metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C3⋊F5, C5⋊Dic3, C15⋊1C4, D5.S3, (C3×D5).1C2, SmallGroup(60,7)
Series: Derived ►Chief ►Lower central ►Upper central
| C15 — C3⋊F5 |
Generators and relations for C3⋊F5
G = < a,b,c | a3=b5=c4=1, ab=ba, cac-1=a-1, cbc-1=b3 >
Character table of C3⋊F5
| class | 1 | 2 | 3 | 4A | 4B | 5 | 6 | 15A | 15B | |
| size | 1 | 5 | 2 | 15 | 15 | 4 | 10 | 4 | 4 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | linear of order 2 |
| ρ3 | 1 | -1 | 1 | i | -i | 1 | -1 | 1 | 1 | linear of order 4 |
| ρ4 | 1 | -1 | 1 | -i | i | 1 | -1 | 1 | 1 | linear of order 4 |
| ρ5 | 2 | 2 | -1 | 0 | 0 | 2 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ6 | 2 | -2 | -1 | 0 | 0 | 2 | 1 | -1 | -1 | symplectic lifted from Dic3, Schur index 2 |
| ρ7 | 4 | 0 | 4 | 0 | 0 | -1 | 0 | -1 | -1 | orthogonal lifted from F5 |
| ρ8 | 4 | 0 | -2 | 0 | 0 | -1 | 0 | 1-√-15/2 | 1+√-15/2 | complex faithful |
| ρ9 | 4 | 0 | -2 | 0 | 0 | -1 | 0 | 1+√-15/2 | 1-√-15/2 | complex faithful |
►On 15 points - transitive group 15T6
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)(6 11)(7 13 10 14)(8 15 9 12)
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)(6,11)(7,13,10,14)(8,15,9,12)>;
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)(6,11)(7,13,10,14)(8,15,9,12) );
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),(6,11),(7,13,10,14),(8,15,9,12)]])
G:=TransitiveGroup(15,6);
►On 30 points - transitive group 30T6
Generators in S30
(1 11 6)(2 12 7)(3 13 8)(4 14 9)(5 15 10)(16 21 26)(17 22 27)(18 23 28)(19 24 29)(20 25 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 17)(2 19 5 20)(3 16 4 18)(6 22)(7 24 10 25)(8 21 9 23)(11 27)(12 29 15 30)(13 26 14 28)
G:=sub<Sym(30)| (1,11,6)(2,12,7)(3,13,8)(4,14,9)(5,15,10)(16,21,26)(17,22,27)(18,23,28)(19,24,29)(20,25,30), (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), (1,17)(2,19,5,20)(3,16,4,18)(6,22)(7,24,10,25)(8,21,9,23)(11,27)(12,29,15,30)(13,26,14,28)>;
G:=Group( (1,11,6)(2,12,7)(3,13,8)(4,14,9)(5,15,10)(16,21,26)(17,22,27)(18,23,28)(19,24,29)(20,25,30), (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), (1,17)(2,19,5,20)(3,16,4,18)(6,22)(7,24,10,25)(8,21,9,23)(11,27)(12,29,15,30)(13,26,14,28) );
G=PermutationGroup([[(1,11,6),(2,12,7),(3,13,8),(4,14,9),(5,15,10),(16,21,26),(17,22,27),(18,23,28),(19,24,29),(20,25,30)], [(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)], [(1,17),(2,19,5,20),(3,16,4,18),(6,22),(7,24,10,25),(8,21,9,23),(11,27),(12,29,15,30),(13,26,14,28)]])
G:=TransitiveGroup(30,6);
C3⋊F5 is a maximal subgroup of
S3×F5 C9⋊F5 C32⋊3F5 A4⋊F5 C75⋊C4 D5.D15 C15⋊F5 C15⋊2F5 ΓL2(𝔽4) C5⋊Dic21
C3⋊F5 is a maximal quotient of
C15⋊C8 C9⋊F5 C32⋊3F5 A4⋊F5 C75⋊C4 D5.D15 C15⋊F5 C15⋊2F5 C5⋊Dic21
| action | f(x) | Disc(f) |
|---|---|---|
| 15T6 | x15-30x10-3708x5-2 | 214·320·530·231110 |
Matrix representation of C3⋊F5 ►in GL4(𝔽2) generated by
| 0 | 1 | 1 | 0 |
| 1 | 0 | 1 | 1 |
| 0 | 1 | 0 | 1 |
| 1 | 1 | 0 | 0 |
| 0 | 1 | 1 | 1 |
| 0 | 0 | 1 | 1 |
| 0 | 0 | 0 | 1 |
| 1 | 1 | 1 | 1 |
| 1 | 0 | 1 | 1 |
| 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 1 |
| 0 | 0 | 1 | 1 |
G:=sub<GL(4,GF(2))| [0,1,0,1,1,0,1,1,1,1,0,0,0,1,1,0],[0,0,0,1,1,0,0,1,1,1,0,1,1,1,1,1],[1,0,0,0,0,0,1,0,1,0,0,1,1,1,1,1] >;
C3⋊F5 in GAP, Magma, Sage, TeX
C_3\rtimes F_5
% in TeX
G:=Group("C3:F5"); // GroupNames label
G:=SmallGroup(60,7);
// by ID
G=gap.SmallGroup(60,7);
# by ID
G:=PCGroup([4,-2,-2,-3,-5,8,98,579,391]);
// Polycyclic
G:=Group<a,b,c|a^3=b^5=c^4=1,a*b=b*a,c*a*c^-1=a^-1,c*b*c^-1=b^3>;
// generators/relations
Export
Subgroup lattice of C3⋊F5 in TeX
Character table of C3⋊F5 in TeX