metabelian, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: D30.C2, D15⋊2C4, C10.3D6, C6.3D10, Dic3○Dic5, Dic5⋊2S3, Dic3⋊2D5, C30.3C22, C5⋊2(C4×S3), C3⋊1(C4×D5), C15⋊6(C2×C4), C2.3(S3×D5), (C5×Dic3)⋊2C2, (C3×Dic5)⋊2C2, SmallGroup(120,10)
Series: Derived ►Chief ►Lower central ►Upper central
| C15 — D30.C2 |
Generators and relations for D30.C2
G = < a,b,c | a30=b2=1, c2=a15, bab=a-1, cac-1=a19, cbc-1=a18b >
Character table of D30.C2
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 5A | 5B | 6 | 10A | 10B | 12A | 12B | 15A | 15B | 20A | 20B | 20C | 20D | 30A | 30B | |
| size | 1 | 1 | 15 | 15 | 2 | 3 | 3 | 5 | 5 | 2 | 2 | 2 | 2 | 2 | 10 | 10 | 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 | -i | i | 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 | i | -i | 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 | i | -i | 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 | -i | i | 1 | 1 | i | -i | -i | i | -1 | -1 | linear of order 4 |
| ρ9 | 2 | 2 | 0 | 0 | -1 | 0 | 0 | -2 | -2 | 2 | 2 | -1 | 2 | 2 | 1 | 1 | -1 | -1 | 0 | 0 | 0 | 0 | -1 | -1 | orthogonal lifted from D6 |
| ρ10 | 2 | 2 | 0 | 0 | -1 | 0 | 0 | 2 | 2 | 2 | 2 | -1 | 2 | 2 | -1 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | -1 | -1 | orthogonal lifted from S3 |
| ρ11 | 2 | 2 | 0 | 0 | 2 | -2 | -2 | 0 | 0 | -1+√5/2 | -1-√5/2 | 2 | -1-√5/2 | -1+√5/2 | 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 | orthogonal lifted from D10 |
| ρ12 | 2 | 2 | 0 | 0 | 2 | 2 | 2 | 0 | 0 | -1-√5/2 | -1+√5/2 | 2 | -1+√5/2 | -1-√5/2 | 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 | orthogonal lifted from D5 |
| ρ13 | 2 | 2 | 0 | 0 | 2 | -2 | -2 | 0 | 0 | -1-√5/2 | -1+√5/2 | 2 | -1+√5/2 | -1-√5/2 | 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 | orthogonal lifted from D10 |
| ρ14 | 2 | 2 | 0 | 0 | 2 | 2 | 2 | 0 | 0 | -1+√5/2 | -1-√5/2 | 2 | -1-√5/2 | -1+√5/2 | 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 | orthogonal lifted from D5 |
| ρ15 | 2 | -2 | 0 | 0 | -1 | 0 | 0 | -2i | 2i | 2 | 2 | 1 | -2 | -2 | i | -i | -1 | -1 | 0 | 0 | 0 | 0 | 1 | 1 | complex lifted from C4×S3 |
| ρ16 | 2 | -2 | 0 | 0 | -1 | 0 | 0 | 2i | -2i | 2 | 2 | 1 | -2 | -2 | -i | i | -1 | -1 | 0 | 0 | 0 | 0 | 1 | 1 | complex lifted from C4×S3 |
| ρ17 | 2 | -2 | 0 | 0 | 2 | 2i | -2i | 0 | 0 | -1+√5/2 | -1-√5/2 | -2 | 1+√5/2 | 1-√5/2 | 0 | 0 | -1-√5/2 | -1+√5/2 | ζ4ζ54+ζ4ζ5 | ζ43ζ53+ζ43ζ52 | ζ43ζ54+ζ43ζ5 | ζ4ζ53+ζ4ζ52 | 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 | 1+√5/2 | 1-√5/2 | 0 | 0 | -1-√5/2 | -1+√5/2 | ζ43ζ54+ζ43ζ5 | ζ4ζ53+ζ4ζ52 | ζ4ζ54+ζ4ζ5 | ζ43ζ53+ζ43ζ52 | 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 | 1-√5/2 | 1+√5/2 | 0 | 0 | -1+√5/2 | -1-√5/2 | ζ43ζ53+ζ43ζ52 | ζ4ζ54+ζ4ζ5 | ζ4ζ53+ζ4ζ52 | ζ43ζ54+ζ43ζ5 | 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 | 1-√5/2 | 1+√5/2 | 0 | 0 | -1+√5/2 | -1-√5/2 | ζ4ζ53+ζ4ζ52 | ζ43ζ54+ζ43ζ5 | ζ43ζ53+ζ43ζ52 | 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 | -1-√5 | -1+√5 | 0 | 0 | 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 | 1+√5 | 1-√5 | 0 | 0 | 1+√5/2 | 1-√5/2 | 0 | 0 | 0 | 0 | -1+√5/2 | -1-√5/2 | orthogonal faithful |
| ρ23 | 4 | -4 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | -1-√5 | -1+√5 | 2 | 1-√5 | 1+√5 | 0 | 0 | 1-√5/2 | 1+√5/2 | 0 | 0 | 0 | 0 | -1-√5/2 | -1+√5/2 | orthogonal faithful |
| ρ24 | 4 | 4 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | -1-√5 | -1+√5 | -2 | -1+√5 | -1-√5 | 0 | 0 | 1-√5/2 | 1+√5/2 | 0 | 0 | 0 | 0 | 1+√5/2 | 1-√5/2 | orthogonal lifted from S3×D5 |
►On 60 points
Generators in S60
(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 30)(2 29)(3 28)(4 27)(5 26)(6 25)(7 24)(8 23)(9 22)(10 21)(11 20)(12 19)(13 18)(14 17)(15 16)(31 50)(32 49)(33 48)(34 47)(35 46)(36 45)(37 44)(38 43)(39 42)(40 41)(51 60)(52 59)(53 58)(54 57)(55 56)
(1 41 16 56)(2 60 17 45)(3 49 18 34)(4 38 19 53)(5 57 20 42)(6 46 21 31)(7 35 22 50)(8 54 23 39)(9 43 24 58)(10 32 25 47)(11 51 26 36)(12 40 27 55)(13 59 28 44)(14 48 29 33)(15 37 30 52)
G:=sub<Sym(60)| (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), (1,30)(2,29)(3,28)(4,27)(5,26)(6,25)(7,24)(8,23)(9,22)(10,21)(11,20)(12,19)(13,18)(14,17)(15,16)(31,50)(32,49)(33,48)(34,47)(35,46)(36,45)(37,44)(38,43)(39,42)(40,41)(51,60)(52,59)(53,58)(54,57)(55,56), (1,41,16,56)(2,60,17,45)(3,49,18,34)(4,38,19,53)(5,57,20,42)(6,46,21,31)(7,35,22,50)(8,54,23,39)(9,43,24,58)(10,32,25,47)(11,51,26,36)(12,40,27,55)(13,59,28,44)(14,48,29,33)(15,37,30,52)>;
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)(31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60), (1,30)(2,29)(3,28)(4,27)(5,26)(6,25)(7,24)(8,23)(9,22)(10,21)(11,20)(12,19)(13,18)(14,17)(15,16)(31,50)(32,49)(33,48)(34,47)(35,46)(36,45)(37,44)(38,43)(39,42)(40,41)(51,60)(52,59)(53,58)(54,57)(55,56), (1,41,16,56)(2,60,17,45)(3,49,18,34)(4,38,19,53)(5,57,20,42)(6,46,21,31)(7,35,22,50)(8,54,23,39)(9,43,24,58)(10,32,25,47)(11,51,26,36)(12,40,27,55)(13,59,28,44)(14,48,29,33)(15,37,30,52) );
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),(31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60)], [(1,30),(2,29),(3,28),(4,27),(5,26),(6,25),(7,24),(8,23),(9,22),(10,21),(11,20),(12,19),(13,18),(14,17),(15,16),(31,50),(32,49),(33,48),(34,47),(35,46),(36,45),(37,44),(38,43),(39,42),(40,41),(51,60),(52,59),(53,58),(54,57),(55,56)], [(1,41,16,56),(2,60,17,45),(3,49,18,34),(4,38,19,53),(5,57,20,42),(6,46,21,31),(7,35,22,50),(8,54,23,39),(9,43,24,58),(10,32,25,47),(11,51,26,36),(12,40,27,55),(13,59,28,44),(14,48,29,33),(15,37,30,52)]])
D30.C2 is a maximal subgroup of
D15⋊C8 Dic3.F5 D60⋊C2 D15⋊Q8 C12.28D10 C4×S3×D5 Dic5.D6 Dic3.D10 D10⋊D6 D90.C2 C30.D6 C6.D30 D30.S3 Dic5.7S4 Dic5⋊2S4
D30.C2 is a maximal quotient of
D15⋊2C8 D30.5C4 Dic3×Dic5 D30⋊4C4 Dic15⋊5C4 D90.C2 C30.D6 C6.D30 D30.S3 Dic5⋊2S4
Matrix representation of D30.C2 ►in GL4(𝔽61) generated by
| 0 | 60 | 0 | 0 |
| 1 | 1 | 0 | 0 |
| 0 | 0 | 0 | 60 |
| 0 | 0 | 1 | 17 |
| 0 | 60 | 0 | 0 |
| 60 | 0 | 0 | 0 |
| 0 | 0 | 44 | 60 |
| 0 | 0 | 44 | 17 |
| 50 | 0 | 0 | 0 |
| 0 | 50 | 0 | 0 |
| 0 | 0 | 44 | 17 |
| 0 | 0 | 1 | 17 |
G:=sub<GL(4,GF(61))| [0,1,0,0,60,1,0,0,0,0,0,1,0,0,60,17],[0,60,0,0,60,0,0,0,0,0,44,44,0,0,60,17],[50,0,0,0,0,50,0,0,0,0,44,1,0,0,17,17] >;
D30.C2 in GAP, Magma, Sage, TeX
D_{30}.C_2 % in TeX
G:=Group("D30.C2"); // GroupNames label
G:=SmallGroup(120,10);
// by ID
G=gap.SmallGroup(120,10);
# by ID
G:=PCGroup([5,-2,-2,-2,-3,-5,20,26,168,2404]);
// Polycyclic
G:=Group<a,b,c|a^30=b^2=1,c^2=a^15,b*a*b=a^-1,c*a*c^-1=a^19,c*b*c^-1=a^18*b>;
// generators/relations
Export
Subgroup lattice of D30.C2 in TeX
Character table of D30.C2 in TeX