metacyclic, supersoluble, monomial, Z-group
Aliases: C35⋊1C12, D5.F7, C7⋊F5⋊C3, C7⋊C3⋊F5, C7⋊(C3×F5), C5⋊(C7⋊C12), (C7×D5).1C6, (C5×C7⋊C3)⋊1C4, (D5×C7⋊C3).1C2, SmallGroup(420,15)
Series: Derived ►Chief ►Lower central ►Upper central
| C35 — C35⋊C12 |
Generators and relations for C35⋊C12
G = < a,b | a35=b12=1, bab-1=a17 >
Character table of C35⋊C12
| class | 1 | 2 | 3A | 3B | 4A | 4B | 5 | 6A | 6B | 7 | 12A | 12B | 12C | 12D | 14 | 15A | 15B | 35A | 35B | |
| size | 1 | 5 | 7 | 7 | 35 | 35 | 4 | 35 | 35 | 6 | 35 | 35 | 35 | 35 | 30 | 28 | 28 | 12 | 12 | |
| ρ1 | 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 | linear of order 2 |
| ρ3 | 1 | 1 | ζ3 | ζ32 | 1 | 1 | 1 | ζ3 | ζ32 | 1 | ζ32 | ζ3 | ζ3 | ζ32 | 1 | ζ3 | ζ32 | 1 | 1 | linear of order 3 |
| ρ4 | 1 | 1 | ζ32 | ζ3 | -1 | -1 | 1 | ζ32 | ζ3 | 1 | ζ65 | ζ6 | ζ6 | ζ65 | 1 | ζ32 | ζ3 | 1 | 1 | linear of order 6 |
| ρ5 | 1 | 1 | ζ32 | ζ3 | 1 | 1 | 1 | ζ32 | ζ3 | 1 | ζ3 | ζ32 | ζ32 | ζ3 | 1 | ζ32 | ζ3 | 1 | 1 | linear of order 3 |
| ρ6 | 1 | 1 | ζ3 | ζ32 | -1 | -1 | 1 | ζ3 | ζ32 | 1 | ζ6 | ζ65 | ζ65 | ζ6 | 1 | ζ3 | ζ32 | 1 | 1 | linear of order 6 |
| ρ7 | 1 | -1 | 1 | 1 | i | -i | 1 | -1 | -1 | 1 | i | -i | i | -i | -1 | 1 | 1 | 1 | 1 | linear of order 4 |
| ρ8 | 1 | -1 | 1 | 1 | -i | i | 1 | -1 | -1 | 1 | -i | i | -i | i | -1 | 1 | 1 | 1 | 1 | linear of order 4 |
| ρ9 | 1 | -1 | ζ3 | ζ32 | i | -i | 1 | ζ65 | ζ6 | 1 | ζ4ζ32 | ζ43ζ3 | ζ4ζ3 | ζ43ζ32 | -1 | ζ3 | ζ32 | 1 | 1 | linear of order 12 |
| ρ10 | 1 | -1 | ζ32 | ζ3 | i | -i | 1 | ζ6 | ζ65 | 1 | ζ4ζ3 | ζ43ζ32 | ζ4ζ32 | ζ43ζ3 | -1 | ζ32 | ζ3 | 1 | 1 | linear of order 12 |
| ρ11 | 1 | -1 | ζ3 | ζ32 | -i | i | 1 | ζ65 | ζ6 | 1 | ζ43ζ32 | ζ4ζ3 | ζ43ζ3 | ζ4ζ32 | -1 | ζ3 | ζ32 | 1 | 1 | linear of order 12 |
| ρ12 | 1 | -1 | ζ32 | ζ3 | -i | i | 1 | ζ6 | ζ65 | 1 | ζ43ζ3 | ζ4ζ32 | ζ43ζ32 | ζ4ζ3 | -1 | ζ32 | ζ3 | 1 | 1 | linear of order 12 |
| ρ13 | 4 | 0 | 4 | 4 | 0 | 0 | -1 | 0 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | orthogonal lifted from F5 |
| ρ14 | 4 | 0 | -2+2√-3 | -2-2√-3 | 0 | 0 | -1 | 0 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | ζ65 | ζ6 | -1 | -1 | complex lifted from C3×F5 |
| ρ15 | 4 | 0 | -2-2√-3 | -2+2√-3 | 0 | 0 | -1 | 0 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | ζ6 | ζ65 | -1 | -1 | complex lifted from C3×F5 |
| ρ16 | 6 | 6 | 0 | 0 | 0 | 0 | 6 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | -1 | -1 | orthogonal lifted from F7 |
| ρ17 | 6 | -6 | 0 | 0 | 0 | 0 | 6 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | -1 | -1 | symplectic lifted from C7⋊C12, Schur index 2 |
| ρ18 | 12 | 0 | 0 | 0 | 0 | 0 | -3 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1+√-35/2 | 1-√-35/2 | complex faithful |
| ρ19 | 12 | 0 | 0 | 0 | 0 | 0 | -3 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1-√-35/2 | 1+√-35/2 | 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)
(2 34 5 28 17 4 30 13 12 14 10 18)(3 32 9 20 33 7 24 25 23 27 19 35)(6 26 21 31 11 16)(8 22 29 15)
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), (2,34,5,28,17,4,30,13,12,14,10,18)(3,32,9,20,33,7,24,25,23,27,19,35)(6,26,21,31,11,16)(8,22,29,15)>;
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), (2,34,5,28,17,4,30,13,12,14,10,18)(3,32,9,20,33,7,24,25,23,27,19,35)(6,26,21,31,11,16)(8,22,29,15) );
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)], [(2,34,5,28,17,4,30,13,12,14,10,18),(3,32,9,20,33,7,24,25,23,27,19,35),(6,26,21,31,11,16),(8,22,29,15)]])
Matrix representation of C35⋊C12 ►in GL10(ℤ)
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| -1 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | -1 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | -1 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | -1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | -1 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| -1 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
G:=sub<GL(10,Integers())| [0,0,0,-1,0,0,0,0,0,0,1,0,0,-1,0,0,0,0,0,0,0,1,0,-1,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,-1,-1,-1,-1,-1,-1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0],[1,0,-1,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0] >;
C35⋊C12 in GAP, Magma, Sage, TeX
C_{35}\rtimes C_{12} % in TeX
G:=Group("C35:C12"); // GroupNames label
G:=SmallGroup(420,15);
// by ID
G=gap.SmallGroup(420,15);
# by ID
G:=PCGroup([5,-2,-3,-2,-5,-7,30,723,173,9004,1509]);
// Polycyclic
G:=Group<a,b|a^35=b^12=1,b*a*b^-1=a^17>;
// generators/relations
Export
Subgroup lattice of C35⋊C12 in TeX
Character table of C35⋊C12 in TeX