metabelian, supersoluble, monomial, A-group
Aliases: C32:3F5, C15:1Dic3, C3:(C3:F5), (C3xC15):2C4, C5:(C3:Dic3), D5.(C3:S3), (C3xD5).3S3, (C32xD5).1C2, SmallGroup(180,22)
Series: Derived ►Chief ►Lower central ►Upper central
| C3xC15 — C32:3F5 |
Generators and relations for C32:3F5
G = < a,b,c,d | a3=b3=c5=d4=1, ab=ba, ac=ca, dad-1=a-1, bc=cb, dbd-1=b-1, dcd-1=c3 >
Quotients: C1, C2, C4, S3, Dic3, C3:S3, F5, C3:Dic3, C3:F5, C32:3F5
Character table of C32:3F5
| class | 1 | 2 | 3A | 3B | 3C | 3D | 4A | 4B | 5 | 6A | 6B | 6C | 6D | 15A | 15B | 15C | 15D | 15E | 15F | 15G | 15H | |
| size | 1 | 5 | 2 | 2 | 2 | 2 | 45 | 45 | 4 | 10 | 10 | 10 | 10 | 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 | trivial |
| ρ2 | 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 | -i | i | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 4 |
| ρ4 | 1 | -1 | 1 | 1 | 1 | 1 | i | -i | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 4 |
| ρ5 | 2 | 2 | -1 | -1 | 2 | -1 | 0 | 0 | 2 | 2 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | 2 | 2 | -1 | -1 | orthogonal lifted from S3 |
| ρ6 | 2 | 2 | -1 | -1 | -1 | 2 | 0 | 0 | 2 | -1 | 2 | -1 | -1 | 2 | -1 | -1 | 2 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ7 | 2 | 2 | -1 | 2 | -1 | -1 | 0 | 0 | 2 | -1 | -1 | -1 | 2 | -1 | 2 | 2 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ8 | 2 | 2 | 2 | -1 | -1 | -1 | 0 | 0 | 2 | -1 | -1 | 2 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | 2 | 2 | orthogonal lifted from S3 |
| ρ9 | 2 | -2 | -1 | 2 | -1 | -1 | 0 | 0 | 2 | 1 | 1 | 1 | -2 | -1 | 2 | 2 | -1 | -1 | -1 | -1 | -1 | symplectic lifted from Dic3, Schur index 2 |
| ρ10 | 2 | -2 | 2 | -1 | -1 | -1 | 0 | 0 | 2 | 1 | 1 | -2 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | 2 | 2 | symplectic lifted from Dic3, Schur index 2 |
| ρ11 | 2 | -2 | -1 | -1 | 2 | -1 | 0 | 0 | 2 | -2 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 2 | 2 | -1 | -1 | symplectic lifted from Dic3, Schur index 2 |
| ρ12 | 2 | -2 | -1 | -1 | -1 | 2 | 0 | 0 | 2 | 1 | -2 | 1 | 1 | 2 | -1 | -1 | 2 | -1 | -1 | -1 | -1 | symplectic lifted from Dic3, Schur index 2 |
| ρ13 | 4 | 0 | 4 | 4 | 4 | 4 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from F5 |
| ρ14 | 4 | 0 | -2 | -2 | 4 | -2 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 1+√-15/2 | 1-√-15/2 | 1+√-15/2 | 1-√-15/2 | -1 | -1 | 1+√-15/2 | 1-√-15/2 | complex lifted from C3:F5 |
| ρ15 | 4 | 0 | -2 | -2 | -2 | 4 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | -1 | 1-√-15/2 | 1+√-15/2 | -1 | 1-√-15/2 | 1+√-15/2 | 1-√-15/2 | 1+√-15/2 | complex lifted from C3:F5 |
| ρ16 | 4 | 0 | -2 | 4 | -2 | -2 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 1+√-15/2 | -1 | -1 | 1-√-15/2 | 1+√-15/2 | 1-√-15/2 | 1-√-15/2 | 1+√-15/2 | complex lifted from C3:F5 |
| ρ17 | 4 | 0 | -2 | -2 | 4 | -2 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 1-√-15/2 | 1+√-15/2 | 1-√-15/2 | 1+√-15/2 | -1 | -1 | 1-√-15/2 | 1+√-15/2 | complex lifted from C3:F5 |
| ρ18 | 4 | 0 | -2 | 4 | -2 | -2 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 1-√-15/2 | -1 | -1 | 1+√-15/2 | 1-√-15/2 | 1+√-15/2 | 1+√-15/2 | 1-√-15/2 | complex lifted from C3:F5 |
| ρ19 | 4 | 0 | 4 | -2 | -2 | -2 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 1+√-15/2 | 1+√-15/2 | 1-√-15/2 | 1-√-15/2 | 1-√-15/2 | 1+√-15/2 | -1 | -1 | complex lifted from C3:F5 |
| ρ20 | 4 | 0 | -2 | -2 | -2 | 4 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | -1 | 1+√-15/2 | 1-√-15/2 | -1 | 1+√-15/2 | 1-√-15/2 | 1+√-15/2 | 1-√-15/2 | complex lifted from C3:F5 |
| ρ21 | 4 | 0 | 4 | -2 | -2 | -2 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 1-√-15/2 | 1-√-15/2 | 1+√-15/2 | 1+√-15/2 | 1+√-15/2 | 1-√-15/2 | -1 | -1 | complex lifted from C3:F5 |
►On 45 points
Generators in S45
(1 31 16)(2 32 17)(3 33 18)(4 34 19)(5 35 20)(6 36 21)(7 37 22)(8 38 23)(9 39 24)(10 40 25)(11 41 26)(12 42 27)(13 43 28)(14 44 29)(15 45 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)(31 41 36)(32 42 37)(33 43 38)(34 44 39)(35 45 40)
(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)(36 37 38 39 40)(41 42 43 44 45)
(2 3 5 4)(6 11)(7 13 10 14)(8 15 9 12)(16 31)(17 33 20 34)(18 35 19 32)(21 41)(22 43 25 44)(23 45 24 42)(26 36)(27 38 30 39)(28 40 29 37)
G:=sub<Sym(45)| (1,31,16)(2,32,17)(3,33,18)(4,34,19)(5,35,20)(6,36,21)(7,37,22)(8,38,23)(9,39,24)(10,40,25)(11,41,26)(12,42,27)(13,43,28)(14,44,29)(15,45,30), (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)(31,41,36)(32,42,37)(33,43,38)(34,44,39)(35,45,40), (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)(36,37,38,39,40)(41,42,43,44,45), (2,3,5,4)(6,11)(7,13,10,14)(8,15,9,12)(16,31)(17,33,20,34)(18,35,19,32)(21,41)(22,43,25,44)(23,45,24,42)(26,36)(27,38,30,39)(28,40,29,37)>;
G:=Group( (1,31,16)(2,32,17)(3,33,18)(4,34,19)(5,35,20)(6,36,21)(7,37,22)(8,38,23)(9,39,24)(10,40,25)(11,41,26)(12,42,27)(13,43,28)(14,44,29)(15,45,30), (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)(31,41,36)(32,42,37)(33,43,38)(34,44,39)(35,45,40), (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)(36,37,38,39,40)(41,42,43,44,45), (2,3,5,4)(6,11)(7,13,10,14)(8,15,9,12)(16,31)(17,33,20,34)(18,35,19,32)(21,41)(22,43,25,44)(23,45,24,42)(26,36)(27,38,30,39)(28,40,29,37) );
G=PermutationGroup([[(1,31,16),(2,32,17),(3,33,18),(4,34,19),(5,35,20),(6,36,21),(7,37,22),(8,38,23),(9,39,24),(10,40,25),(11,41,26),(12,42,27),(13,43,28),(14,44,29),(15,45,30)], [(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),(31,41,36),(32,42,37),(33,43,38),(34,44,39),(35,45,40)], [(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),(36,37,38,39,40),(41,42,43,44,45)], [(2,3,5,4),(6,11),(7,13,10,14),(8,15,9,12),(16,31),(17,33,20,34),(18,35,19,32),(21,41),(22,43,25,44),(23,45,24,42),(26,36),(27,38,30,39),(28,40,29,37)]])
C32:3F5 is a maximal subgroup of
C3:S3xF5 S3xC3:F5
C32:3F5 is a maximal quotient of C30.Dic3
Matrix representation of C32:3F5 ►in GL6(F61)
| 0 | 60 | 0 | 0 | 0 | 0 |
| 1 | 60 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 60 | 1 | 0 | 0 | 0 | 0 |
| 60 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 33 | 0 | 6 | 6 |
| 0 | 0 | 55 | 27 | 55 | 0 |
| 0 | 0 | 0 | 55 | 27 | 55 |
| 0 | 0 | 6 | 6 | 0 | 33 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 60 | 60 | 60 | 60 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 11 | 0 | 0 | 0 | 0 | 0 |
| 11 | 50 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 60 | 60 | 60 | 60 |
G:=sub<GL(6,GF(61))| [0,1,0,0,0,0,60,60,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[60,60,0,0,0,0,1,0,0,0,0,0,0,0,33,55,0,6,0,0,0,27,55,6,0,0,6,55,27,0,0,0,6,0,55,33],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,60,1,0,0,0,0,60,0,1,0,0,0,60,0,0,1,0,0,60,0,0,0],[11,11,0,0,0,0,0,50,0,0,0,0,0,0,1,0,0,60,0,0,0,0,1,60,0,0,0,0,0,60,0,0,0,1,0,60] >;
C32:3F5 in GAP, Magma, Sage, TeX
C_3^2\rtimes_3F_5
% in TeX
G:=Group("C3^2:3F5"); // GroupNames label
G:=SmallGroup(180,22);
// by ID
G=gap.SmallGroup(180,22);
# by ID
G:=PCGroup([5,-2,-2,-3,-3,-5,10,122,483,2704,1809]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^3=c^5=d^4=1,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=c^3>;
// generators/relations
Export
Subgroup lattice of C32:3F5 in TeX
Character table of C32:3F5 in TeX