non-abelian, soluble, monomial, A-group
Aliases: C52⋊(C3⋊S3), C3⋊(C52⋊S3), (C5×C15)⋊1S3, C52⋊C3⋊1S3, (C3×C52⋊C3)⋊2C2, SmallGroup(450,21)
Series: Derived ►Chief ►Lower central ►Upper central
| C3×C52⋊C3 — C52⋊(C3⋊S3) |
Generators and relations for C52⋊(C3⋊S3)
G = < a,b,c,d,e | a5=b5=c3=d3=e2=1, ab=ba, ac=ca, dad-1=ab3, ae=ea, bc=cb, dbd-1=a-1b3, ebe=a-1b-1, cd=dc, ece=c-1, ede=d-1 >
Character table of C52⋊(C3⋊S3)
| class | 1 | 2 | 3A | 3B | 3C | 3D | 5A | 5B | 5C | 5D | 5E | 5F | 10A | 10B | 10C | 10D | 15A | 15B | 15C | 15D | 15E | 15F | 15G | 15H | |
| size | 1 | 45 | 2 | 50 | 50 | 50 | 3 | 3 | 3 | 3 | 6 | 6 | 45 | 45 | 45 | 45 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | |
| 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 | 2 | 0 | -1 | -1 | 2 | -1 | 2 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ4 | 2 | 0 | 2 | -1 | -1 | -1 | 2 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | orthogonal lifted from S3 |
| ρ5 | 2 | 0 | -1 | 2 | -1 | -1 | 2 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ6 | 2 | 0 | -1 | -1 | -1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ7 | 3 | -1 | 3 | 0 | 0 | 0 | ζ54+2ζ53 | 2ζ52+ζ5 | ζ53+2ζ5 | 2ζ54+ζ52 | 1-√5/2 | 1+√5/2 | -ζ5 | -ζ53 | -ζ52 | -ζ54 | 2ζ54+ζ52 | ζ54+2ζ53 | 2ζ52+ζ5 | ζ53+2ζ5 | 1+√5/2 | 1-√5/2 | 1+√5/2 | 1-√5/2 | complex lifted from C52⋊S3 |
| ρ8 | 3 | 1 | 3 | 0 | 0 | 0 | ζ54+2ζ53 | 2ζ52+ζ5 | ζ53+2ζ5 | 2ζ54+ζ52 | 1-√5/2 | 1+√5/2 | ζ5 | ζ53 | ζ52 | ζ54 | 2ζ54+ζ52 | ζ54+2ζ53 | 2ζ52+ζ5 | ζ53+2ζ5 | 1+√5/2 | 1-√5/2 | 1+√5/2 | 1-√5/2 | complex lifted from C52⋊S3 |
| ρ9 | 3 | 1 | 3 | 0 | 0 | 0 | ζ53+2ζ5 | 2ζ54+ζ52 | 2ζ52+ζ5 | ζ54+2ζ53 | 1+√5/2 | 1-√5/2 | ζ52 | ζ5 | ζ54 | ζ53 | ζ54+2ζ53 | ζ53+2ζ5 | 2ζ54+ζ52 | 2ζ52+ζ5 | 1-√5/2 | 1+√5/2 | 1-√5/2 | 1+√5/2 | complex lifted from C52⋊S3 |
| ρ10 | 3 | 1 | 3 | 0 | 0 | 0 | 2ζ54+ζ52 | ζ53+2ζ5 | ζ54+2ζ53 | 2ζ52+ζ5 | 1+√5/2 | 1-√5/2 | ζ53 | ζ54 | ζ5 | ζ52 | 2ζ52+ζ5 | 2ζ54+ζ52 | ζ53+2ζ5 | ζ54+2ζ53 | 1-√5/2 | 1+√5/2 | 1-√5/2 | 1+√5/2 | complex lifted from C52⋊S3 |
| ρ11 | 3 | 1 | 3 | 0 | 0 | 0 | 2ζ52+ζ5 | ζ54+2ζ53 | 2ζ54+ζ52 | ζ53+2ζ5 | 1-√5/2 | 1+√5/2 | ζ54 | ζ52 | ζ53 | ζ5 | ζ53+2ζ5 | 2ζ52+ζ5 | ζ54+2ζ53 | 2ζ54+ζ52 | 1+√5/2 | 1-√5/2 | 1+√5/2 | 1-√5/2 | complex lifted from C52⋊S3 |
| ρ12 | 3 | -1 | 3 | 0 | 0 | 0 | ζ53+2ζ5 | 2ζ54+ζ52 | 2ζ52+ζ5 | ζ54+2ζ53 | 1+√5/2 | 1-√5/2 | -ζ52 | -ζ5 | -ζ54 | -ζ53 | ζ54+2ζ53 | ζ53+2ζ5 | 2ζ54+ζ52 | 2ζ52+ζ5 | 1-√5/2 | 1+√5/2 | 1-√5/2 | 1+√5/2 | complex lifted from C52⋊S3 |
| ρ13 | 3 | -1 | 3 | 0 | 0 | 0 | 2ζ54+ζ52 | ζ53+2ζ5 | ζ54+2ζ53 | 2ζ52+ζ5 | 1+√5/2 | 1-√5/2 | -ζ53 | -ζ54 | -ζ5 | -ζ52 | 2ζ52+ζ5 | 2ζ54+ζ52 | ζ53+2ζ5 | ζ54+2ζ53 | 1-√5/2 | 1+√5/2 | 1-√5/2 | 1+√5/2 | complex lifted from C52⋊S3 |
| ρ14 | 3 | -1 | 3 | 0 | 0 | 0 | 2ζ52+ζ5 | ζ54+2ζ53 | 2ζ54+ζ52 | ζ53+2ζ5 | 1-√5/2 | 1+√5/2 | -ζ54 | -ζ52 | -ζ53 | -ζ5 | ζ53+2ζ5 | 2ζ52+ζ5 | ζ54+2ζ53 | 2ζ54+ζ52 | 1+√5/2 | 1-√5/2 | 1+√5/2 | 1-√5/2 | complex lifted from C52⋊S3 |
| ρ15 | 6 | 0 | 6 | 0 | 0 | 0 | 1+√5 | 1+√5 | 1-√5 | 1-√5 | -3-√5/2 | -3+√5/2 | 0 | 0 | 0 | 0 | 1-√5 | 1+√5 | 1+√5 | 1-√5 | -3+√5/2 | -3-√5/2 | -3+√5/2 | -3-√5/2 | orthogonal lifted from C52⋊S3 |
| ρ16 | 6 | 0 | -3 | 0 | 0 | 0 | 1-√5 | 1-√5 | 1+√5 | 1+√5 | -3+√5/2 | -3-√5/2 | 0 | 0 | 0 | 0 | -1-√5/2 | -1+√5/2 | -1+√5/2 | -1-√5/2 | -2ζ3ζ54-3ζ3ζ53+ζ3ζ52-ζ3-ζ54-2ζ53 | -3ζ32ζ54-2ζ32ζ52+ζ32ζ5-ζ32-2ζ54-ζ52 | ζ3ζ53-3ζ3ζ52-2ζ3ζ5-ζ3-2ζ52-ζ5 | -3ζ3ζ54-2ζ3ζ52+ζ3ζ5-ζ3-2ζ54-ζ52 | orthogonal faithful |
| ρ17 | 6 | 0 | -3 | 0 | 0 | 0 | 1+√5 | 1+√5 | 1-√5 | 1-√5 | -3-√5/2 | -3+√5/2 | 0 | 0 | 0 | 0 | -1+√5/2 | -1-√5/2 | -1-√5/2 | -1+√5/2 | -3ζ3ζ54-2ζ3ζ52+ζ3ζ5-ζ3-2ζ54-ζ52 | -2ζ3ζ54-3ζ3ζ53+ζ3ζ52-ζ3-ζ54-2ζ53 | -3ζ32ζ54-2ζ32ζ52+ζ32ζ5-ζ32-2ζ54-ζ52 | ζ3ζ53-3ζ3ζ52-2ζ3ζ5-ζ3-2ζ52-ζ5 | orthogonal faithful |
| ρ18 | 6 | 0 | 6 | 0 | 0 | 0 | 1-√5 | 1-√5 | 1+√5 | 1+√5 | -3+√5/2 | -3-√5/2 | 0 | 0 | 0 | 0 | 1+√5 | 1-√5 | 1-√5 | 1+√5 | -3-√5/2 | -3+√5/2 | -3-√5/2 | -3+√5/2 | orthogonal lifted from C52⋊S3 |
| ρ19 | 6 | 0 | -3 | 0 | 0 | 0 | 1+√5 | 1+√5 | 1-√5 | 1-√5 | -3-√5/2 | -3+√5/2 | 0 | 0 | 0 | 0 | -1+√5/2 | -1-√5/2 | -1-√5/2 | -1+√5/2 | -3ζ32ζ54-2ζ32ζ52+ζ32ζ5-ζ32-2ζ54-ζ52 | ζ3ζ53-3ζ3ζ52-2ζ3ζ5-ζ3-2ζ52-ζ5 | -3ζ3ζ54-2ζ3ζ52+ζ3ζ5-ζ3-2ζ54-ζ52 | -2ζ3ζ54-3ζ3ζ53+ζ3ζ52-ζ3-ζ54-2ζ53 | orthogonal faithful |
| ρ20 | 6 | 0 | -3 | 0 | 0 | 0 | 1-√5 | 1-√5 | 1+√5 | 1+√5 | -3+√5/2 | -3-√5/2 | 0 | 0 | 0 | 0 | -1-√5/2 | -1+√5/2 | -1+√5/2 | -1-√5/2 | ζ3ζ53-3ζ3ζ52-2ζ3ζ5-ζ3-2ζ52-ζ5 | -3ζ3ζ54-2ζ3ζ52+ζ3ζ5-ζ3-2ζ54-ζ52 | -2ζ3ζ54-3ζ3ζ53+ζ3ζ52-ζ3-ζ54-2ζ53 | -3ζ32ζ54-2ζ32ζ52+ζ32ζ5-ζ32-2ζ54-ζ52 | orthogonal faithful |
| ρ21 | 6 | 0 | -3 | 0 | 0 | 0 | 2ζ54+4ζ53 | 4ζ52+2ζ5 | 2ζ53+4ζ5 | 4ζ54+2ζ52 | 1-√5 | 1+√5 | 0 | 0 | 0 | 0 | -2ζ54-ζ52 | -ζ54-2ζ53 | -2ζ52-ζ5 | -ζ53-2ζ5 | -1-√5/2 | -1+√5/2 | -1-√5/2 | -1+√5/2 | complex faithful |
| ρ22 | 6 | 0 | -3 | 0 | 0 | 0 | 4ζ54+2ζ52 | 2ζ53+4ζ5 | 2ζ54+4ζ53 | 4ζ52+2ζ5 | 1+√5 | 1-√5 | 0 | 0 | 0 | 0 | -2ζ52-ζ5 | -2ζ54-ζ52 | -ζ53-2ζ5 | -ζ54-2ζ53 | -1+√5/2 | -1-√5/2 | -1+√5/2 | -1-√5/2 | complex faithful |
| ρ23 | 6 | 0 | -3 | 0 | 0 | 0 | 4ζ52+2ζ5 | 2ζ54+4ζ53 | 4ζ54+2ζ52 | 2ζ53+4ζ5 | 1-√5 | 1+√5 | 0 | 0 | 0 | 0 | -ζ53-2ζ5 | -2ζ52-ζ5 | -ζ54-2ζ53 | -2ζ54-ζ52 | -1-√5/2 | -1+√5/2 | -1-√5/2 | -1+√5/2 | complex faithful |
| ρ24 | 6 | 0 | -3 | 0 | 0 | 0 | 2ζ53+4ζ5 | 4ζ54+2ζ52 | 4ζ52+2ζ5 | 2ζ54+4ζ53 | 1+√5 | 1-√5 | 0 | 0 | 0 | 0 | -ζ54-2ζ53 | -ζ53-2ζ5 | -2ζ54-ζ52 | -2ζ52-ζ5 | -1+√5/2 | -1-√5/2 | -1+√5/2 | -1-√5/2 | complex faithful |
►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 3 5 2 4)(6 8 10 7 9)(11 13 15 12 14)(31 35 34 33 32)(36 40 39 38 37)(41 45 44 43 42)
(1 13 8)(2 14 9)(3 15 10)(4 11 6)(5 12 7)(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 19 34)(2 17 32)(3 20 35)(4 18 33)(5 16 31)(6 23 38)(7 21 36)(8 24 39)(9 22 37)(10 25 40)(11 28 43)(12 26 41)(13 29 44)(14 27 42)(15 30 45)
(6 11)(7 12)(8 13)(9 14)(10 15)(16 31)(17 32)(18 33)(19 34)(20 35)(21 41)(22 42)(23 43)(24 44)(25 45)(26 36)(27 37)(28 38)(29 39)(30 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), (1,3,5,2,4)(6,8,10,7,9)(11,13,15,12,14)(31,35,34,33,32)(36,40,39,38,37)(41,45,44,43,42), (1,13,8)(2,14,9)(3,15,10)(4,11,6)(5,12,7)(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), (1,19,34)(2,17,32)(3,20,35)(4,18,33)(5,16,31)(6,23,38)(7,21,36)(8,24,39)(9,22,37)(10,25,40)(11,28,43)(12,26,41)(13,29,44)(14,27,42)(15,30,45), (6,11)(7,12)(8,13)(9,14)(10,15)(16,31)(17,32)(18,33)(19,34)(20,35)(21,41)(22,42)(23,43)(24,44)(25,45)(26,36)(27,37)(28,38)(29,39)(30,40)>;
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), (1,3,5,2,4)(6,8,10,7,9)(11,13,15,12,14)(31,35,34,33,32)(36,40,39,38,37)(41,45,44,43,42), (1,13,8)(2,14,9)(3,15,10)(4,11,6)(5,12,7)(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), (1,19,34)(2,17,32)(3,20,35)(4,18,33)(5,16,31)(6,23,38)(7,21,36)(8,24,39)(9,22,37)(10,25,40)(11,28,43)(12,26,41)(13,29,44)(14,27,42)(15,30,45), (6,11)(7,12)(8,13)(9,14)(10,15)(16,31)(17,32)(18,33)(19,34)(20,35)(21,41)(22,42)(23,43)(24,44)(25,45)(26,36)(27,37)(28,38)(29,39)(30,40) );
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)], [(1,3,5,2,4),(6,8,10,7,9),(11,13,15,12,14),(31,35,34,33,32),(36,40,39,38,37),(41,45,44,43,42)], [(1,13,8),(2,14,9),(3,15,10),(4,11,6),(5,12,7),(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)], [(1,19,34),(2,17,32),(3,20,35),(4,18,33),(5,16,31),(6,23,38),(7,21,36),(8,24,39),(9,22,37),(10,25,40),(11,28,43),(12,26,41),(13,29,44),(14,27,42),(15,30,45)], [(6,11),(7,12),(8,13),(9,14),(10,15),(16,31),(17,32),(18,33),(19,34),(20,35),(21,41),(22,42),(23,43),(24,44),(25,45),(26,36),(27,37),(28,38),(29,39),(30,40)]])
Matrix representation of C52⋊(C3⋊S3) ►in GL5(𝔽31)
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 2 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 4 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 8 |
| 29 | 30 | 0 | 0 | 0 |
| 3 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 30 | 30 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 1 | 0 |
G:=sub<GL(5,GF(31))| [1,0,0,0,0,0,1,0,0,0,0,0,2,0,0,0,0,0,4,0,0,0,0,0,4],[1,0,0,0,0,0,1,0,0,0,0,0,4,0,0,0,0,0,1,0,0,0,0,0,8],[29,3,0,0,0,30,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0],[30,0,0,0,0,30,1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,1,0] >;
C52⋊(C3⋊S3) in GAP, Magma, Sage, TeX
C_5^2\rtimes (C_3\rtimes S_3)
% in TeX
G:=Group("C5^2:(C3:S3)"); // GroupNames label
G:=SmallGroup(450,21);
// by ID
G=gap.SmallGroup(450,21);
# by ID
G:=PCGroup([5,-2,-3,-3,-5,5,41,182,2888,10804,4284]);
// Polycyclic
G:=Group<a,b,c,d,e|a^5=b^5=c^3=d^3=e^2=1,a*b=b*a,a*c=c*a,d*a*d^-1=a*b^3,a*e=e*a,b*c=c*b,d*b*d^-1=a^-1*b^3,e*b*e=a^-1*b^-1,c*d=d*c,e*c*e=c^-1,e*d*e=d^-1>;
// generators/relations
Export
Subgroup lattice of C52⋊(C3⋊S3) in TeX
Character table of C52⋊(C3⋊S3) in TeX