non-abelian, soluble, monomial, A-group
Aliases: C52⋊D9, (C5×C15).S3, C52⋊C9⋊1C2, C3.(C52⋊S3), SmallGroup(450,11)
Series: Derived ►Chief ►Lower central ►Upper central
| C52⋊C9 — C52⋊D9 |
Generators and relations for C52⋊D9
G = < a,b,c,d | a5=b5=c9=d2=1, cbc-1=ab=ba, cac-1=dad=a3b2, dbd=ab2, dcd=c-1 >
Character table of C52⋊D9
| class | 1 | 2 | 3 | 5A | 5B | 5C | 5D | 5E | 5F | 9A | 9B | 9C | 10A | 10B | 10C | 10D | 15A | 15B | 15C | 15D | 15E | 15F | 15G | 15H | |
| size | 1 | 45 | 2 | 3 | 3 | 3 | 3 | 6 | 6 | 50 | 50 | 50 | 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 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | orthogonal lifted from S3 |
| ρ4 | 2 | 0 | -1 | 2 | 2 | 2 | 2 | 2 | 2 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from D9 |
| ρ5 | 2 | 0 | -1 | 2 | 2 | 2 | 2 | 2 | 2 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from D9 |
| ρ6 | 2 | 0 | -1 | 2 | 2 | 2 | 2 | 2 | 2 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from D9 |
| ρ7 | 3 | 1 | 3 | ζ53+2ζ5 | 2ζ52+ζ5 | 2ζ54+ζ52 | ζ54+2ζ53 | 1-√5/2 | 1+√5/2 | 0 | 0 | 0 | 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 |
| ρ8 | 3 | -1 | 3 | 2ζ52+ζ5 | 2ζ54+ζ52 | ζ54+2ζ53 | ζ53+2ζ5 | 1+√5/2 | 1-√5/2 | 0 | 0 | 0 | -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 |
| ρ9 | 3 | 1 | 3 | ζ54+2ζ53 | ζ53+2ζ5 | 2ζ52+ζ5 | 2ζ54+ζ52 | 1+√5/2 | 1-√5/2 | 0 | 0 | 0 | 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 |
| ρ10 | 3 | -1 | 3 | 2ζ54+ζ52 | ζ54+2ζ53 | ζ53+2ζ5 | 2ζ52+ζ5 | 1-√5/2 | 1+√5/2 | 0 | 0 | 0 | -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 | ζ54+2ζ53 | ζ53+2ζ5 | 2ζ52+ζ5 | 2ζ54+ζ52 | 1+√5/2 | 1-√5/2 | 0 | 0 | 0 | -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 |
| ρ12 | 3 | 1 | 3 | 2ζ54+ζ52 | ζ54+2ζ53 | ζ53+2ζ5 | 2ζ52+ζ5 | 1-√5/2 | 1+√5/2 | 0 | 0 | 0 | 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 |
| ρ13 | 3 | -1 | 3 | ζ53+2ζ5 | 2ζ52+ζ5 | 2ζ54+ζ52 | ζ54+2ζ53 | 1-√5/2 | 1+√5/2 | 0 | 0 | 0 | -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 |
| ρ14 | 3 | 1 | 3 | 2ζ52+ζ5 | 2ζ54+ζ52 | ζ54+2ζ53 | ζ53+2ζ5 | 1+√5/2 | 1-√5/2 | 0 | 0 | 0 | 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 | 1+√5 | 1-√5 | 1+√5 | 1-√5 | -3+√5/2 | -3-√5/2 | 0 | 0 | 0 | 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 | 1+√5 | 1-√5 | 1+√5 | 1-√5 | -3+√5/2 | -3-√5/2 | 0 | 0 | 0 | 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 |
| ρ17 | 6 | 0 | -3 | 1+√5 | 1-√5 | 1+√5 | 1-√5 | -3+√5/2 | -3-√5/2 | 0 | 0 | 0 | 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 |
| ρ18 | 6 | 0 | -3 | 1-√5 | 1+√5 | 1-√5 | 1+√5 | -3-√5/2 | -3+√5/2 | 0 | 0 | 0 | 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 |
| ρ19 | 6 | 0 | 6 | 1-√5 | 1+√5 | 1-√5 | 1+√5 | -3-√5/2 | -3+√5/2 | 0 | 0 | 0 | 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 |
| ρ20 | 6 | 0 | -3 | 1-√5 | 1+√5 | 1-√5 | 1+√5 | -3-√5/2 | -3+√5/2 | 0 | 0 | 0 | 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 | 4ζ52+2ζ5 | 4ζ54+2ζ52 | 2ζ54+4ζ53 | 2ζ53+4ζ5 | 1+√5 | 1-√5 | 0 | 0 | 0 | 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 |
| ρ22 | 6 | 0 | -3 | 2ζ54+4ζ53 | 2ζ53+4ζ5 | 4ζ52+2ζ5 | 4ζ54+2ζ52 | 1+√5 | 1-√5 | 0 | 0 | 0 | 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 |
| ρ23 | 6 | 0 | -3 | 4ζ54+2ζ52 | 2ζ54+4ζ53 | 2ζ53+4ζ5 | 4ζ52+2ζ5 | 1-√5 | 1+√5 | 0 | 0 | 0 | 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 |
| ρ24 | 6 | 0 | -3 | 2ζ53+4ζ5 | 4ζ52+2ζ5 | 4ζ54+2ζ52 | 2ζ54+4ζ53 | 1-√5 | 1+√5 | 0 | 0 | 0 | 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 42 32 11 26)(2 33 27 43 12)(3 34 19 44 13)(4 45 35 14 20)(5 36 21 37 15)(6 28 22 38 16)(7 39 29 17 23)(8 30 24 40 18)(9 31 25 41 10)
(1 32 26 42 11)(2 12 43 27 33)(4 35 20 45 14)(5 15 37 21 36)(7 29 23 39 17)(8 18 40 24 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 31 32 33 34 35 36)(37 38 39 40 41 42 43 44 45)
(1 9)(2 8)(3 7)(4 6)(10 11)(12 18)(13 17)(14 16)(19 23)(20 22)(24 27)(25 26)(28 35)(29 34)(30 33)(31 32)(38 45)(39 44)(40 43)(41 42)
G:=sub<Sym(45)| (1,42,32,11,26)(2,33,27,43,12)(3,34,19,44,13)(4,45,35,14,20)(5,36,21,37,15)(6,28,22,38,16)(7,39,29,17,23)(8,30,24,40,18)(9,31,25,41,10), (1,32,26,42,11)(2,12,43,27,33)(4,35,20,45,14)(5,15,37,21,36)(7,29,23,39,17)(8,18,40,24,30), (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)(28,29,30,31,32,33,34,35,36)(37,38,39,40,41,42,43,44,45), (1,9)(2,8)(3,7)(4,6)(10,11)(12,18)(13,17)(14,16)(19,23)(20,22)(24,27)(25,26)(28,35)(29,34)(30,33)(31,32)(38,45)(39,44)(40,43)(41,42)>;
G:=Group( (1,42,32,11,26)(2,33,27,43,12)(3,34,19,44,13)(4,45,35,14,20)(5,36,21,37,15)(6,28,22,38,16)(7,39,29,17,23)(8,30,24,40,18)(9,31,25,41,10), (1,32,26,42,11)(2,12,43,27,33)(4,35,20,45,14)(5,15,37,21,36)(7,29,23,39,17)(8,18,40,24,30), (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)(28,29,30,31,32,33,34,35,36)(37,38,39,40,41,42,43,44,45), (1,9)(2,8)(3,7)(4,6)(10,11)(12,18)(13,17)(14,16)(19,23)(20,22)(24,27)(25,26)(28,35)(29,34)(30,33)(31,32)(38,45)(39,44)(40,43)(41,42) );
G=PermutationGroup([[(1,42,32,11,26),(2,33,27,43,12),(3,34,19,44,13),(4,45,35,14,20),(5,36,21,37,15),(6,28,22,38,16),(7,39,29,17,23),(8,30,24,40,18),(9,31,25,41,10)], [(1,32,26,42,11),(2,12,43,27,33),(4,35,20,45,14),(5,15,37,21,36),(7,29,23,39,17),(8,18,40,24,30)], [(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),(28,29,30,31,32,33,34,35,36),(37,38,39,40,41,42,43,44,45)], [(1,9),(2,8),(3,7),(4,6),(10,11),(12,18),(13,17),(14,16),(19,23),(20,22),(24,27),(25,26),(28,35),(29,34),(30,33),(31,32),(38,45),(39,44),(40,43),(41,42)]])
Matrix representation of C52⋊D9 ►in GL5(𝔽181)
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 135 | 0 | 102 |
| 0 | 0 | 0 | 125 | 0 |
| 0 | 0 | 0 | 0 | 125 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 125 | 0 | 14 |
| 0 | 0 | 0 | 1 | 33 |
| 0 | 0 | 0 | 0 | 42 |
| 127 | 131 | 0 | 0 | 0 |
| 50 | 177 | 0 | 0 | 0 |
| 0 | 0 | 161 | 1 | 38 |
| 0 | 0 | 33 | 0 | 100 |
| 0 | 0 | 41 | 0 | 20 |
| 50 | 54 | 0 | 0 | 0 |
| 4 | 131 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 87 |
| 0 | 0 | 1 | 0 | 94 |
| 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(5,GF(181))| [1,0,0,0,0,0,1,0,0,0,0,0,135,0,0,0,0,0,125,0,0,0,102,0,125],[1,0,0,0,0,0,1,0,0,0,0,0,125,0,0,0,0,0,1,0,0,0,14,33,42],[127,50,0,0,0,131,177,0,0,0,0,0,161,33,41,0,0,1,0,0,0,0,38,100,20],[50,4,0,0,0,54,131,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,87,94,1] >;
C52⋊D9 in GAP, Magma, Sage, TeX
C_5^2\rtimes D_9
% in TeX
G:=Group("C5^2:D9"); // GroupNames label
G:=SmallGroup(450,11);
// by ID
G=gap.SmallGroup(450,11);
# by ID
G:=PCGroup([5,-2,-3,-3,-5,5,101,66,182,2888,10804,4284]);
// Polycyclic
G:=Group<a,b,c,d|a^5=b^5=c^9=d^2=1,c*b*c^-1=a*b=b*a,c*a*c^-1=d*a*d=a^3*b^2,d*b*d=a*b^2,d*c*d=c^-1>;
// generators/relations
Export
Subgroup lattice of C52⋊D9 in TeX
Character table of C52⋊D9 in TeX