direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C5×S3, C3⋊C10, C15⋊3C2, SmallGroup(30,1)
Series: Derived ►Chief ►Lower central ►Upper central
| C3 — C5×S3 |
Generators and relations for C5×S3
G = < a,b,c | a5=b3=c2=1, ab=ba, ac=ca, cbc=b-1 >
Character table of C5×S3
| class | 1 | 2 | 3 | 5A | 5B | 5C | 5D | 10A | 10B | 10C | 10D | 15A | 15B | 15C | 15D | |
| size | 1 | 3 | 2 | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 | 2 | 2 | 2 | 2 | |
| ρ1 | 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 | linear of order 2 |
| ρ3 | 1 | -1 | 1 | ζ5 | ζ54 | ζ52 | ζ53 | -ζ52 | -ζ5 | -ζ54 | -ζ53 | ζ54 | ζ5 | ζ52 | ζ53 | linear of order 10 |
| ρ4 | 1 | -1 | 1 | ζ52 | ζ53 | ζ54 | ζ5 | -ζ54 | -ζ52 | -ζ53 | -ζ5 | ζ53 | ζ52 | ζ54 | ζ5 | linear of order 10 |
| ρ5 | 1 | 1 | 1 | ζ52 | ζ53 | ζ54 | ζ5 | ζ54 | ζ52 | ζ53 | ζ5 | ζ53 | ζ52 | ζ54 | ζ5 | linear of order 5 |
| ρ6 | 1 | 1 | 1 | ζ5 | ζ54 | ζ52 | ζ53 | ζ52 | ζ5 | ζ54 | ζ53 | ζ54 | ζ5 | ζ52 | ζ53 | linear of order 5 |
| ρ7 | 1 | -1 | 1 | ζ53 | ζ52 | ζ5 | ζ54 | -ζ5 | -ζ53 | -ζ52 | -ζ54 | ζ52 | ζ53 | ζ5 | ζ54 | linear of order 10 |
| ρ8 | 1 | 1 | 1 | ζ53 | ζ52 | ζ5 | ζ54 | ζ5 | ζ53 | ζ52 | ζ54 | ζ52 | ζ53 | ζ5 | ζ54 | linear of order 5 |
| ρ9 | 1 | 1 | 1 | ζ54 | ζ5 | ζ53 | ζ52 | ζ53 | ζ54 | ζ5 | ζ52 | ζ5 | ζ54 | ζ53 | ζ52 | linear of order 5 |
| ρ10 | 1 | -1 | 1 | ζ54 | ζ5 | ζ53 | ζ52 | -ζ53 | -ζ54 | -ζ5 | -ζ52 | ζ5 | ζ54 | ζ53 | ζ52 | linear of order 10 |
| ρ11 | 2 | 0 | -1 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ12 | 2 | 0 | -1 | 2ζ54 | 2ζ5 | 2ζ53 | 2ζ52 | 0 | 0 | 0 | 0 | -ζ5 | -ζ54 | -ζ53 | -ζ52 | complex faithful |
| ρ13 | 2 | 0 | -1 | 2ζ53 | 2ζ52 | 2ζ5 | 2ζ54 | 0 | 0 | 0 | 0 | -ζ52 | -ζ53 | -ζ5 | -ζ54 | complex faithful |
| ρ14 | 2 | 0 | -1 | 2ζ52 | 2ζ53 | 2ζ54 | 2ζ5 | 0 | 0 | 0 | 0 | -ζ53 | -ζ52 | -ζ54 | -ζ5 | complex faithful |
| ρ15 | 2 | 0 | -1 | 2ζ5 | 2ζ54 | 2ζ52 | 2ζ53 | 0 | 0 | 0 | 0 | -ζ54 | -ζ5 | -ζ52 | -ζ53 | complex faithful |
►On 15 points - transitive group 15T4
Generators in S15
(1 2 3 4 5)(6 7 8 9 10)(11 12 13 14 15)
(1 8 12)(2 9 13)(3 10 14)(4 6 15)(5 7 11)
(6 15)(7 11)(8 12)(9 13)(10 14)
G:=sub<Sym(15)| (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15), (1,8,12)(2,9,13)(3,10,14)(4,6,15)(5,7,11), (6,15)(7,11)(8,12)(9,13)(10,14)>;
G:=Group( (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15), (1,8,12)(2,9,13)(3,10,14)(4,6,15)(5,7,11), (6,15)(7,11)(8,12)(9,13)(10,14) );
G=PermutationGroup([[(1,2,3,4,5),(6,7,8,9,10),(11,12,13,14,15)], [(1,8,12),(2,9,13),(3,10,14),(4,6,15),(5,7,11)], [(6,15),(7,11),(8,12),(9,13),(10,14)]])
G:=TransitiveGroup(15,4);
►Regular action on 30 points - transitive group 30T2
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 23 17)(2 24 18)(3 25 19)(4 21 20)(5 22 16)(6 15 29)(7 11 30)(8 12 26)(9 13 27)(10 14 28)
(1 8)(2 9)(3 10)(4 6)(5 7)(11 16)(12 17)(13 18)(14 19)(15 20)(21 29)(22 30)(23 26)(24 27)(25 28)
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), (1,23,17)(2,24,18)(3,25,19)(4,21,20)(5,22,16)(6,15,29)(7,11,30)(8,12,26)(9,13,27)(10,14,28), (1,8)(2,9)(3,10)(4,6)(5,7)(11,16)(12,17)(13,18)(14,19)(15,20)(21,29)(22,30)(23,26)(24,27)(25,28)>;
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), (1,23,17)(2,24,18)(3,25,19)(4,21,20)(5,22,16)(6,15,29)(7,11,30)(8,12,26)(9,13,27)(10,14,28), (1,8)(2,9)(3,10)(4,6)(5,7)(11,16)(12,17)(13,18)(14,19)(15,20)(21,29)(22,30)(23,26)(24,27)(25,28) );
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)], [(1,23,17),(2,24,18),(3,25,19),(4,21,20),(5,22,16),(6,15,29),(7,11,30),(8,12,26),(9,13,27),(10,14,28)], [(1,8),(2,9),(3,10),(4,6),(5,7),(11,16),(12,17),(13,18),(14,19),(15,20),(21,29),(22,30),(23,26),(24,27),(25,28)]])
G:=TransitiveGroup(30,2);
C5×S3 is a maximal subgroup of
C3⋊F11
C5×S3 is a maximal quotient of C3⋊F11
| action | f(x) | Disc(f) |
|---|---|---|
| 15T4 | x15+x14-4x13+x12-18x11-24x10-5x9-58x8+79x7-32x6+97x5+35x4+42x3+15x2+1 | -1112·315·672·1312·11872·4676992 |
Matrix representation of C5×S3 ►in GL2(𝔽11) generated by
| 5 | 0 |
| 0 | 5 |
| 0 | 8 |
| 4 | 10 |
| 1 | 8 |
| 0 | 10 |
G:=sub<GL(2,GF(11))| [5,0,0,5],[0,4,8,10],[1,0,8,10] >;
C5×S3 in GAP, Magma, Sage, TeX
C_5\times S_3
% in TeX
G:=Group("C5xS3"); // GroupNames label
G:=SmallGroup(30,1);
// by ID
G=gap.SmallGroup(30,1);
# by ID
G:=PCGroup([3,-2,-5,-3,182]);
// Polycyclic
G:=Group<a,b,c|a^5=b^3=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export
Subgroup lattice of C5×S3 in TeX
Character table of C5×S3 in TeX