direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C6×F5, C10⋊C12, D5⋊C12, C30⋊2C4, D10.C6, C5⋊(C2×C12), C15⋊3(C2×C4), D5.(C2×C6), (C3×D5)⋊3C4, (C6×D5).3C2, (C3×D5).3C22, SmallGroup(120,40)
Series: Derived ►Chief ►Lower central ►Upper central
| C5 — C6×F5 |
Generators and relations for C6×F5
G = < a,b,c | a6=b5=c4=1, ab=ba, ac=ca, cbc-1=b3 >
Character table of C6×F5
| class | 1 | 2A | 2B | 2C | 3A | 3B | 4A | 4B | 4C | 4D | 5 | 6A | 6B | 6C | 6D | 6E | 6F | 10 | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 15A | 15B | 30A | 30B | |
| size | 1 | 1 | 5 | 5 | 1 | 1 | 5 | 5 | 5 | 5 | 4 | 1 | 1 | 5 | 5 | 5 | 5 | 4 | 5 | 5 | 5 | 5 | 5 | 5 | 5 | 5 | 4 | 4 | 4 | 4 | |
| 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 | 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 | 1 | 1 | 1 | -1 | -1 | linear of order 2 |
| ρ3 | 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 | -1 | 1 | 1 | -1 | -1 | linear of order 2 |
| ρ4 | 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 | -1 | 1 | 1 | 1 | 1 | linear of order 2 |
| ρ5 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | -1 | -1 | -1 | -1 | 1 | ζ32 | ζ3 | ζ3 | ζ32 | ζ32 | ζ3 | 1 | ζ6 | ζ65 | ζ65 | ζ6 | ζ6 | ζ65 | ζ65 | ζ6 | ζ3 | ζ32 | ζ32 | ζ3 | linear of order 6 |
| ρ6 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | -1 | -1 | -1 | -1 | 1 | ζ3 | ζ32 | ζ32 | ζ3 | ζ3 | ζ32 | 1 | ζ65 | ζ6 | ζ6 | ζ65 | ζ65 | ζ6 | ζ6 | ζ65 | ζ32 | ζ3 | ζ3 | ζ32 | linear of order 6 |
| ρ7 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | 1 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | ζ3 | ζ32 | ζ32 | ζ3 | 1 | ζ32 | ζ3 | ζ3 | ζ32 | ζ32 | ζ3 | ζ3 | ζ32 | ζ3 | ζ32 | ζ32 | ζ3 | linear of order 3 |
| ρ8 | 1 | -1 | -1 | 1 | ζ32 | ζ3 | 1 | -1 | -1 | 1 | 1 | ζ65 | ζ6 | ζ6 | ζ3 | ζ65 | ζ32 | -1 | ζ3 | ζ6 | ζ6 | ζ65 | ζ65 | ζ32 | ζ32 | ζ3 | ζ32 | ζ3 | ζ65 | ζ6 | linear of order 6 |
| ρ9 | 1 | -1 | -1 | 1 | ζ3 | ζ32 | 1 | -1 | -1 | 1 | 1 | ζ6 | ζ65 | ζ65 | ζ32 | ζ6 | ζ3 | -1 | ζ32 | ζ65 | ζ65 | ζ6 | ζ6 | ζ3 | ζ3 | ζ32 | ζ3 | ζ32 | ζ6 | ζ65 | linear of order 6 |
| ρ10 | 1 | -1 | -1 | 1 | ζ3 | ζ32 | -1 | 1 | 1 | -1 | 1 | ζ6 | ζ65 | ζ65 | ζ32 | ζ6 | ζ3 | -1 | ζ6 | ζ3 | ζ3 | ζ32 | ζ32 | ζ65 | ζ65 | ζ6 | ζ3 | ζ32 | ζ6 | ζ65 | linear of order 6 |
| ρ11 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | 1 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | ζ32 | ζ3 | ζ3 | ζ32 | 1 | ζ3 | ζ32 | ζ32 | ζ3 | ζ3 | ζ32 | ζ32 | ζ3 | ζ32 | ζ3 | ζ3 | ζ32 | linear of order 3 |
| ρ12 | 1 | -1 | -1 | 1 | ζ32 | ζ3 | -1 | 1 | 1 | -1 | 1 | ζ65 | ζ6 | ζ6 | ζ3 | ζ65 | ζ32 | -1 | ζ65 | ζ32 | ζ32 | ζ3 | ζ3 | ζ6 | ζ6 | ζ65 | ζ32 | ζ3 | ζ65 | ζ6 | linear of order 6 |
| ρ13 | 1 | 1 | -1 | -1 | 1 | 1 | -i | i | -i | i | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | -i | i | -i | i | -i | i | -i | i | 1 | 1 | 1 | 1 | linear of order 4 |
| ρ14 | 1 | 1 | -1 | -1 | 1 | 1 | i | -i | i | -i | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | i | -i | i | -i | i | -i | i | -i | 1 | 1 | 1 | 1 | linear of order 4 |
| ρ15 | 1 | -1 | 1 | -1 | 1 | 1 | -i | -i | i | i | 1 | -1 | -1 | 1 | -1 | 1 | -1 | -1 | -i | -i | i | -i | i | i | -i | i | 1 | 1 | -1 | -1 | linear of order 4 |
| ρ16 | 1 | -1 | 1 | -1 | 1 | 1 | i | i | -i | -i | 1 | -1 | -1 | 1 | -1 | 1 | -1 | -1 | i | i | -i | i | -i | -i | i | -i | 1 | 1 | -1 | -1 | linear of order 4 |
| ρ17 | 1 | -1 | 1 | -1 | ζ3 | ζ32 | i | i | -i | -i | 1 | ζ6 | ζ65 | ζ3 | ζ6 | ζ32 | ζ65 | -1 | ζ4ζ32 | ζ4ζ3 | ζ43ζ3 | ζ4ζ32 | ζ43ζ32 | ζ43ζ3 | ζ4ζ3 | ζ43ζ32 | ζ3 | ζ32 | ζ6 | ζ65 | linear of order 12 |
| ρ18 | 1 | 1 | -1 | -1 | ζ32 | ζ3 | i | -i | i | -i | 1 | ζ3 | ζ32 | ζ6 | ζ65 | ζ65 | ζ6 | 1 | ζ4ζ3 | ζ43ζ32 | ζ4ζ32 | ζ43ζ3 | ζ4ζ3 | ζ43ζ32 | ζ4ζ32 | ζ43ζ3 | ζ32 | ζ3 | ζ3 | ζ32 | linear of order 12 |
| ρ19 | 1 | 1 | -1 | -1 | ζ32 | ζ3 | -i | i | -i | i | 1 | ζ3 | ζ32 | ζ6 | ζ65 | ζ65 | ζ6 | 1 | ζ43ζ3 | ζ4ζ32 | ζ43ζ32 | ζ4ζ3 | ζ43ζ3 | ζ4ζ32 | ζ43ζ32 | ζ4ζ3 | ζ32 | ζ3 | ζ3 | ζ32 | linear of order 12 |
| ρ20 | 1 | 1 | -1 | -1 | ζ3 | ζ32 | -i | i | -i | i | 1 | ζ32 | ζ3 | ζ65 | ζ6 | ζ6 | ζ65 | 1 | ζ43ζ32 | ζ4ζ3 | ζ43ζ3 | ζ4ζ32 | ζ43ζ32 | ζ4ζ3 | ζ43ζ3 | ζ4ζ32 | ζ3 | ζ32 | ζ32 | ζ3 | linear of order 12 |
| ρ21 | 1 | -1 | 1 | -1 | ζ32 | ζ3 | -i | -i | i | i | 1 | ζ65 | ζ6 | ζ32 | ζ65 | ζ3 | ζ6 | -1 | ζ43ζ3 | ζ43ζ32 | ζ4ζ32 | ζ43ζ3 | ζ4ζ3 | ζ4ζ32 | ζ43ζ32 | ζ4ζ3 | ζ32 | ζ3 | ζ65 | ζ6 | linear of order 12 |
| ρ22 | 1 | -1 | 1 | -1 | ζ32 | ζ3 | i | i | -i | -i | 1 | ζ65 | ζ6 | ζ32 | ζ65 | ζ3 | ζ6 | -1 | ζ4ζ3 | ζ4ζ32 | ζ43ζ32 | ζ4ζ3 | ζ43ζ3 | ζ43ζ32 | ζ4ζ32 | ζ43ζ3 | ζ32 | ζ3 | ζ65 | ζ6 | linear of order 12 |
| ρ23 | 1 | 1 | -1 | -1 | ζ3 | ζ32 | i | -i | i | -i | 1 | ζ32 | ζ3 | ζ65 | ζ6 | ζ6 | ζ65 | 1 | ζ4ζ32 | ζ43ζ3 | ζ4ζ3 | ζ43ζ32 | ζ4ζ32 | ζ43ζ3 | ζ4ζ3 | ζ43ζ32 | ζ3 | ζ32 | ζ32 | ζ3 | linear of order 12 |
| ρ24 | 1 | -1 | 1 | -1 | ζ3 | ζ32 | -i | -i | i | i | 1 | ζ6 | ζ65 | ζ3 | ζ6 | ζ32 | ζ65 | -1 | ζ43ζ32 | ζ43ζ3 | ζ4ζ3 | ζ43ζ32 | ζ4ζ32 | ζ4ζ3 | ζ43ζ3 | ζ4ζ32 | ζ3 | ζ32 | ζ6 | ζ65 | linear of order 12 |
| ρ25 | 4 | 4 | 0 | 0 | 4 | 4 | 0 | 0 | 0 | 0 | -1 | 4 | 4 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | orthogonal lifted from F5 |
| ρ26 | 4 | -4 | 0 | 0 | 4 | 4 | 0 | 0 | 0 | 0 | -1 | -4 | -4 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | 1 | 1 | orthogonal lifted from C2×F5 |
| ρ27 | 4 | 4 | 0 | 0 | -2+2√-3 | -2-2√-3 | 0 | 0 | 0 | 0 | -1 | -2-2√-3 | -2+2√-3 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ65 | ζ6 | ζ6 | ζ65 | complex lifted from C3×F5 |
| ρ28 | 4 | -4 | 0 | 0 | -2+2√-3 | -2-2√-3 | 0 | 0 | 0 | 0 | -1 | 2+2√-3 | 2-2√-3 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ65 | ζ6 | ζ32 | ζ3 | complex faithful |
| ρ29 | 4 | -4 | 0 | 0 | -2-2√-3 | -2+2√-3 | 0 | 0 | 0 | 0 | -1 | 2-2√-3 | 2+2√-3 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ6 | ζ65 | ζ3 | ζ32 | complex faithful |
| ρ30 | 4 | 4 | 0 | 0 | -2-2√-3 | -2+2√-3 | 0 | 0 | 0 | 0 | -1 | -2+2√-3 | -2-2√-3 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ6 | ζ65 | ζ65 | ζ6 | complex lifted from C3×F5 |
►On 30 points - transitive group 30T26
Generators in S30
(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 14 8 25 22)(2 15 9 26 23)(3 16 10 27 24)(4 17 11 28 19)(5 18 12 29 20)(6 13 7 30 21)
(1 4)(2 5)(3 6)(7 24 30 16)(8 19 25 17)(9 20 26 18)(10 21 27 13)(11 22 28 14)(12 23 29 15)
G:=sub<Sym(30)| (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), (1,14,8,25,22)(2,15,9,26,23)(3,16,10,27,24)(4,17,11,28,19)(5,18,12,29,20)(6,13,7,30,21), (1,4)(2,5)(3,6)(7,24,30,16)(8,19,25,17)(9,20,26,18)(10,21,27,13)(11,22,28,14)(12,23,29,15)>;
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), (1,14,8,25,22)(2,15,9,26,23)(3,16,10,27,24)(4,17,11,28,19)(5,18,12,29,20)(6,13,7,30,21), (1,4)(2,5)(3,6)(7,24,30,16)(8,19,25,17)(9,20,26,18)(10,21,27,13)(11,22,28,14)(12,23,29,15) );
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)], [(1,14,8,25,22),(2,15,9,26,23),(3,16,10,27,24),(4,17,11,28,19),(5,18,12,29,20),(6,13,7,30,21)], [(1,4),(2,5),(3,6),(7,24,30,16),(8,19,25,17),(9,20,26,18),(10,21,27,13),(11,22,28,14),(12,23,29,15)]])
G:=TransitiveGroup(30,26);
C6×F5 is a maximal subgroup of
D6⋊F5 Dic3⋊F5
Matrix representation of C6×F5 ►in GL4(𝔽7) generated by
| 5 | 0 | 0 | 0 |
| 0 | 5 | 0 | 0 |
| 0 | 0 | 5 | 0 |
| 0 | 0 | 0 | 5 |
| 5 | 6 | 4 | 5 |
| 1 | 5 | 1 | 6 |
| 3 | 6 | 1 | 4 |
| 4 | 2 | 0 | 2 |
| 6 | 0 | 4 | 0 |
| 0 | 2 | 6 | 5 |
| 0 | 5 | 3 | 4 |
| 0 | 3 | 6 | 3 |
G:=sub<GL(4,GF(7))| [5,0,0,0,0,5,0,0,0,0,5,0,0,0,0,5],[5,1,3,4,6,5,6,2,4,1,1,0,5,6,4,2],[6,0,0,0,0,2,5,3,4,6,3,6,0,5,4,3] >;
C6×F5 in GAP, Magma, Sage, TeX
C_6\times F_5
% in TeX
G:=Group("C6xF5"); // GroupNames label
G:=SmallGroup(120,40);
// by ID
G=gap.SmallGroup(120,40);
# by ID
G:=PCGroup([5,-2,-2,-3,-2,-5,60,1204,219]);
// Polycyclic
G:=Group<a,b,c|a^6=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 C6×F5 in TeX
Character table of C6×F5 in TeX