direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C3×F5, C5⋊C12, C15⋊2C4, D5.C6, (C3×D5).2C2, SmallGroup(60,6)
Series: Derived ►Chief ►Lower central ►Upper central
| C5 — C3×F5 |
Generators and relations for C3×F5
G = < a,b,c | a3=b5=c4=1, ab=ba, ac=ca, cbc-1=b3 >
Character table of C3×F5
| class | 1 | 2 | 3A | 3B | 4A | 4B | 5 | 6A | 6B | 12A | 12B | 12C | 12D | 15A | 15B | |
| size | 1 | 5 | 1 | 1 | 5 | 5 | 4 | 5 | 5 | 5 | 5 | 5 | 5 | 4 | 4 | |
| ρ1 | 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 | linear of order 2 |
| ρ3 | 1 | 1 | ζ32 | ζ3 | 1 | 1 | 1 | ζ32 | ζ3 | ζ3 | ζ32 | ζ3 | ζ32 | ζ3 | ζ32 | linear of order 3 |
| ρ4 | 1 | 1 | ζ3 | ζ32 | -1 | -1 | 1 | ζ3 | ζ32 | ζ6 | ζ65 | ζ6 | ζ65 | ζ32 | ζ3 | linear of order 6 |
| ρ5 | 1 | 1 | ζ32 | ζ3 | -1 | -1 | 1 | ζ32 | ζ3 | ζ65 | ζ6 | ζ65 | ζ6 | ζ3 | ζ32 | linear of order 6 |
| ρ6 | 1 | 1 | ζ3 | ζ32 | 1 | 1 | 1 | ζ3 | ζ32 | ζ32 | ζ3 | ζ32 | ζ3 | ζ32 | ζ3 | linear of order 3 |
| ρ7 | 1 | -1 | 1 | 1 | i | -i | 1 | -1 | -1 | i | i | -i | -i | 1 | 1 | linear of order 4 |
| ρ8 | 1 | -1 | 1 | 1 | -i | i | 1 | -1 | -1 | -i | -i | i | i | 1 | 1 | linear of order 4 |
| ρ9 | 1 | -1 | ζ32 | ζ3 | -i | i | 1 | ζ6 | ζ65 | ζ43ζ3 | ζ43ζ32 | ζ4ζ3 | ζ4ζ32 | ζ3 | ζ32 | linear of order 12 |
| ρ10 | 1 | -1 | ζ3 | ζ32 | -i | i | 1 | ζ65 | ζ6 | ζ43ζ32 | ζ43ζ3 | ζ4ζ32 | ζ4ζ3 | ζ32 | ζ3 | linear of order 12 |
| ρ11 | 1 | -1 | ζ32 | ζ3 | i | -i | 1 | ζ6 | ζ65 | ζ4ζ3 | ζ4ζ32 | ζ43ζ3 | ζ43ζ32 | ζ3 | ζ32 | linear of order 12 |
| ρ12 | 1 | -1 | ζ3 | ζ32 | i | -i | 1 | ζ65 | ζ6 | ζ4ζ32 | ζ4ζ3 | ζ43ζ32 | ζ43ζ3 | ζ32 | ζ3 | linear of order 12 |
| ρ13 | 4 | 0 | 4 | 4 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | orthogonal lifted from F5 |
| ρ14 | 4 | 0 | -2-2√-3 | -2+2√-3 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | ζ65 | ζ6 | complex faithful |
| ρ15 | 4 | 0 | -2+2√-3 | -2-2√-3 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | ζ6 | ζ65 | complex faithful |
►On 15 points - transitive group 15T8
Generators in S15
(1 11 6)(2 12 7)(3 13 8)(4 14 9)(5 15 10)
(1 2 3 4 5)(6 7 8 9 10)(11 12 13 14 15)
(2 3 5 4)(7 8 10 9)(12 13 15 14)
G:=sub<Sym(15)| (1,11,6)(2,12,7)(3,13,8)(4,14,9)(5,15,10), (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15), (2,3,5,4)(7,8,10,9)(12,13,15,14)>;
G:=Group( (1,11,6)(2,12,7)(3,13,8)(4,14,9)(5,15,10), (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15), (2,3,5,4)(7,8,10,9)(12,13,15,14) );
G=PermutationGroup([[(1,11,6),(2,12,7),(3,13,8),(4,14,9),(5,15,10)], [(1,2,3,4,5),(6,7,8,9,10),(11,12,13,14,15)], [(2,3,5,4),(7,8,10,9),(12,13,15,14)]])
G:=TransitiveGroup(15,8);
►On 30 points - transitive group 30T7
Generators in S30
(1 11 6)(2 12 7)(3 13 8)(4 14 9)(5 15 10)(16 26 21)(17 27 22)(18 28 23)(19 29 24)(20 30 25)
(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)
(1 18)(2 20 5 16)(3 17 4 19)(6 23)(7 25 10 21)(8 22 9 24)(11 28)(12 30 15 26)(13 27 14 29)
G:=sub<Sym(30)| (1,11,6)(2,12,7)(3,13,8)(4,14,9)(5,15,10)(16,26,21)(17,27,22)(18,28,23)(19,29,24)(20,30,25), (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), (1,18)(2,20,5,16)(3,17,4,19)(6,23)(7,25,10,21)(8,22,9,24)(11,28)(12,30,15,26)(13,27,14,29)>;
G:=Group( (1,11,6)(2,12,7)(3,13,8)(4,14,9)(5,15,10)(16,26,21)(17,27,22)(18,28,23)(19,29,24)(20,30,25), (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), (1,18)(2,20,5,16)(3,17,4,19)(6,23)(7,25,10,21)(8,22,9,24)(11,28)(12,30,15,26)(13,27,14,29) );
G=PermutationGroup([[(1,11,6),(2,12,7),(3,13,8),(4,14,9),(5,15,10),(16,26,21),(17,27,22),(18,28,23),(19,29,24),(20,30,25)], [(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)], [(1,18),(2,20,5,16),(3,17,4,19),(6,23),(7,25,10,21),(8,22,9,24),(11,28),(12,30,15,26),(13,27,14,29)]])
G:=TransitiveGroup(30,7);
C3×F5 is a maximal subgroup of
C35⋊C12
C3×F5 is a maximal quotient of C35⋊C12
| action | f(x) | Disc(f) |
|---|---|---|
| 15T8 | x15+32x10-228x5-8 | 242·515·710·18110 |
Matrix representation of C3×F5 ►in GL4(𝔽7) generated by
| 4 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 4 |
| 0 | 3 | 2 | 3 |
| 2 | 0 | 3 | 2 |
| 5 | 6 | 5 | 0 |
| 2 | 3 | 4 | 1 |
| 1 | 0 | 3 | 0 |
| 0 | 5 | 1 | 2 |
| 0 | 2 | 4 | 3 |
| 0 | 4 | 1 | 4 |
G:=sub<GL(4,GF(7))| [4,0,0,0,0,4,0,0,0,0,4,0,0,0,0,4],[0,2,5,2,3,0,6,3,2,3,5,4,3,2,0,1],[1,0,0,0,0,5,2,4,3,1,4,1,0,2,3,4] >;
C3×F5 in GAP, Magma, Sage, TeX
C_3\times F_5
% in TeX
G:=Group("C3xF5"); // GroupNames label
G:=SmallGroup(60,6);
// by ID
G=gap.SmallGroup(60,6);
# by ID
G:=PCGroup([4,-2,-3,-2,-5,24,387,139]);
// Polycyclic
G:=Group<a,b,c|a^3=b^5=c^4=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^3>;
// generators/relations
Export
Subgroup lattice of C3×F5 in TeX
Character table of C3×F5 in TeX