direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: D5×Dic3, C10.1D6, C6.1D10, D10.2S3, Dic15⋊2C2, C30.1C22, C3⋊3(C4×D5), C15⋊4(C2×C4), (C3×D5)⋊1C4, C2.1(S3×D5), C5⋊2(C2×Dic3), (C6×D5).1C2, (C5×Dic3)⋊1C2, SmallGroup(120,8)
Series: Derived ►Chief ►Lower central ►Upper central
| C15 — D5×Dic3 |
Generators and relations for D5×Dic3
G = < a,b,c,d | a5=b2=c6=1, d2=c3, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >
Character table of D5×Dic3
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 5A | 5B | 6A | 6B | 6C | 10A | 10B | 15A | 15B | 20A | 20B | 20C | 20D | 30A | 30B | |
| size | 1 | 1 | 5 | 5 | 2 | 3 | 3 | 15 | 15 | 2 | 2 | 2 | 10 | 10 | 2 | 2 | 4 | 4 | 6 | 6 | 6 | 6 | 4 | 4 | |
| 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 | 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 |
| ρ4 | 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 |
| ρ5 | 1 | -1 | -1 | 1 | 1 | -i | i | i | -i | 1 | 1 | -1 | -1 | 1 | -1 | -1 | 1 | 1 | i | -i | -i | i | -1 | -1 | linear of order 4 |
| ρ6 | 1 | -1 | 1 | -1 | 1 | -i | i | -i | i | 1 | 1 | -1 | 1 | -1 | -1 | -1 | 1 | 1 | i | -i | -i | i | -1 | -1 | linear of order 4 |
| ρ7 | 1 | -1 | 1 | -1 | 1 | i | -i | i | -i | 1 | 1 | -1 | 1 | -1 | -1 | -1 | 1 | 1 | -i | i | i | -i | -1 | -1 | linear of order 4 |
| ρ8 | 1 | -1 | -1 | 1 | 1 | i | -i | -i | i | 1 | 1 | -1 | -1 | 1 | -1 | -1 | 1 | 1 | -i | i | i | -i | -1 | -1 | linear of order 4 |
| ρ9 | 2 | 2 | -2 | -2 | -1 | 0 | 0 | 0 | 0 | 2 | 2 | -1 | 1 | 1 | 2 | 2 | -1 | -1 | 0 | 0 | 0 | 0 | -1 | -1 | orthogonal lifted from D6 |
| ρ10 | 2 | 2 | 0 | 0 | 2 | -2 | -2 | 0 | 0 | -1+√5/2 | -1-√5/2 | 2 | 0 | 0 | -1+√5/2 | -1-√5/2 | -1+√5/2 | -1-√5/2 | 1-√5/2 | 1-√5/2 | 1+√5/2 | 1+√5/2 | -1+√5/2 | -1-√5/2 | orthogonal lifted from D10 |
| ρ11 | 2 | 2 | 0 | 0 | 2 | 2 | 2 | 0 | 0 | -1-√5/2 | -1+√5/2 | 2 | 0 | 0 | -1-√5/2 | -1+√5/2 | -1-√5/2 | -1+√5/2 | -1-√5/2 | -1-√5/2 | -1+√5/2 | -1+√5/2 | -1-√5/2 | -1+√5/2 | orthogonal lifted from D5 |
| ρ12 | 2 | 2 | 2 | 2 | -1 | 0 | 0 | 0 | 0 | 2 | 2 | -1 | -1 | -1 | 2 | 2 | -1 | -1 | 0 | 0 | 0 | 0 | -1 | -1 | orthogonal lifted from S3 |
| ρ13 | 2 | 2 | 0 | 0 | 2 | -2 | -2 | 0 | 0 | -1-√5/2 | -1+√5/2 | 2 | 0 | 0 | -1-√5/2 | -1+√5/2 | -1-√5/2 | -1+√5/2 | 1+√5/2 | 1+√5/2 | 1-√5/2 | 1-√5/2 | -1-√5/2 | -1+√5/2 | orthogonal lifted from D10 |
| ρ14 | 2 | 2 | 0 | 0 | 2 | 2 | 2 | 0 | 0 | -1+√5/2 | -1-√5/2 | 2 | 0 | 0 | -1+√5/2 | -1-√5/2 | -1+√5/2 | -1-√5/2 | -1+√5/2 | -1+√5/2 | -1-√5/2 | -1-√5/2 | -1+√5/2 | -1-√5/2 | orthogonal lifted from D5 |
| ρ15 | 2 | -2 | -2 | 2 | -1 | 0 | 0 | 0 | 0 | 2 | 2 | 1 | 1 | -1 | -2 | -2 | -1 | -1 | 0 | 0 | 0 | 0 | 1 | 1 | symplectic lifted from Dic3, Schur index 2 |
| ρ16 | 2 | -2 | 2 | -2 | -1 | 0 | 0 | 0 | 0 | 2 | 2 | 1 | -1 | 1 | -2 | -2 | -1 | -1 | 0 | 0 | 0 | 0 | 1 | 1 | symplectic lifted from Dic3, Schur index 2 |
| ρ17 | 2 | -2 | 0 | 0 | 2 | 2i | -2i | 0 | 0 | -1-√5/2 | -1+√5/2 | -2 | 0 | 0 | 1+√5/2 | 1-√5/2 | -1-√5/2 | -1+√5/2 | ζ43ζ53+ζ43ζ52 | ζ4ζ53+ζ4ζ52 | ζ4ζ54+ζ4ζ5 | ζ43ζ54+ζ43ζ5 | 1+√5/2 | 1-√5/2 | complex lifted from C4×D5 |
| ρ18 | 2 | -2 | 0 | 0 | 2 | -2i | 2i | 0 | 0 | -1+√5/2 | -1-√5/2 | -2 | 0 | 0 | 1-√5/2 | 1+√5/2 | -1+√5/2 | -1-√5/2 | ζ4ζ54+ζ4ζ5 | ζ43ζ54+ζ43ζ5 | ζ43ζ53+ζ43ζ52 | ζ4ζ53+ζ4ζ52 | 1-√5/2 | 1+√5/2 | complex lifted from C4×D5 |
| ρ19 | 2 | -2 | 0 | 0 | 2 | 2i | -2i | 0 | 0 | -1+√5/2 | -1-√5/2 | -2 | 0 | 0 | 1-√5/2 | 1+√5/2 | -1+√5/2 | -1-√5/2 | ζ43ζ54+ζ43ζ5 | ζ4ζ54+ζ4ζ5 | ζ4ζ53+ζ4ζ52 | ζ43ζ53+ζ43ζ52 | 1-√5/2 | 1+√5/2 | complex lifted from C4×D5 |
| ρ20 | 2 | -2 | 0 | 0 | 2 | -2i | 2i | 0 | 0 | -1-√5/2 | -1+√5/2 | -2 | 0 | 0 | 1+√5/2 | 1-√5/2 | -1-√5/2 | -1+√5/2 | ζ4ζ53+ζ4ζ52 | ζ43ζ53+ζ43ζ52 | ζ43ζ54+ζ43ζ5 | ζ4ζ54+ζ4ζ5 | 1+√5/2 | 1-√5/2 | complex lifted from C4×D5 |
| ρ21 | 4 | 4 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | -1-√5 | -1+√5 | -2 | 0 | 0 | -1-√5 | -1+√5 | 1+√5/2 | 1-√5/2 | 0 | 0 | 0 | 0 | 1+√5/2 | 1-√5/2 | orthogonal lifted from S3×D5 |
| ρ22 | 4 | 4 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | -1+√5 | -1-√5 | -2 | 0 | 0 | -1+√5 | -1-√5 | 1-√5/2 | 1+√5/2 | 0 | 0 | 0 | 0 | 1-√5/2 | 1+√5/2 | orthogonal lifted from S3×D5 |
| ρ23 | 4 | -4 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | -1-√5 | -1+√5 | 2 | 0 | 0 | 1+√5 | 1-√5 | 1+√5/2 | 1-√5/2 | 0 | 0 | 0 | 0 | -1-√5/2 | -1+√5/2 | symplectic faithful, Schur index 2 |
| ρ24 | 4 | -4 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | -1+√5 | -1-√5 | 2 | 0 | 0 | 1-√5 | 1+√5 | 1-√5/2 | 1+√5/2 | 0 | 0 | 0 | 0 | -1+√5/2 | -1-√5/2 | symplectic faithful, Schur index 2 |
►On 60 points
Generators in S60
(1 34 46 53 42)(2 35 47 54 37)(3 36 48 49 38)(4 31 43 50 39)(5 32 44 51 40)(6 33 45 52 41)(7 24 55 26 14)(8 19 56 27 15)(9 20 57 28 16)(10 21 58 29 17)(11 22 59 30 18)(12 23 60 25 13)
(1 42)(2 37)(3 38)(4 39)(5 40)(6 41)(7 26)(8 27)(9 28)(10 29)(11 30)(12 25)(19 56)(20 57)(21 58)(22 59)(23 60)(24 55)(31 50)(32 51)(33 52)(34 53)(35 54)(36 49)
(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 46 47 48)(49 50 51 52 53 54)(55 56 57 58 59 60)
(1 59 4 56)(2 58 5 55)(3 57 6 60)(7 54 10 51)(8 53 11 50)(9 52 12 49)(13 48 16 45)(14 47 17 44)(15 46 18 43)(19 42 22 39)(20 41 23 38)(21 40 24 37)(25 36 28 33)(26 35 29 32)(27 34 30 31)
G:=sub<Sym(60)| (1,34,46,53,42)(2,35,47,54,37)(3,36,48,49,38)(4,31,43,50,39)(5,32,44,51,40)(6,33,45,52,41)(7,24,55,26,14)(8,19,56,27,15)(9,20,57,28,16)(10,21,58,29,17)(11,22,59,30,18)(12,23,60,25,13), (1,42)(2,37)(3,38)(4,39)(5,40)(6,41)(7,26)(8,27)(9,28)(10,29)(11,30)(12,25)(19,56)(20,57)(21,58)(22,59)(23,60)(24,55)(31,50)(32,51)(33,52)(34,53)(35,54)(36,49), (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)(43,44,45,46,47,48)(49,50,51,52,53,54)(55,56,57,58,59,60), (1,59,4,56)(2,58,5,55)(3,57,6,60)(7,54,10,51)(8,53,11,50)(9,52,12,49)(13,48,16,45)(14,47,17,44)(15,46,18,43)(19,42,22,39)(20,41,23,38)(21,40,24,37)(25,36,28,33)(26,35,29,32)(27,34,30,31)>;
G:=Group( (1,34,46,53,42)(2,35,47,54,37)(3,36,48,49,38)(4,31,43,50,39)(5,32,44,51,40)(6,33,45,52,41)(7,24,55,26,14)(8,19,56,27,15)(9,20,57,28,16)(10,21,58,29,17)(11,22,59,30,18)(12,23,60,25,13), (1,42)(2,37)(3,38)(4,39)(5,40)(6,41)(7,26)(8,27)(9,28)(10,29)(11,30)(12,25)(19,56)(20,57)(21,58)(22,59)(23,60)(24,55)(31,50)(32,51)(33,52)(34,53)(35,54)(36,49), (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)(43,44,45,46,47,48)(49,50,51,52,53,54)(55,56,57,58,59,60), (1,59,4,56)(2,58,5,55)(3,57,6,60)(7,54,10,51)(8,53,11,50)(9,52,12,49)(13,48,16,45)(14,47,17,44)(15,46,18,43)(19,42,22,39)(20,41,23,38)(21,40,24,37)(25,36,28,33)(26,35,29,32)(27,34,30,31) );
G=PermutationGroup([[(1,34,46,53,42),(2,35,47,54,37),(3,36,48,49,38),(4,31,43,50,39),(5,32,44,51,40),(6,33,45,52,41),(7,24,55,26,14),(8,19,56,27,15),(9,20,57,28,16),(10,21,58,29,17),(11,22,59,30,18),(12,23,60,25,13)], [(1,42),(2,37),(3,38),(4,39),(5,40),(6,41),(7,26),(8,27),(9,28),(10,29),(11,30),(12,25),(19,56),(20,57),(21,58),(22,59),(23,60),(24,55),(31,50),(32,51),(33,52),(34,53),(35,54),(36,49)], [(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),(43,44,45,46,47,48),(49,50,51,52,53,54),(55,56,57,58,59,60)], [(1,59,4,56),(2,58,5,55),(3,57,6,60),(7,54,10,51),(8,53,11,50),(9,52,12,49),(13,48,16,45),(14,47,17,44),(15,46,18,43),(19,42,22,39),(20,41,23,38),(21,40,24,37),(25,36,28,33),(26,35,29,32),(27,34,30,31)]])
D5×Dic3 is a maximal subgroup of
Dic3⋊F5 D20⋊5S3 D20⋊S3 C4×S3×D5 Dic5.D6 C30.C23 D30.S3
D5×Dic3 is a maximal quotient of C20.32D6 D10⋊Dic3 C30.Q8 D30.S3
Matrix representation of D5×Dic3 ►in GL4(𝔽61) generated by
| 0 | 1 | 0 | 0 |
| 60 | 17 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 60 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 |
| 0 | 0 | 60 | 1 |
| 0 | 0 | 60 | 0 |
| 50 | 0 | 0 | 0 |
| 0 | 50 | 0 | 0 |
| 0 | 0 | 15 | 23 |
| 0 | 0 | 38 | 46 |
G:=sub<GL(4,GF(61))| [0,60,0,0,1,17,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[60,0,0,0,0,60,0,0,0,0,60,60,0,0,1,0],[50,0,0,0,0,50,0,0,0,0,15,38,0,0,23,46] >;
D5×Dic3 in GAP, Magma, Sage, TeX
D_5\times {\rm Dic}_3 % in TeX
G:=Group("D5xDic3"); // GroupNames label
G:=SmallGroup(120,8);
// by ID
G=gap.SmallGroup(120,8);
# by ID
G:=PCGroup([5,-2,-2,-2,-3,-5,26,168,2404]);
// Polycyclic
G:=Group<a,b,c,d|a^5=b^2=c^6=1,d^2=c^3,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations
Export
Subgroup lattice of D5×Dic3 in TeX
Character table of D5×Dic3 in TeX