direct product, metabelian, supersoluble, monomial, A-group
Aliases: C5×C3⋊S3, C15⋊3S3, C32⋊2C10, C3⋊(C5×S3), (C3×C15)⋊5C2, SmallGroup(90,8)
Series: Derived ►Chief ►Lower central ►Upper central
| C32 — C5×C3⋊S3 |
Generators and relations for C5×C3⋊S3
G = < a,b,c,d | a5=b3=c3=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=b-1, dcd=c-1 >
Character table of C5×C3⋊S3
| class | 1 | 2 | 3A | 3B | 3C | 3D | 5A | 5B | 5C | 5D | 10A | 10B | 10C | 10D | 15A | 15B | 15C | 15D | 15E | 15F | 15G | 15H | 15I | 15J | 15K | 15L | 15M | 15N | 15O | 15P | |
| size | 1 | 9 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 9 | 9 | 9 | 9 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | |
| 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 | 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 | 1 | 1 | 1 | linear of order 2 |
| ρ3 | 1 | 1 | 1 | 1 | 1 | 1 | ζ53 | ζ5 | ζ54 | ζ52 | ζ54 | ζ5 | ζ52 | ζ53 | ζ5 | ζ5 | ζ53 | ζ53 | ζ53 | ζ53 | ζ5 | ζ54 | ζ54 | ζ54 | ζ54 | ζ5 | ζ52 | ζ52 | ζ52 | ζ52 | linear of order 5 |
| ρ4 | 1 | -1 | 1 | 1 | 1 | 1 | ζ54 | ζ53 | ζ52 | ζ5 | -ζ52 | -ζ53 | -ζ5 | -ζ54 | ζ53 | ζ53 | ζ54 | ζ54 | ζ54 | ζ54 | ζ53 | ζ52 | ζ52 | ζ52 | ζ52 | ζ53 | ζ5 | ζ5 | ζ5 | ζ5 | linear of order 10 |
| ρ5 | 1 | -1 | 1 | 1 | 1 | 1 | ζ5 | ζ52 | ζ53 | ζ54 | -ζ53 | -ζ52 | -ζ54 | -ζ5 | ζ52 | ζ52 | ζ5 | ζ5 | ζ5 | ζ5 | ζ52 | ζ53 | ζ53 | ζ53 | ζ53 | ζ52 | ζ54 | ζ54 | ζ54 | ζ54 | linear of order 10 |
| ρ6 | 1 | -1 | 1 | 1 | 1 | 1 | ζ52 | ζ54 | ζ5 | ζ53 | -ζ5 | -ζ54 | -ζ53 | -ζ52 | ζ54 | ζ54 | ζ52 | ζ52 | ζ52 | ζ52 | ζ54 | ζ5 | ζ5 | ζ5 | ζ5 | ζ54 | ζ53 | ζ53 | ζ53 | ζ53 | linear of order 10 |
| ρ7 | 1 | 1 | 1 | 1 | 1 | 1 | ζ5 | ζ52 | ζ53 | ζ54 | ζ53 | ζ52 | ζ54 | ζ5 | ζ52 | ζ52 | ζ5 | ζ5 | ζ5 | ζ5 | ζ52 | ζ53 | ζ53 | ζ53 | ζ53 | ζ52 | ζ54 | ζ54 | ζ54 | ζ54 | linear of order 5 |
| ρ8 | 1 | 1 | 1 | 1 | 1 | 1 | ζ54 | ζ53 | ζ52 | ζ5 | ζ52 | ζ53 | ζ5 | ζ54 | ζ53 | ζ53 | ζ54 | ζ54 | ζ54 | ζ54 | ζ53 | ζ52 | ζ52 | ζ52 | ζ52 | ζ53 | ζ5 | ζ5 | ζ5 | ζ5 | linear of order 5 |
| ρ9 | 1 | 1 | 1 | 1 | 1 | 1 | ζ52 | ζ54 | ζ5 | ζ53 | ζ5 | ζ54 | ζ53 | ζ52 | ζ54 | ζ54 | ζ52 | ζ52 | ζ52 | ζ52 | ζ54 | ζ5 | ζ5 | ζ5 | ζ5 | ζ54 | ζ53 | ζ53 | ζ53 | ζ53 | linear of order 5 |
| ρ10 | 1 | -1 | 1 | 1 | 1 | 1 | ζ53 | ζ5 | ζ54 | ζ52 | -ζ54 | -ζ5 | -ζ52 | -ζ53 | ζ5 | ζ5 | ζ53 | ζ53 | ζ53 | ζ53 | ζ5 | ζ54 | ζ54 | ζ54 | ζ54 | ζ5 | ζ52 | ζ52 | ζ52 | ζ52 | linear of order 10 |
| ρ11 | 2 | 0 | 2 | -1 | -1 | -1 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | -1 | 2 | -1 | 2 | -1 | -1 | -1 | -1 | 2 | -1 | -1 | -1 | -1 | 2 | -1 | -1 | orthogonal lifted from S3 |
| ρ12 | 2 | 0 | -1 | -1 | 2 | -1 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | 2 | -1 | -1 | -1 | -1 | 2 | 2 | -1 | -1 | -1 | 2 | orthogonal lifted from S3 |
| ρ13 | 2 | 0 | -1 | -1 | -1 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | -1 | -1 | 2 | -1 | -1 | -1 | 2 | 2 | -1 | -1 | -1 | -1 | 2 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ14 | 2 | 0 | -1 | 2 | -1 | -1 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 2 | -1 | -1 | -1 | 2 | -1 | -1 | -1 | -1 | 2 | -1 | -1 | -1 | -1 | 2 | -1 | orthogonal lifted from S3 |
| ρ15 | 2 | 0 | -1 | -1 | 2 | -1 | 2ζ53 | 2ζ5 | 2ζ54 | 2ζ52 | 0 | 0 | 0 | 0 | -ζ5 | -ζ5 | -ζ53 | -ζ53 | -ζ53 | 2ζ53 | -ζ5 | -ζ54 | -ζ54 | -ζ54 | 2ζ54 | 2ζ5 | -ζ52 | -ζ52 | -ζ52 | 2ζ52 | complex lifted from C5×S3 |
| ρ16 | 2 | 0 | -1 | -1 | 2 | -1 | 2ζ54 | 2ζ53 | 2ζ52 | 2ζ5 | 0 | 0 | 0 | 0 | -ζ53 | -ζ53 | -ζ54 | -ζ54 | -ζ54 | 2ζ54 | -ζ53 | -ζ52 | -ζ52 | -ζ52 | 2ζ52 | 2ζ53 | -ζ5 | -ζ5 | -ζ5 | 2ζ5 | complex lifted from C5×S3 |
| ρ17 | 2 | 0 | 2 | -1 | -1 | -1 | 2ζ5 | 2ζ52 | 2ζ53 | 2ζ54 | 0 | 0 | 0 | 0 | -ζ52 | 2ζ52 | -ζ5 | 2ζ5 | -ζ5 | -ζ5 | -ζ52 | -ζ53 | 2ζ53 | -ζ53 | -ζ53 | -ζ52 | -ζ54 | 2ζ54 | -ζ54 | -ζ54 | complex lifted from C5×S3 |
| ρ18 | 2 | 0 | -1 | -1 | -1 | 2 | 2ζ5 | 2ζ52 | 2ζ53 | 2ζ54 | 0 | 0 | 0 | 0 | -ζ52 | -ζ52 | 2ζ5 | -ζ5 | -ζ5 | -ζ5 | 2ζ52 | 2ζ53 | -ζ53 | -ζ53 | -ζ53 | -ζ52 | 2ζ54 | -ζ54 | -ζ54 | -ζ54 | complex lifted from C5×S3 |
| ρ19 | 2 | 0 | -1 | 2 | -1 | -1 | 2ζ53 | 2ζ5 | 2ζ54 | 2ζ52 | 0 | 0 | 0 | 0 | 2ζ5 | -ζ5 | -ζ53 | -ζ53 | 2ζ53 | -ζ53 | -ζ5 | -ζ54 | -ζ54 | 2ζ54 | -ζ54 | -ζ5 | -ζ52 | -ζ52 | 2ζ52 | -ζ52 | complex lifted from C5×S3 |
| ρ20 | 2 | 0 | -1 | 2 | -1 | -1 | 2ζ5 | 2ζ52 | 2ζ53 | 2ζ54 | 0 | 0 | 0 | 0 | 2ζ52 | -ζ52 | -ζ5 | -ζ5 | 2ζ5 | -ζ5 | -ζ52 | -ζ53 | -ζ53 | 2ζ53 | -ζ53 | -ζ52 | -ζ54 | -ζ54 | 2ζ54 | -ζ54 | complex lifted from C5×S3 |
| ρ21 | 2 | 0 | -1 | -1 | 2 | -1 | 2ζ5 | 2ζ52 | 2ζ53 | 2ζ54 | 0 | 0 | 0 | 0 | -ζ52 | -ζ52 | -ζ5 | -ζ5 | -ζ5 | 2ζ5 | -ζ52 | -ζ53 | -ζ53 | -ζ53 | 2ζ53 | 2ζ52 | -ζ54 | -ζ54 | -ζ54 | 2ζ54 | complex lifted from C5×S3 |
| ρ22 | 2 | 0 | -1 | -1 | -1 | 2 | 2ζ53 | 2ζ5 | 2ζ54 | 2ζ52 | 0 | 0 | 0 | 0 | -ζ5 | -ζ5 | 2ζ53 | -ζ53 | -ζ53 | -ζ53 | 2ζ5 | 2ζ54 | -ζ54 | -ζ54 | -ζ54 | -ζ5 | 2ζ52 | -ζ52 | -ζ52 | -ζ52 | complex lifted from C5×S3 |
| ρ23 | 2 | 0 | 2 | -1 | -1 | -1 | 2ζ54 | 2ζ53 | 2ζ52 | 2ζ5 | 0 | 0 | 0 | 0 | -ζ53 | 2ζ53 | -ζ54 | 2ζ54 | -ζ54 | -ζ54 | -ζ53 | -ζ52 | 2ζ52 | -ζ52 | -ζ52 | -ζ53 | -ζ5 | 2ζ5 | -ζ5 | -ζ5 | complex lifted from C5×S3 |
| ρ24 | 2 | 0 | -1 | -1 | -1 | 2 | 2ζ54 | 2ζ53 | 2ζ52 | 2ζ5 | 0 | 0 | 0 | 0 | -ζ53 | -ζ53 | 2ζ54 | -ζ54 | -ζ54 | -ζ54 | 2ζ53 | 2ζ52 | -ζ52 | -ζ52 | -ζ52 | -ζ53 | 2ζ5 | -ζ5 | -ζ5 | -ζ5 | complex lifted from C5×S3 |
| ρ25 | 2 | 0 | -1 | 2 | -1 | -1 | 2ζ54 | 2ζ53 | 2ζ52 | 2ζ5 | 0 | 0 | 0 | 0 | 2ζ53 | -ζ53 | -ζ54 | -ζ54 | 2ζ54 | -ζ54 | -ζ53 | -ζ52 | -ζ52 | 2ζ52 | -ζ52 | -ζ53 | -ζ5 | -ζ5 | 2ζ5 | -ζ5 | complex lifted from C5×S3 |
| ρ26 | 2 | 0 | -1 | -1 | -1 | 2 | 2ζ52 | 2ζ54 | 2ζ5 | 2ζ53 | 0 | 0 | 0 | 0 | -ζ54 | -ζ54 | 2ζ52 | -ζ52 | -ζ52 | -ζ52 | 2ζ54 | 2ζ5 | -ζ5 | -ζ5 | -ζ5 | -ζ54 | 2ζ53 | -ζ53 | -ζ53 | -ζ53 | complex lifted from C5×S3 |
| ρ27 | 2 | 0 | -1 | 2 | -1 | -1 | 2ζ52 | 2ζ54 | 2ζ5 | 2ζ53 | 0 | 0 | 0 | 0 | 2ζ54 | -ζ54 | -ζ52 | -ζ52 | 2ζ52 | -ζ52 | -ζ54 | -ζ5 | -ζ5 | 2ζ5 | -ζ5 | -ζ54 | -ζ53 | -ζ53 | 2ζ53 | -ζ53 | complex lifted from C5×S3 |
| ρ28 | 2 | 0 | 2 | -1 | -1 | -1 | 2ζ52 | 2ζ54 | 2ζ5 | 2ζ53 | 0 | 0 | 0 | 0 | -ζ54 | 2ζ54 | -ζ52 | 2ζ52 | -ζ52 | -ζ52 | -ζ54 | -ζ5 | 2ζ5 | -ζ5 | -ζ5 | -ζ54 | -ζ53 | 2ζ53 | -ζ53 | -ζ53 | complex lifted from C5×S3 |
| ρ29 | 2 | 0 | 2 | -1 | -1 | -1 | 2ζ53 | 2ζ5 | 2ζ54 | 2ζ52 | 0 | 0 | 0 | 0 | -ζ5 | 2ζ5 | -ζ53 | 2ζ53 | -ζ53 | -ζ53 | -ζ5 | -ζ54 | 2ζ54 | -ζ54 | -ζ54 | -ζ5 | -ζ52 | 2ζ52 | -ζ52 | -ζ52 | complex lifted from C5×S3 |
| ρ30 | 2 | 0 | -1 | -1 | 2 | -1 | 2ζ52 | 2ζ54 | 2ζ5 | 2ζ53 | 0 | 0 | 0 | 0 | -ζ54 | -ζ54 | -ζ52 | -ζ52 | -ζ52 | 2ζ52 | -ζ54 | -ζ5 | -ζ5 | -ζ5 | 2ζ5 | 2ζ54 | -ζ53 | -ζ53 | -ζ53 | 2ζ53 | complex lifted from C5×S3 |
►On 45 points
Generators in S45
(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)
(1 8 33)(2 9 34)(3 10 35)(4 6 31)(5 7 32)(11 23 20)(12 24 16)(13 25 17)(14 21 18)(15 22 19)(26 42 36)(27 43 37)(28 44 38)(29 45 39)(30 41 40)
(1 18 42)(2 19 43)(3 20 44)(4 16 45)(5 17 41)(6 12 39)(7 13 40)(8 14 36)(9 15 37)(10 11 38)(21 26 33)(22 27 34)(23 28 35)(24 29 31)(25 30 32)
(6 31)(7 32)(8 33)(9 34)(10 35)(11 28)(12 29)(13 30)(14 26)(15 27)(16 45)(17 41)(18 42)(19 43)(20 44)(21 36)(22 37)(23 38)(24 39)(25 40)
G:=sub<Sym(45)| (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), (1,8,33)(2,9,34)(3,10,35)(4,6,31)(5,7,32)(11,23,20)(12,24,16)(13,25,17)(14,21,18)(15,22,19)(26,42,36)(27,43,37)(28,44,38)(29,45,39)(30,41,40), (1,18,42)(2,19,43)(3,20,44)(4,16,45)(5,17,41)(6,12,39)(7,13,40)(8,14,36)(9,15,37)(10,11,38)(21,26,33)(22,27,34)(23,28,35)(24,29,31)(25,30,32), (6,31)(7,32)(8,33)(9,34)(10,35)(11,28)(12,29)(13,30)(14,26)(15,27)(16,45)(17,41)(18,42)(19,43)(20,44)(21,36)(22,37)(23,38)(24,39)(25,40)>;
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)(31,32,33,34,35)(36,37,38,39,40)(41,42,43,44,45), (1,8,33)(2,9,34)(3,10,35)(4,6,31)(5,7,32)(11,23,20)(12,24,16)(13,25,17)(14,21,18)(15,22,19)(26,42,36)(27,43,37)(28,44,38)(29,45,39)(30,41,40), (1,18,42)(2,19,43)(3,20,44)(4,16,45)(5,17,41)(6,12,39)(7,13,40)(8,14,36)(9,15,37)(10,11,38)(21,26,33)(22,27,34)(23,28,35)(24,29,31)(25,30,32), (6,31)(7,32)(8,33)(9,34)(10,35)(11,28)(12,29)(13,30)(14,26)(15,27)(16,45)(17,41)(18,42)(19,43)(20,44)(21,36)(22,37)(23,38)(24,39)(25,40) );
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),(31,32,33,34,35),(36,37,38,39,40),(41,42,43,44,45)], [(1,8,33),(2,9,34),(3,10,35),(4,6,31),(5,7,32),(11,23,20),(12,24,16),(13,25,17),(14,21,18),(15,22,19),(26,42,36),(27,43,37),(28,44,38),(29,45,39),(30,41,40)], [(1,18,42),(2,19,43),(3,20,44),(4,16,45),(5,17,41),(6,12,39),(7,13,40),(8,14,36),(9,15,37),(10,11,38),(21,26,33),(22,27,34),(23,28,35),(24,29,31),(25,30,32)], [(6,31),(7,32),(8,33),(9,34),(10,35),(11,28),(12,29),(13,30),(14,26),(15,27),(16,45),(17,41),(18,42),(19,43),(20,44),(21,36),(22,37),(23,38),(24,39),(25,40)]])
C5×C3⋊S3 is a maximal subgroup of
C32⋊Dic5 C5×S32 D15⋊S3
Matrix representation of C5×C3⋊S3 ►in GL4(𝔽31) generated by
| 8 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 |
| 0 | 0 | 2 | 0 |
| 0 | 0 | 0 | 2 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 30 | 1 |
| 0 | 0 | 30 | 0 |
| 0 | 30 | 0 | 0 |
| 1 | 30 | 0 | 0 |
| 0 | 0 | 0 | 30 |
| 0 | 0 | 1 | 30 |
| 1 | 30 | 0 | 0 |
| 0 | 30 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
G:=sub<GL(4,GF(31))| [8,0,0,0,0,8,0,0,0,0,2,0,0,0,0,2],[1,0,0,0,0,1,0,0,0,0,30,30,0,0,1,0],[0,1,0,0,30,30,0,0,0,0,0,1,0,0,30,30],[1,0,0,0,30,30,0,0,0,0,0,1,0,0,1,0] >;
C5×C3⋊S3 in GAP, Magma, Sage, TeX
C_5\times C_3\rtimes S_3
% in TeX
G:=Group("C5xC3:S3"); // GroupNames label
G:=SmallGroup(90,8);
// by ID
G=gap.SmallGroup(90,8);
# by ID
G:=PCGroup([4,-2,-5,-3,-3,242,963]);
// Polycyclic
G:=Group<a,b,c,d|a^5=b^3=c^3=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations
Export
Subgroup lattice of C5×C3⋊S3 in TeX
Character table of C5×C3⋊S3 in TeX