metabelian, supersoluble, monomial, A-group
Aliases: C15⋊2F5, C52⋊6Dic3, (C5×C15)⋊4C4, C3⋊(C52⋊C4), C5⋊2(C3⋊F5), C5⋊D5.3S3, (C3×C5⋊D5).2C2, SmallGroup(300,35)
Series: Derived ►Chief ►Lower central ►Upper central
| C5×C15 — C15⋊2F5 |
Generators and relations for C15⋊2F5
G = < a,b,c | a15=b5=c4=1, ab=ba, cac-1=a2, cbc-1=b3 >
Character table of C15⋊2F5
| class | 1 | 2 | 3 | 4A | 4B | 5A | 5B | 5C | 5D | 5E | 5F | 6 | 15A | 15B | 15C | 15D | 15E | 15F | 15G | 15H | 15I | 15J | 15K | 15L | |
| size | 1 | 25 | 2 | 75 | 75 | 4 | 4 | 4 | 4 | 4 | 4 | 50 | 4 | 4 | 4 | 4 | 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 | 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 | linear of order 2 |
| ρ3 | 1 | -1 | 1 | -i | i | 1 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 4 |
| ρ4 | 1 | -1 | 1 | i | -i | 1 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 4 |
| ρ5 | 2 | 2 | -1 | 0 | 0 | 2 | 2 | 2 | 2 | 2 | 2 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ6 | 2 | -2 | -1 | 0 | 0 | 2 | 2 | 2 | 2 | 2 | 2 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | symplectic lifted from Dic3, Schur index 2 |
| ρ7 | 4 | 0 | 4 | 0 | 0 | -1 | -1 | -1 | 4 | -1 | -1 | 0 | -1 | -1 | -1 | 4 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | 4 | orthogonal lifted from F5 |
| ρ8 | 4 | 0 | 4 | 0 | 0 | -1 | -1 | 4 | -1 | -1 | -1 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | 4 | -1 | 4 | -1 | orthogonal lifted from F5 |
| ρ9 | 4 | 0 | 4 | 0 | 0 | 3+√5/2 | -1-√5 | -1 | -1 | -1+√5 | 3-√5/2 | 0 | -1+√5 | 3+√5/2 | -1-√5 | -1 | 3-√5/2 | -1-√5 | -1+√5 | 3+√5/2 | -1 | 3-√5/2 | -1 | -1 | orthogonal lifted from C52⋊C4 |
| ρ10 | 4 | 0 | 4 | 0 | 0 | -1-√5 | 3-√5/2 | -1 | -1 | 3+√5/2 | -1+√5 | 0 | 3+√5/2 | -1-√5 | 3-√5/2 | -1 | -1+√5 | 3-√5/2 | 3+√5/2 | -1-√5 | -1 | -1+√5 | -1 | -1 | orthogonal lifted from C52⋊C4 |
| ρ11 | 4 | 0 | 4 | 0 | 0 | 3-√5/2 | -1+√5 | -1 | -1 | -1-√5 | 3+√5/2 | 0 | -1-√5 | 3-√5/2 | -1+√5 | -1 | 3+√5/2 | -1+√5 | -1-√5 | 3-√5/2 | -1 | 3+√5/2 | -1 | -1 | orthogonal lifted from C52⋊C4 |
| ρ12 | 4 | 0 | 4 | 0 | 0 | -1+√5 | 3+√5/2 | -1 | -1 | 3-√5/2 | -1-√5 | 0 | 3-√5/2 | -1+√5 | 3+√5/2 | -1 | -1-√5 | 3+√5/2 | 3-√5/2 | -1+√5 | -1 | -1-√5 | -1 | -1 | orthogonal lifted from C52⋊C4 |
| ρ13 | 4 | 0 | -2 | 0 | 0 | 3-√5/2 | -1+√5 | -1 | -1 | -1-√5 | 3+√5/2 | 0 | 1+√5/2 | ζ32ζ53+ζ32ζ52-2ζ32-2 | 1-√5/2 | 1+√-15/2 | ζ32ζ54+ζ32ζ5-2ζ32-2 | 1-√5/2 | 1+√5/2 | ζ3ζ53+ζ3ζ52-2ζ3-2 | 1+√-15/2 | ζ3ζ54+ζ3ζ5-2ζ3-2 | 1-√-15/2 | 1-√-15/2 | complex faithful |
| ρ14 | 4 | 0 | -2 | 0 | 0 | -1+√5 | 3+√5/2 | -1 | -1 | 3-√5/2 | -1-√5 | 0 | ζ3ζ53+ζ3ζ52-2ζ3-2 | 1-√5/2 | ζ32ζ54+ζ32ζ5-2ζ32-2 | 1-√-15/2 | 1+√5/2 | ζ3ζ54+ζ3ζ5-2ζ3-2 | ζ32ζ53+ζ32ζ52-2ζ32-2 | 1-√5/2 | 1+√-15/2 | 1+√5/2 | 1-√-15/2 | 1+√-15/2 | complex faithful |
| ρ15 | 4 | 0 | -2 | 0 | 0 | 3+√5/2 | -1-√5 | -1 | -1 | -1+√5 | 3-√5/2 | 0 | 1-√5/2 | ζ3ζ54+ζ3ζ5-2ζ3-2 | 1+√5/2 | 1+√-15/2 | ζ3ζ53+ζ3ζ52-2ζ3-2 | 1+√5/2 | 1-√5/2 | ζ32ζ54+ζ32ζ5-2ζ32-2 | 1+√-15/2 | ζ32ζ53+ζ32ζ52-2ζ32-2 | 1-√-15/2 | 1-√-15/2 | complex faithful |
| ρ16 | 4 | 0 | -2 | 0 | 0 | -1-√5 | 3-√5/2 | -1 | -1 | 3+√5/2 | -1+√5 | 0 | ζ3ζ54+ζ3ζ5-2ζ3-2 | 1+√5/2 | ζ32ζ53+ζ32ζ52-2ζ32-2 | 1+√-15/2 | 1-√5/2 | ζ3ζ53+ζ3ζ52-2ζ3-2 | ζ32ζ54+ζ32ζ5-2ζ32-2 | 1+√5/2 | 1-√-15/2 | 1-√5/2 | 1+√-15/2 | 1-√-15/2 | complex faithful |
| ρ17 | 4 | 0 | -2 | 0 | 0 | -1 | -1 | -1 | 4 | -1 | -1 | 0 | 1+√-15/2 | 1-√-15/2 | 1+√-15/2 | -2 | 1+√-15/2 | 1-√-15/2 | 1-√-15/2 | 1+√-15/2 | 1+√-15/2 | 1-√-15/2 | 1-√-15/2 | -2 | complex lifted from C3⋊F5 |
| ρ18 | 4 | 0 | -2 | 0 | 0 | -1 | -1 | 4 | -1 | -1 | -1 | 0 | 1+√-15/2 | 1+√-15/2 | 1+√-15/2 | 1-√-15/2 | 1-√-15/2 | 1-√-15/2 | 1-√-15/2 | 1-√-15/2 | -2 | 1+√-15/2 | -2 | 1+√-15/2 | complex lifted from C3⋊F5 |
| ρ19 | 4 | 0 | -2 | 0 | 0 | 3+√5/2 | -1-√5 | -1 | -1 | -1+√5 | 3-√5/2 | 0 | 1-√5/2 | ζ32ζ54+ζ32ζ5-2ζ32-2 | 1+√5/2 | 1-√-15/2 | ζ32ζ53+ζ32ζ52-2ζ32-2 | 1+√5/2 | 1-√5/2 | ζ3ζ54+ζ3ζ5-2ζ3-2 | 1-√-15/2 | ζ3ζ53+ζ3ζ52-2ζ3-2 | 1+√-15/2 | 1+√-15/2 | complex faithful |
| ρ20 | 4 | 0 | -2 | 0 | 0 | -1-√5 | 3-√5/2 | -1 | -1 | 3+√5/2 | -1+√5 | 0 | ζ32ζ54+ζ32ζ5-2ζ32-2 | 1+√5/2 | ζ3ζ53+ζ3ζ52-2ζ3-2 | 1-√-15/2 | 1-√5/2 | ζ32ζ53+ζ32ζ52-2ζ32-2 | ζ3ζ54+ζ3ζ5-2ζ3-2 | 1+√5/2 | 1+√-15/2 | 1-√5/2 | 1-√-15/2 | 1+√-15/2 | complex faithful |
| ρ21 | 4 | 0 | -2 | 0 | 0 | -1 | -1 | 4 | -1 | -1 | -1 | 0 | 1-√-15/2 | 1-√-15/2 | 1-√-15/2 | 1+√-15/2 | 1+√-15/2 | 1+√-15/2 | 1+√-15/2 | 1+√-15/2 | -2 | 1-√-15/2 | -2 | 1-√-15/2 | complex lifted from C3⋊F5 |
| ρ22 | 4 | 0 | -2 | 0 | 0 | -1+√5 | 3+√5/2 | -1 | -1 | 3-√5/2 | -1-√5 | 0 | ζ32ζ53+ζ32ζ52-2ζ32-2 | 1-√5/2 | ζ3ζ54+ζ3ζ5-2ζ3-2 | 1+√-15/2 | 1+√5/2 | ζ32ζ54+ζ32ζ5-2ζ32-2 | ζ3ζ53+ζ3ζ52-2ζ3-2 | 1-√5/2 | 1-√-15/2 | 1+√5/2 | 1+√-15/2 | 1-√-15/2 | complex faithful |
| ρ23 | 4 | 0 | -2 | 0 | 0 | 3-√5/2 | -1+√5 | -1 | -1 | -1-√5 | 3+√5/2 | 0 | 1+√5/2 | ζ3ζ53+ζ3ζ52-2ζ3-2 | 1-√5/2 | 1-√-15/2 | ζ3ζ54+ζ3ζ5-2ζ3-2 | 1-√5/2 | 1+√5/2 | ζ32ζ53+ζ32ζ52-2ζ32-2 | 1-√-15/2 | ζ32ζ54+ζ32ζ5-2ζ32-2 | 1+√-15/2 | 1+√-15/2 | complex faithful |
| ρ24 | 4 | 0 | -2 | 0 | 0 | -1 | -1 | -1 | 4 | -1 | -1 | 0 | 1-√-15/2 | 1+√-15/2 | 1-√-15/2 | -2 | 1-√-15/2 | 1+√-15/2 | 1+√-15/2 | 1-√-15/2 | 1-√-15/2 | 1+√-15/2 | 1+√-15/2 | -2 | complex lifted from C3⋊F5 |
►On 30 points - transitive group 30T76
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 4 7 10 13)(2 5 8 11 14)(3 6 9 12 15)(16 28 25 22 19)(17 29 26 23 20)(18 30 27 24 21)
(1 20)(2 28 5 22)(3 21 9 24)(4 29 13 26)(6 30)(7 23 10 17)(8 16 14 19)(11 25)(12 18 15 27)
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), (1,4,7,10,13)(2,5,8,11,14)(3,6,9,12,15)(16,28,25,22,19)(17,29,26,23,20)(18,30,27,24,21), (1,20)(2,28,5,22)(3,21,9,24)(4,29,13,26)(6,30)(7,23,10,17)(8,16,14,19)(11,25)(12,18,15,27)>;
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), (1,4,7,10,13)(2,5,8,11,14)(3,6,9,12,15)(16,28,25,22,19)(17,29,26,23,20)(18,30,27,24,21), (1,20)(2,28,5,22)(3,21,9,24)(4,29,13,26)(6,30)(7,23,10,17)(8,16,14,19)(11,25)(12,18,15,27) );
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)], [(1,4,7,10,13),(2,5,8,11,14),(3,6,9,12,15),(16,28,25,22,19),(17,29,26,23,20),(18,30,27,24,21)], [(1,20),(2,28,5,22),(3,21,9,24),(4,29,13,26),(6,30),(7,23,10,17),(8,16,14,19),(11,25),(12,18,15,27)]])
G:=TransitiveGroup(30,76);
Matrix representation of C15⋊2F5 ►in GL4(𝔽61) generated by
| 0 | 53 | 40 | 2 |
| 55 | 8 | 26 | 39 |
| 0 | 0 | 38 | 13 |
| 0 | 0 | 48 | 0 |
| 0 | 18 | 43 | 60 |
| 44 | 43 | 18 | 17 |
| 0 | 0 | 0 | 60 |
| 0 | 0 | 1 | 17 |
| 17 | 1 | 0 | 1 |
| 60 | 43 | 0 | 0 |
| 16 | 43 | 1 | 1 |
| 60 | 44 | 0 | 0 |
G:=sub<GL(4,GF(61))| [0,55,0,0,53,8,0,0,40,26,38,48,2,39,13,0],[0,44,0,0,18,43,0,0,43,18,0,1,60,17,60,17],[17,60,16,60,1,43,43,44,0,0,1,0,1,0,1,0] >;
C15⋊2F5 in GAP, Magma, Sage, TeX
C_{15}\rtimes_2F_5 % in TeX
G:=Group("C15:2F5"); // GroupNames label
G:=SmallGroup(300,35);
// by ID
G=gap.SmallGroup(300,35);
# by ID
G:=PCGroup([5,-2,-2,-3,-5,-5,10,122,483,488,4504,3009]);
// Polycyclic
G:=Group<a,b,c|a^15=b^5=c^4=1,a*b=b*a,c*a*c^-1=a^2,c*b*c^-1=b^3>;
// generators/relations
Export
Subgroup lattice of C15⋊2F5 in TeX
Character table of C15⋊2F5 in TeX