7⋊C3
G = < a,b,c,d | a5=b4=c7=d3=1, bab-1=a3, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c4 >
Character table of F5×C7⋊C3
| class | 1 | 2 | 3A | 3B | 4A | 4B | 5 | 6A | 6B | 7A | 7B | 12A | 12B | 12C | 12D | 14A | 14B | 15A | 15B | 28A | 28B | 28C | 28D | 35A | 35B | |
| size | 1 | 5 | 7 | 7 | 5 | 5 | 4 | 35 | 35 | 3 | 3 | 35 | 35 | 35 | 35 | 15 | 15 | 28 | 28 | 15 | 15 | 15 | 15 | 12 | 12 | |
| ρ1 | 1 | 1 | 1 | 1 | 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 | -1 | -1 | 1 | 1 | linear of order 2 |
| ρ3 | 1 | 1 | ζ32 | ζ3 | 1 | 1 | 1 | ζ3 | ζ32 | 1 | 1 | ζ32 | ζ3 | ζ3 | ζ32 | 1 | 1 | ζ3 | ζ32 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 3 |
| ρ4 | 1 | 1 | ζ32 | ζ3 | -1 | -1 | 1 | ζ3 | ζ32 | 1 | 1 | ζ6 | ζ65 | ζ65 | ζ6 | 1 | 1 | ζ3 | ζ32 | -1 | -1 | -1 | -1 | 1 | 1 | linear of order 6 |
| ρ5 | 1 | 1 | ζ3 | ζ32 | -1 | -1 | 1 | ζ32 | ζ3 | 1 | 1 | ζ65 | ζ6 | ζ6 | ζ65 | 1 | 1 | ζ32 | ζ3 | -1 | -1 | -1 | -1 | 1 | 1 | linear of order 6 |
| ρ6 | 1 | 1 | ζ3 | ζ32 | 1 | 1 | 1 | ζ32 | ζ3 | 1 | 1 | ζ3 | ζ32 | ζ32 | ζ3 | 1 | 1 | ζ32 | ζ3 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 3 |
| ρ7 | 1 | -1 | 1 | 1 | i | -i | 1 | -1 | -1 | 1 | 1 | i | -i | i | -i | -1 | -1 | 1 | 1 | i | -i | -i | i | 1 | 1 | linear of order 4 |
| ρ8 | 1 | -1 | 1 | 1 | -i | i | 1 | -1 | -1 | 1 | 1 | -i | i | -i | i | -1 | -1 | 1 | 1 | -i | i | i | -i | 1 | 1 | linear of order 4 |
| ρ9 | 1 | -1 | ζ3 | ζ32 | -i | i | 1 | ζ6 | ζ65 | 1 | 1 | ζ43ζ3 | ζ4ζ32 | ζ43ζ32 | ζ4ζ3 | -1 | -1 | ζ32 | ζ3 | -i | i | i | -i | 1 | 1 | linear of order 12 |
| ρ10 | 1 | -1 | ζ32 | ζ3 | -i | i | 1 | ζ65 | ζ6 | 1 | 1 | ζ43ζ32 | ζ4ζ3 | ζ43ζ3 | ζ4ζ32 | -1 | -1 | ζ3 | ζ32 | -i | i | i | -i | 1 | 1 | linear of order 12 |
| ρ11 | 1 | -1 | ζ32 | ζ3 | i | -i | 1 | ζ65 | ζ6 | 1 | 1 | ζ4ζ32 | ζ43ζ3 | ζ4ζ3 | ζ43ζ32 | -1 | -1 | ζ3 | ζ32 | i | -i | -i | i | 1 | 1 | linear of order 12 |
| ρ12 | 1 | -1 | ζ3 | ζ32 | i | -i | 1 | ζ6 | ζ65 | 1 | 1 | ζ4ζ3 | ζ43ζ32 | ζ4ζ32 | ζ43ζ3 | -1 | -1 | ζ32 | ζ3 | i | -i | -i | i | 1 | 1 | linear of order 12 |
| ρ13 | 3 | 3 | 0 | 0 | -3 | -3 | 3 | 0 | 0 | -1+√-7/2 | -1-√-7/2 | 0 | 0 | 0 | 0 | -1-√-7/2 | -1+√-7/2 | 0 | 0 | 1-√-7/2 | 1-√-7/2 | 1+√-7/2 | 1+√-7/2 | -1+√-7/2 | -1-√-7/2 | complex lifted from C2×C7⋊C3 |
| ρ14 | 3 | 3 | 0 | 0 | 3 | 3 | 3 | 0 | 0 | -1-√-7/2 | -1+√-7/2 | 0 | 0 | 0 | 0 | -1+√-7/2 | -1-√-7/2 | 0 | 0 | -1-√-7/2 | -1-√-7/2 | -1+√-7/2 | -1+√-7/2 | -1-√-7/2 | -1+√-7/2 | complex lifted from C7⋊C3 |
| ρ15 | 3 | 3 | 0 | 0 | -3 | -3 | 3 | 0 | 0 | -1-√-7/2 | -1+√-7/2 | 0 | 0 | 0 | 0 | -1+√-7/2 | -1-√-7/2 | 0 | 0 | 1+√-7/2 | 1+√-7/2 | 1-√-7/2 | 1-√-7/2 | -1-√-7/2 | -1+√-7/2 | complex lifted from C2×C7⋊C3 |
| ρ16 | 3 | 3 | 0 | 0 | 3 | 3 | 3 | 0 | 0 | -1+√-7/2 | -1-√-7/2 | 0 | 0 | 0 | 0 | -1-√-7/2 | -1+√-7/2 | 0 | 0 | -1+√-7/2 | -1+√-7/2 | -1-√-7/2 | -1-√-7/2 | -1+√-7/2 | -1-√-7/2 | complex lifted from C7⋊C3 |
| ρ17 | 3 | -3 | 0 | 0 | 3i | -3i | 3 | 0 | 0 | -1-√-7/2 | -1+√-7/2 | 0 | 0 | 0 | 0 | 1-√-7/2 | 1+√-7/2 | 0 | 0 | ζ4ζ76+ζ4ζ75+ζ4ζ73 | ζ43ζ76+ζ43ζ75+ζ43ζ73 | ζ43ζ74+ζ43ζ72+ζ43ζ7 | ζ4ζ74+ζ4ζ72+ζ4ζ7 | -1-√-7/2 | -1+√-7/2 | complex lifted from C4×C7⋊C3 |
| ρ18 | 3 | -3 | 0 | 0 | -3i | 3i | 3 | 0 | 0 | -1-√-7/2 | -1+√-7/2 | 0 | 0 | 0 | 0 | 1-√-7/2 | 1+√-7/2 | 0 | 0 | ζ43ζ76+ζ43ζ75+ζ43ζ73 | ζ4ζ76+ζ4ζ75+ζ4ζ73 | ζ4ζ74+ζ4ζ72+ζ4ζ7 | ζ43ζ74+ζ43ζ72+ζ43ζ7 | -1-√-7/2 | -1+√-7/2 | complex lifted from C4×C7⋊C3 |
| ρ19 | 3 | -3 | 0 | 0 | 3i | -3i | 3 | 0 | 0 | -1+√-7/2 | -1-√-7/2 | 0 | 0 | 0 | 0 | 1+√-7/2 | 1-√-7/2 | 0 | 0 | ζ4ζ74+ζ4ζ72+ζ4ζ7 | ζ43ζ74+ζ43ζ72+ζ43ζ7 | ζ43ζ76+ζ43ζ75+ζ43ζ73 | ζ4ζ76+ζ4ζ75+ζ4ζ73 | -1+√-7/2 | -1-√-7/2 | complex lifted from C4×C7⋊C3 |
| ρ20 | 3 | -3 | 0 | 0 | -3i | 3i | 3 | 0 | 0 | -1+√-7/2 | -1-√-7/2 | 0 | 0 | 0 | 0 | 1+√-7/2 | 1-√-7/2 | 0 | 0 | ζ43ζ74+ζ43ζ72+ζ43ζ7 | ζ4ζ74+ζ4ζ72+ζ4ζ7 | ζ4ζ76+ζ4ζ75+ζ4ζ73 | ζ43ζ76+ζ43ζ75+ζ43ζ73 | -1+√-7/2 | -1-√-7/2 | complex lifted from C4×C7⋊C3 |
| ρ21 | 4 | 0 | 4 | 4 | 0 | 0 | -1 | 0 | 0 | 4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | 0 | 0 | 0 | 0 | -1 | -1 | orthogonal lifted from F5 |
| ρ22 | 4 | 0 | -2+2√-3 | -2-2√-3 | 0 | 0 | -1 | 0 | 0 | 4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | ζ6 | ζ65 | 0 | 0 | 0 | 0 | -1 | -1 | complex lifted from C3×F5 |
| ρ23 | 4 | 0 | -2-2√-3 | -2+2√-3 | 0 | 0 | -1 | 0 | 0 | 4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | ζ65 | ζ6 | 0 | 0 | 0 | 0 | -1 | -1 | complex lifted from C3×F5 |
| ρ24 | 12 | 0 | 0 | 0 | 0 | 0 | -3 | 0 | 0 | -2-2√-7 | -2+2√-7 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1+√-7/2 | 1-√-7/2 | complex faithful |
| ρ25 | 12 | 0 | 0 | 0 | 0 | 0 | -3 | 0 | 0 | -2+2√-7 | -2-2√-7 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1-√-7/2 | 1+√-7/2 | complex faithful |
►On 35 points
Generators in S35
(1 8 15 22 29)(2 9 16 23 30)(3 10 17 24 31)(4 11 18 25 32)(5 12 19 26 33)(6 13 20 27 34)(7 14 21 28 35)
(8 15 29 22)(9 16 30 23)(10 17 31 24)(11 18 32 25)(12 19 33 26)(13 20 34 27)(14 21 35 28)
(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 3 5)(4 7 6)(9 10 12)(11 14 13)(16 17 19)(18 21 20)(23 24 26)(25 28 27)(30 31 33)(32 35 34)
G:=sub<Sym(35)| (1,8,15,22,29)(2,9,16,23,30)(3,10,17,24,31)(4,11,18,25,32)(5,12,19,26,33)(6,13,20,27,34)(7,14,21,28,35), (8,15,29,22)(9,16,30,23)(10,17,31,24)(11,18,32,25)(12,19,33,26)(13,20,34,27)(14,21,35,28), (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), (2,3,5)(4,7,6)(9,10,12)(11,14,13)(16,17,19)(18,21,20)(23,24,26)(25,28,27)(30,31,33)(32,35,34)>;
G:=Group( (1,8,15,22,29)(2,9,16,23,30)(3,10,17,24,31)(4,11,18,25,32)(5,12,19,26,33)(6,13,20,27,34)(7,14,21,28,35), (8,15,29,22)(9,16,30,23)(10,17,31,24)(11,18,32,25)(12,19,33,26)(13,20,34,27)(14,21,35,28), (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), (2,3,5)(4,7,6)(9,10,12)(11,14,13)(16,17,19)(18,21,20)(23,24,26)(25,28,27)(30,31,33)(32,35,34) );
G=PermutationGroup([[(1,8,15,22,29),(2,9,16,23,30),(3,10,17,24,31),(4,11,18,25,32),(5,12,19,26,33),(6,13,20,27,34),(7,14,21,28,35)], [(8,15,29,22),(9,16,30,23),(10,17,31,24),(11,18,32,25),(12,19,33,26),(13,20,34,27),(14,21,35,28)], [(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)], [(2,3,5),(4,7,6),(9,10,12),(11,14,13),(16,17,19),(18,21,20),(23,24,26),(25,28,27),(30,31,33),(32,35,34)]])
Matrix representation of F5×C7⋊C3 ►in GL7(𝔽421)
| 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 420 |
| 0 | 0 | 0 | 1 | 0 | 0 | 420 |
| 0 | 0 | 0 | 0 | 1 | 0 | 420 |
| 0 | 0 | 0 | 0 | 0 | 1 | 420 |
| 420 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 420 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 420 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 177 | 0 | 0 | 0 | 0 |
| 0 | 1 | 176 | 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 |
| 176 | 244 | 0 | 0 | 0 | 0 | 0 |
| 420 | 245 | 1 | 0 | 0 | 0 | 0 |
| 420 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 20 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 20 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 20 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 20 |
G:=sub<GL(7,GF(421))| [1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,420,420,420,420],[420,0,0,0,0,0,0,0,420,0,0,0,0,0,0,0,420,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,0,0,1,0],[0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,177,176,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[176,420,420,0,0,0,0,244,245,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,20,0,0,0,0,0,0,0,20,0,0,0,0,0,0,0,20,0,0,0,0,0,0,0,20] >;
F5×C7⋊C3 in GAP, Magma, Sage, TeX
F_5\times C_7\rtimes C_3
% in TeX
G:=Group("F5xC7:C3"); // GroupNames label
G:=SmallGroup(420,14);
// by ID
G=gap.SmallGroup(420,14);
# by ID
G:=PCGroup([5,-2,-3,-2,-5,-7,30,483,173,1509]);
// Polycyclic
G:=Group<a,b,c,d|a^5=b^4=c^7=d^3=1,b*a*b^-1=a^3,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^4>;
// generators/relations
Export
Subgroup lattice of F5×C7⋊C3 in TeX
Character table of F5×C7⋊C3 in TeX