metabelian, soluble, monomial, A-group
Aliases: C5⋊D15⋊C3, C3⋊(C52⋊C6), (C5×C15)⋊1C6, C52⋊C3⋊2S3, C52⋊2(C3×S3), (C3×C52⋊C3)⋊1C2, SmallGroup(450,24)
Series: Derived ►Chief ►Lower central ►Upper central
| C5×C15 — C5⋊D15⋊C3 |
Generators and relations for C5⋊D15⋊C3
G = < a,b,c,d | a5=b15=c2=d3=1, dbd-1=ab=ba, cac=a-1, dad-1=a3b12, cbc=b-1, dcd-1=ac >
Character table of C5⋊D15⋊C3
| class | 1 | 2 | 3A | 3B | 3C | 3D | 3E | 5A | 5B | 5C | 5D | 6A | 6B | 15A | 15B | 15C | 15D | 15E | 15F | 15G | 15H | |
| size | 1 | 75 | 2 | 25 | 25 | 50 | 50 | 6 | 6 | 6 | 6 | 75 | 75 | 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 | trivial |
| ρ2 | 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 | ζ32 | ζ3 | ζ32 | ζ3 | 1 | 1 | 1 | 1 | ζ65 | ζ6 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 6 |
| ρ4 | 1 | -1 | 1 | ζ3 | ζ32 | ζ3 | ζ32 | 1 | 1 | 1 | 1 | ζ6 | ζ65 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 6 |
| ρ5 | 1 | 1 | 1 | ζ32 | ζ3 | ζ32 | ζ3 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 3 |
| ρ6 | 1 | 1 | 1 | ζ3 | ζ32 | ζ3 | ζ32 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 3 |
| ρ7 | 2 | 0 | -1 | 2 | 2 | -1 | -1 | 2 | 2 | 2 | 2 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ8 | 2 | 0 | -1 | -1-√-3 | -1+√-3 | ζ6 | ζ65 | 2 | 2 | 2 | 2 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | complex lifted from C3×S3 |
| ρ9 | 2 | 0 | -1 | -1+√-3 | -1-√-3 | ζ65 | ζ6 | 2 | 2 | 2 | 2 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | complex lifted from C3×S3 |
| ρ10 | 6 | 0 | 6 | 0 | 0 | 0 | 0 | 1-√5 | -3+√5/2 | -3-√5/2 | 1+√5 | 0 | 0 | -3+√5/2 | 1+√5 | 1-√5 | 1-√5 | 1+√5 | -3-√5/2 | -3+√5/2 | -3-√5/2 | orthogonal lifted from C52⋊C6 |
| ρ11 | 6 | 0 | 6 | 0 | 0 | 0 | 0 | -3-√5/2 | 1-√5 | 1+√5 | -3+√5/2 | 0 | 0 | 1-√5 | -3+√5/2 | -3-√5/2 | -3-√5/2 | -3+√5/2 | 1+√5 | 1-√5 | 1+√5 | orthogonal lifted from C52⋊C6 |
| ρ12 | 6 | 0 | 6 | 0 | 0 | 0 | 0 | -3+√5/2 | 1+√5 | 1-√5 | -3-√5/2 | 0 | 0 | 1+√5 | -3-√5/2 | -3+√5/2 | -3+√5/2 | -3-√5/2 | 1-√5 | 1+√5 | 1-√5 | orthogonal lifted from C52⋊C6 |
| ρ13 | 6 | 0 | 6 | 0 | 0 | 0 | 0 | 1+√5 | -3-√5/2 | -3+√5/2 | 1-√5 | 0 | 0 | -3-√5/2 | 1-√5 | 1+√5 | 1+√5 | 1-√5 | -3+√5/2 | -3-√5/2 | -3+√5/2 | orthogonal lifted from C52⋊C6 |
| ρ14 | 6 | 0 | -3 | 0 | 0 | 0 | 0 | -3+√5/2 | 1+√5 | 1-√5 | -3-√5/2 | 0 | 0 | -1-√5/2 | -2ζ3ζ54-3ζ3ζ53+ζ3ζ52-ζ3-ζ54-2ζ53 | -3ζ32ζ54-2ζ32ζ52+ζ32ζ5-ζ32-2ζ54-ζ52 | -3ζ3ζ54-2ζ3ζ52+ζ3ζ5-ζ3-2ζ54-ζ52 | ζ3ζ53-3ζ3ζ52-2ζ3ζ5-ζ3-2ζ52-ζ5 | -1+√5/2 | -1-√5/2 | -1+√5/2 | orthogonal faithful |
| ρ15 | 6 | 0 | -3 | 0 | 0 | 0 | 0 | -3-√5/2 | 1-√5 | 1+√5 | -3+√5/2 | 0 | 0 | -1+√5/2 | -3ζ32ζ54-2ζ32ζ52+ζ32ζ5-ζ32-2ζ54-ζ52 | ζ3ζ53-3ζ3ζ52-2ζ3ζ5-ζ3-2ζ52-ζ5 | -2ζ3ζ54-3ζ3ζ53+ζ3ζ52-ζ3-ζ54-2ζ53 | -3ζ3ζ54-2ζ3ζ52+ζ3ζ5-ζ3-2ζ54-ζ52 | -1-√5/2 | -1+√5/2 | -1-√5/2 | orthogonal faithful |
| ρ16 | 6 | 0 | -3 | 0 | 0 | 0 | 0 | 1+√5 | -3-√5/2 | -3+√5/2 | 1-√5 | 0 | 0 | ζ3ζ53-3ζ3ζ52-2ζ3ζ5-ζ3-2ζ52-ζ5 | -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 | orthogonal faithful |
| ρ17 | 6 | 0 | -3 | 0 | 0 | 0 | 0 | 1-√5 | -3+√5/2 | -3-√5/2 | 1+√5 | 0 | 0 | -3ζ3ζ54-2ζ3ζ52+ζ3ζ5-ζ3-2ζ54-ζ52 | -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 | orthogonal faithful |
| ρ18 | 6 | 0 | -3 | 0 | 0 | 0 | 0 | -3+√5/2 | 1+√5 | 1-√5 | -3-√5/2 | 0 | 0 | -1-√5/2 | ζ3ζ53-3ζ3ζ52-2ζ3ζ5-ζ3-2ζ52-ζ5 | -3ζ3ζ54-2ζ3ζ52+ζ3ζ5-ζ3-2ζ54-ζ52 | -3ζ32ζ54-2ζ32ζ52+ζ32ζ5-ζ32-2ζ54-ζ52 | -2ζ3ζ54-3ζ3ζ53+ζ3ζ52-ζ3-ζ54-2ζ53 | -1+√5/2 | -1-√5/2 | -1+√5/2 | orthogonal faithful |
| ρ19 | 6 | 0 | -3 | 0 | 0 | 0 | 0 | 1+√5 | -3-√5/2 | -3+√5/2 | 1-√5 | 0 | 0 | -2ζ3ζ54-3ζ3ζ53+ζ3ζ52-ζ3-ζ54-2ζ53 | -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 | orthogonal faithful |
| ρ20 | 6 | 0 | -3 | 0 | 0 | 0 | 0 | -3-√5/2 | 1-√5 | 1+√5 | -3+√5/2 | 0 | 0 | -1+√5/2 | -3ζ3ζ54-2ζ3ζ52+ζ3ζ5-ζ3-2ζ54-ζ52 | -2ζ3ζ54-3ζ3ζ53+ζ3ζ52-ζ3-ζ54-2ζ53 | ζ3ζ53-3ζ3ζ52-2ζ3ζ5-ζ3-2ζ52-ζ5 | -3ζ32ζ54-2ζ32ζ52+ζ32ζ5-ζ32-2ζ54-ζ52 | -1-√5/2 | -1+√5/2 | -1-√5/2 | orthogonal faithful |
| ρ21 | 6 | 0 | -3 | 0 | 0 | 0 | 0 | 1-√5 | -3+√5/2 | -3-√5/2 | 1+√5 | 0 | 0 | -3ζ32ζ54-2ζ32ζ52+ζ32ζ5-ζ32-2ζ54-ζ52 | -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 | orthogonal faithful |
►On 45 points
Generators in S45
(1 15 5 11 8)(2 13 6 12 9)(3 14 4 10 7)(16 25 19 28 22)(17 26 20 29 23)(18 27 21 30 24)(31 34 37 40 43)(32 35 38 41 44)(33 36 39 42 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 11)(2 10)(3 12)(4 13)(5 15)(6 14)(7 9)(16 20)(17 19)(21 30)(22 29)(23 28)(24 27)(25 26)(31 42)(32 41)(33 40)(34 39)(35 38)(36 37)(43 45)
(1 38 30)(2 33 25)(3 43 20)(4 40 23)(5 35 18)(6 45 28)(7 37 26)(8 32 21)(9 42 16)(10 31 17)(11 41 27)(12 36 22)(13 39 19)(14 34 29)(15 44 24)
G:=sub<Sym(45)| (1,15,5,11,8)(2,13,6,12,9)(3,14,4,10,7)(16,25,19,28,22)(17,26,20,29,23)(18,27,21,30,24)(31,34,37,40,43)(32,35,38,41,44)(33,36,39,42,45), (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), (1,11)(2,10)(3,12)(4,13)(5,15)(6,14)(7,9)(16,20)(17,19)(21,30)(22,29)(23,28)(24,27)(25,26)(31,42)(32,41)(33,40)(34,39)(35,38)(36,37)(43,45), (1,38,30)(2,33,25)(3,43,20)(4,40,23)(5,35,18)(6,45,28)(7,37,26)(8,32,21)(9,42,16)(10,31,17)(11,41,27)(12,36,22)(13,39,19)(14,34,29)(15,44,24)>;
G:=Group( (1,15,5,11,8)(2,13,6,12,9)(3,14,4,10,7)(16,25,19,28,22)(17,26,20,29,23)(18,27,21,30,24)(31,34,37,40,43)(32,35,38,41,44)(33,36,39,42,45), (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), (1,11)(2,10)(3,12)(4,13)(5,15)(6,14)(7,9)(16,20)(17,19)(21,30)(22,29)(23,28)(24,27)(25,26)(31,42)(32,41)(33,40)(34,39)(35,38)(36,37)(43,45), (1,38,30)(2,33,25)(3,43,20)(4,40,23)(5,35,18)(6,45,28)(7,37,26)(8,32,21)(9,42,16)(10,31,17)(11,41,27)(12,36,22)(13,39,19)(14,34,29)(15,44,24) );
G=PermutationGroup([[(1,15,5,11,8),(2,13,6,12,9),(3,14,4,10,7),(16,25,19,28,22),(17,26,20,29,23),(18,27,21,30,24),(31,34,37,40,43),(32,35,38,41,44),(33,36,39,42,45)], [(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)], [(1,11),(2,10),(3,12),(4,13),(5,15),(6,14),(7,9),(16,20),(17,19),(21,30),(22,29),(23,28),(24,27),(25,26),(31,42),(32,41),(33,40),(34,39),(35,38),(36,37),(43,45)], [(1,38,30),(2,33,25),(3,43,20),(4,40,23),(5,35,18),(6,45,28),(7,37,26),(8,32,21),(9,42,16),(10,31,17),(11,41,27),(12,36,22),(13,39,19),(14,34,29),(15,44,24)]])
Matrix representation of C5⋊D15⋊C3 ►in GL6(𝔽31)
| 12 | 1 | 0 | 0 | 0 | 0 |
| 30 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 1 | 0 | 0 |
| 0 | 0 | 30 | 0 | 0 | 0 |
| 19 | 18 | 19 | 18 | 18 | 19 |
| 1 | 13 | 1 | 13 | 13 | 0 |
| 16 | 26 | 0 | 0 | 0 | 0 |
| 5 | 14 | 0 | 0 | 0 | 0 |
| 0 | 0 | 26 | 17 | 0 | 0 |
| 0 | 0 | 14 | 8 | 0 | 0 |
| 0 | 5 | 5 | 19 | 27 | 16 |
| 15 | 17 | 17 | 4 | 24 | 20 |
| 8 | 20 | 0 | 0 | 0 | 0 |
| 17 | 23 | 0 | 0 | 0 | 0 |
| 0 | 0 | 20 | 15 | 0 | 0 |
| 0 | 0 | 23 | 11 | 0 | 0 |
| 0 | 8 | 8 | 28 | 12 | 17 |
| 23 | 20 | 20 | 28 | 8 | 19 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 1 |
| 30 | 30 | 30 | 30 | 29 | 30 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 1 | 0 | 0 | 0 | 19 | 0 |
G:=sub<GL(6,GF(31))| [12,30,0,0,19,1,1,0,0,0,18,13,0,0,12,30,19,1,0,0,1,0,18,13,0,0,0,0,18,13,0,0,0,0,19,0],[16,5,0,0,0,15,26,14,0,0,5,17,0,0,26,14,5,17,0,0,17,8,19,4,0,0,0,0,27,24,0,0,0,0,16,20],[8,17,0,0,0,23,20,23,0,0,8,20,0,0,20,23,8,20,0,0,15,11,28,28,0,0,0,0,12,8,0,0,0,0,17,19],[0,0,0,30,0,1,0,0,0,30,0,0,1,0,0,30,0,0,0,1,0,30,0,0,0,0,12,29,1,19,0,0,1,30,0,0] >;
C5⋊D15⋊C3 in GAP, Magma, Sage, TeX
C_5\rtimes D_{15}\rtimes C_3 % in TeX
G:=Group("C5:D15:C3"); // GroupNames label
G:=SmallGroup(450,24);
// by ID
G=gap.SmallGroup(450,24);
# by ID
G:=PCGroup([5,-2,-3,-3,-5,5,182,1443,2348,9004,1359]);
// Polycyclic
G:=Group<a,b,c,d|a^5=b^15=c^2=d^3=1,d*b*d^-1=a*b=b*a,c*a*c=a^-1,d*a*d^-1=a^3*b^12,c*b*c=b^-1,d*c*d^-1=a*c>;
// generators/relations
Export
Subgroup lattice of C5⋊D15⋊C3 in TeX
Character table of C5⋊D15⋊C3 in TeX