direct product, metacyclic, supersoluble, monomial, A-group
Aliases: C3×C3⋊F5, C15⋊1C12, C32⋊2F5, C15⋊2Dic3, C3⋊(C3×F5), C5⋊(C3×Dic3), (C3×C15)⋊3C4, D5.(C3×S3), (C3×D5).4S3, (C3×D5).1C6, (C32×D5).2C2, SmallGroup(180,21)
Series: Derived ►Chief ►Lower central ►Upper central
| C15 — C3×C3⋊F5 |
Generators and relations for C3×C3⋊F5
G = < a,b,c,d | a3=b3=c5=d4=1, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=c3 >
Character table of C3×C3⋊F5
| class | 1 | 2 | 3A | 3B | 3C | 3D | 3E | 4A | 4B | 5 | 6A | 6B | 6C | 6D | 6E | 12A | 12B | 12C | 12D | 15A | 15B | 15C | 15D | 15E | 15F | 15G | 15H | |
| size | 1 | 5 | 1 | 1 | 2 | 2 | 2 | 15 | 15 | 4 | 5 | 5 | 10 | 10 | 10 | 15 | 15 | 15 | 15 | 4 | 4 | 4 | 4 | 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 | 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 | linear of order 2 |
| ρ3 | 1 | 1 | ζ32 | ζ3 | ζ3 | 1 | ζ32 | 1 | 1 | 1 | ζ32 | ζ3 | ζ3 | 1 | ζ32 | ζ32 | ζ3 | ζ3 | ζ32 | ζ3 | 1 | ζ32 | 1 | ζ3 | ζ32 | ζ32 | ζ3 | linear of order 3 |
| ρ4 | 1 | 1 | ζ3 | ζ32 | ζ32 | 1 | ζ3 | 1 | 1 | 1 | ζ3 | ζ32 | ζ32 | 1 | ζ3 | ζ3 | ζ32 | ζ32 | ζ3 | ζ32 | 1 | ζ3 | 1 | ζ32 | ζ3 | ζ3 | ζ32 | linear of order 3 |
| ρ5 | 1 | 1 | ζ3 | ζ32 | ζ32 | 1 | ζ3 | -1 | -1 | 1 | ζ3 | ζ32 | ζ32 | 1 | ζ3 | ζ65 | ζ6 | ζ6 | ζ65 | ζ32 | 1 | ζ3 | 1 | ζ32 | ζ3 | ζ3 | ζ32 | linear of order 6 |
| ρ6 | 1 | 1 | ζ32 | ζ3 | ζ3 | 1 | ζ32 | -1 | -1 | 1 | ζ32 | ζ3 | ζ3 | 1 | ζ32 | ζ6 | ζ65 | ζ65 | ζ6 | ζ3 | 1 | ζ32 | 1 | ζ3 | ζ32 | ζ32 | ζ3 | linear of order 6 |
| ρ7 | 1 | -1 | 1 | 1 | 1 | 1 | 1 | i | -i | 1 | -1 | -1 | -1 | -1 | -1 | i | -i | i | -i | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 4 |
| ρ8 | 1 | -1 | 1 | 1 | 1 | 1 | 1 | -i | i | 1 | -1 | -1 | -1 | -1 | -1 | -i | i | -i | i | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 4 |
| ρ9 | 1 | -1 | ζ3 | ζ32 | ζ32 | 1 | ζ3 | -i | i | 1 | ζ65 | ζ6 | ζ6 | -1 | ζ65 | ζ43ζ3 | ζ4ζ32 | ζ43ζ32 | ζ4ζ3 | ζ32 | 1 | ζ3 | 1 | ζ32 | ζ3 | ζ3 | ζ32 | linear of order 12 |
| ρ10 | 1 | -1 | ζ32 | ζ3 | ζ3 | 1 | ζ32 | i | -i | 1 | ζ6 | ζ65 | ζ65 | -1 | ζ6 | ζ4ζ32 | ζ43ζ3 | ζ4ζ3 | ζ43ζ32 | ζ3 | 1 | ζ32 | 1 | ζ3 | ζ32 | ζ32 | ζ3 | linear of order 12 |
| ρ11 | 1 | -1 | ζ3 | ζ32 | ζ32 | 1 | ζ3 | i | -i | 1 | ζ65 | ζ6 | ζ6 | -1 | ζ65 | ζ4ζ3 | ζ43ζ32 | ζ4ζ32 | ζ43ζ3 | ζ32 | 1 | ζ3 | 1 | ζ32 | ζ3 | ζ3 | ζ32 | linear of order 12 |
| ρ12 | 1 | -1 | ζ32 | ζ3 | ζ3 | 1 | ζ32 | -i | i | 1 | ζ6 | ζ65 | ζ65 | -1 | ζ6 | ζ43ζ32 | ζ4ζ3 | ζ43ζ3 | ζ4ζ32 | ζ3 | 1 | ζ32 | 1 | ζ3 | ζ32 | ζ32 | ζ3 | linear of order 12 |
| ρ13 | 2 | 2 | 2 | 2 | -1 | -1 | -1 | 0 | 0 | 2 | 2 | 2 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 2 | -1 | 2 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ14 | 2 | -2 | 2 | 2 | -1 | -1 | -1 | 0 | 0 | 2 | -2 | -2 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 2 | -1 | 2 | -1 | -1 | -1 | -1 | -1 | symplectic lifted from Dic3, Schur index 2 |
| ρ15 | 2 | -2 | -1+√-3 | -1-√-3 | ζ6 | -1 | ζ65 | 0 | 0 | 2 | 1-√-3 | 1+√-3 | ζ32 | 1 | ζ3 | 0 | 0 | 0 | 0 | -1-√-3 | -1 | -1+√-3 | -1 | ζ6 | ζ65 | ζ65 | ζ6 | complex lifted from C3×Dic3 |
| ρ16 | 2 | -2 | -1-√-3 | -1+√-3 | ζ65 | -1 | ζ6 | 0 | 0 | 2 | 1+√-3 | 1-√-3 | ζ3 | 1 | ζ32 | 0 | 0 | 0 | 0 | -1+√-3 | -1 | -1-√-3 | -1 | ζ65 | ζ6 | ζ6 | ζ65 | complex lifted from C3×Dic3 |
| ρ17 | 2 | 2 | -1-√-3 | -1+√-3 | ζ65 | -1 | ζ6 | 0 | 0 | 2 | -1-√-3 | -1+√-3 | ζ65 | -1 | ζ6 | 0 | 0 | 0 | 0 | -1+√-3 | -1 | -1-√-3 | -1 | ζ65 | ζ6 | ζ6 | ζ65 | complex lifted from C3×S3 |
| ρ18 | 2 | 2 | -1+√-3 | -1-√-3 | ζ6 | -1 | ζ65 | 0 | 0 | 2 | -1+√-3 | -1-√-3 | ζ6 | -1 | ζ65 | 0 | 0 | 0 | 0 | -1-√-3 | -1 | -1+√-3 | -1 | ζ6 | ζ65 | ζ65 | ζ6 | complex lifted from C3×S3 |
| ρ19 | 4 | 0 | 4 | 4 | 4 | 4 | 4 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from F5 |
| ρ20 | 4 | 0 | -2-2√-3 | -2+2√-3 | -2+2√-3 | 4 | -2-2√-3 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ65 | -1 | ζ6 | -1 | ζ65 | ζ6 | ζ6 | ζ65 | complex lifted from C3×F5 |
| ρ21 | 4 | 0 | -2+2√-3 | -2-2√-3 | -2-2√-3 | 4 | -2+2√-3 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ6 | -1 | ζ65 | -1 | ζ6 | ζ65 | ζ65 | ζ6 | complex lifted from C3×F5 |
| ρ22 | 4 | 0 | -2+2√-3 | -2-2√-3 | 1+√-3 | -2 | 1-√-3 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ6 | 1-√-15/2 | ζ65 | 1+√-15/2 | -ζ3ζ53-ζ3ζ52-ζ3+ζ53+ζ52 | -ζ32ζ54-ζ32ζ5-ζ32+ζ54+ζ5 | -ζ32ζ53-ζ32ζ52-ζ32+ζ53+ζ52 | -ζ3ζ54-ζ3ζ5-ζ3+ζ54+ζ5 | complex faithful |
| ρ23 | 4 | 0 | -2-2√-3 | -2+2√-3 | 1-√-3 | -2 | 1+√-3 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ65 | 1-√-15/2 | ζ6 | 1+√-15/2 | -ζ32ζ54-ζ32ζ5-ζ32+ζ54+ζ5 | -ζ3ζ53-ζ3ζ52-ζ3+ζ53+ζ52 | -ζ3ζ54-ζ3ζ5-ζ3+ζ54+ζ5 | -ζ32ζ53-ζ32ζ52-ζ32+ζ53+ζ52 | complex faithful |
| ρ24 | 4 | 0 | -2+2√-3 | -2-2√-3 | 1+√-3 | -2 | 1-√-3 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ6 | 1+√-15/2 | ζ65 | 1-√-15/2 | -ζ3ζ54-ζ3ζ5-ζ3+ζ54+ζ5 | -ζ32ζ53-ζ32ζ52-ζ32+ζ53+ζ52 | -ζ32ζ54-ζ32ζ5-ζ32+ζ54+ζ5 | -ζ3ζ53-ζ3ζ52-ζ3+ζ53+ζ52 | complex faithful |
| ρ25 | 4 | 0 | -2-2√-3 | -2+2√-3 | 1-√-3 | -2 | 1+√-3 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ65 | 1+√-15/2 | ζ6 | 1-√-15/2 | -ζ32ζ53-ζ32ζ52-ζ32+ζ53+ζ52 | -ζ3ζ54-ζ3ζ5-ζ3+ζ54+ζ5 | -ζ3ζ53-ζ3ζ52-ζ3+ζ53+ζ52 | -ζ32ζ54-ζ32ζ5-ζ32+ζ54+ζ5 | complex faithful |
| ρ26 | 4 | 0 | 4 | 4 | -2 | -2 | -2 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 1-√-15/2 | -1 | 1+√-15/2 | 1-√-15/2 | 1-√-15/2 | 1+√-15/2 | 1+√-15/2 | complex lifted from C3⋊F5 |
| ρ27 | 4 | 0 | 4 | 4 | -2 | -2 | -2 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 1+√-15/2 | -1 | 1-√-15/2 | 1+√-15/2 | 1+√-15/2 | 1-√-15/2 | 1-√-15/2 | complex lifted from C3⋊F5 |
►On 30 points - transitive group 30T47
Generators in S30
(1 9 14)(2 10 15)(3 6 11)(4 7 12)(5 8 13)(16 21 26)(17 22 27)(18 23 28)(19 24 29)(20 25 30)
(1 9 14)(2 10 15)(3 6 11)(4 7 12)(5 8 13)(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 22 7 24)(8 21 10 25)(9 23)(11 27 12 29)(13 26 15 30)(14 28)
G:=sub<Sym(30)| (1,9,14)(2,10,15)(3,6,11)(4,7,12)(5,8,13)(16,21,26)(17,22,27)(18,23,28)(19,24,29)(20,25,30), (1,9,14)(2,10,15)(3,6,11)(4,7,12)(5,8,13)(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)(6,22,7,24)(8,21,10,25)(9,23)(11,27,12,29)(13,26,15,30)(14,28)>;
G:=Group( (1,9,14)(2,10,15)(3,6,11)(4,7,12)(5,8,13)(16,21,26)(17,22,27)(18,23,28)(19,24,29)(20,25,30), (1,9,14)(2,10,15)(3,6,11)(4,7,12)(5,8,13)(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)(6,22,7,24)(8,21,10,25)(9,23)(11,27,12,29)(13,26,15,30)(14,28) );
G=PermutationGroup([[(1,9,14),(2,10,15),(3,6,11),(4,7,12),(5,8,13),(16,21,26),(17,22,27),(18,23,28),(19,24,29),(20,25,30)], [(1,9,14),(2,10,15),(3,6,11),(4,7,12),(5,8,13),(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),(6,22,7,24),(8,21,10,25),(9,23),(11,27,12,29),(13,26,15,30),(14,28)]])
G:=TransitiveGroup(30,47);
C3×C3⋊F5 is a maximal subgroup of
C3×S3×F5 C3⋊F5⋊S3
Matrix representation of C3×C3⋊F5 ►in GL4(𝔽61) generated by
| 13 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 |
| 0 | 0 | 13 | 0 |
| 0 | 0 | 0 | 13 |
| 13 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 |
| 55 | 20 | 47 | 0 |
| 55 | 20 | 0 | 47 |
| 60 | 1 | 0 | 0 |
| 16 | 44 | 0 | 0 |
| 2 | 17 | 18 | 18 |
| 44 | 60 | 43 | 60 |
| 43 | 60 | 42 | 60 |
| 1 | 17 | 17 | 18 |
| 0 | 0 | 1 | 0 |
| 1 | 0 | 1 | 0 |
G:=sub<GL(4,GF(61))| [13,0,0,0,0,13,0,0,0,0,13,0,0,0,0,13],[13,0,55,55,0,13,20,20,0,0,47,0,0,0,0,47],[60,16,2,44,1,44,17,60,0,0,18,43,0,0,18,60],[43,1,0,1,60,17,0,0,42,17,1,1,60,18,0,0] >;
C3×C3⋊F5 in GAP, Magma, Sage, TeX
C_3\times C_3\rtimes F_5
% in TeX
G:=Group("C3xC3:F5"); // GroupNames label
G:=SmallGroup(180,21);
// by ID
G=gap.SmallGroup(180,21);
# by ID
G:=PCGroup([5,-2,-3,-2,-3,-5,30,483,2704,614]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^3=c^5=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=c^3>;
// generators/relations
Export
Subgroup lattice of C3×C3⋊F5 in TeX
Character table of C3×C3⋊F5 in TeX