direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C5×Dic3, C3⋊C20, C15⋊5C4, C6.C10, C30.3C2, C10.2S3, C2.(C5×S3), SmallGroup(60,1)
Series: Derived ►Chief ►Lower central ►Upper central
| C3 — C5×Dic3 |
Generators and relations for C5×Dic3
G = < a,b,c | a5=b6=1, c2=b3, ab=ba, ac=ca, cbc-1=b-1 >
Character table of C5×Dic3
| class | 1 | 2 | 3 | 4A | 4B | 5A | 5B | 5C | 5D | 6 | 10A | 10B | 10C | 10D | 15A | 15B | 15C | 15D | 20A | 20B | 20C | 20D | 20E | 20F | 20G | 20H | 30A | 30B | 30C | 30D | |
| size | 1 | 1 | 2 | 3 | 3 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 2 | 2 | 2 | 2 | |
| 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 | 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 | -1 | -1 | 1 | 1 | 1 | 1 | linear of order 2 |
| ρ3 | 1 | -1 | 1 | -i | i | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | -i | -i | i | -i | i | -i | i | i | -1 | -1 | -1 | -1 | linear of order 4 |
| ρ4 | 1 | -1 | 1 | i | -i | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | i | i | -i | i | -i | i | -i | -i | -1 | -1 | -1 | -1 | linear of order 4 |
| ρ5 | 1 | 1 | 1 | -1 | -1 | ζ53 | ζ5 | ζ54 | ζ52 | 1 | ζ54 | ζ5 | ζ52 | ζ53 | ζ54 | ζ5 | ζ53 | ζ52 | -ζ52 | -ζ53 | -ζ5 | -ζ5 | -ζ54 | -ζ54 | -ζ52 | -ζ53 | ζ54 | ζ52 | ζ5 | ζ53 | linear of order 10 |
| ρ6 | 1 | 1 | 1 | -1 | -1 | ζ5 | ζ52 | ζ53 | ζ54 | 1 | ζ53 | ζ52 | ζ54 | ζ5 | ζ53 | ζ52 | ζ5 | ζ54 | -ζ54 | -ζ5 | -ζ52 | -ζ52 | -ζ53 | -ζ53 | -ζ54 | -ζ5 | ζ53 | ζ54 | ζ52 | ζ5 | linear of order 10 |
| ρ7 | 1 | 1 | 1 | 1 | 1 | ζ52 | ζ54 | ζ5 | ζ53 | 1 | ζ5 | ζ54 | ζ53 | ζ52 | ζ5 | ζ54 | ζ52 | ζ53 | ζ53 | ζ52 | ζ54 | ζ54 | ζ5 | ζ5 | ζ53 | ζ52 | ζ5 | ζ53 | ζ54 | ζ52 | linear of order 5 |
| ρ8 | 1 | 1 | 1 | 1 | 1 | ζ5 | ζ52 | ζ53 | ζ54 | 1 | ζ53 | ζ52 | ζ54 | ζ5 | ζ53 | ζ52 | ζ5 | ζ54 | ζ54 | ζ5 | ζ52 | ζ52 | ζ53 | ζ53 | ζ54 | ζ5 | ζ53 | ζ54 | ζ52 | ζ5 | linear of order 5 |
| ρ9 | 1 | 1 | 1 | 1 | 1 | ζ54 | ζ53 | ζ52 | ζ5 | 1 | ζ52 | ζ53 | ζ5 | ζ54 | ζ52 | ζ53 | ζ54 | ζ5 | ζ5 | ζ54 | ζ53 | ζ53 | ζ52 | ζ52 | ζ5 | ζ54 | ζ52 | ζ5 | ζ53 | ζ54 | linear of order 5 |
| ρ10 | 1 | 1 | 1 | 1 | 1 | ζ53 | ζ5 | ζ54 | ζ52 | 1 | ζ54 | ζ5 | ζ52 | ζ53 | ζ54 | ζ5 | ζ53 | ζ52 | ζ52 | ζ53 | ζ5 | ζ5 | ζ54 | ζ54 | ζ52 | ζ53 | ζ54 | ζ52 | ζ5 | ζ53 | linear of order 5 |
| ρ11 | 1 | 1 | 1 | -1 | -1 | ζ52 | ζ54 | ζ5 | ζ53 | 1 | ζ5 | ζ54 | ζ53 | ζ52 | ζ5 | ζ54 | ζ52 | ζ53 | -ζ53 | -ζ52 | -ζ54 | -ζ54 | -ζ5 | -ζ5 | -ζ53 | -ζ52 | ζ5 | ζ53 | ζ54 | ζ52 | linear of order 10 |
| ρ12 | 1 | 1 | 1 | -1 | -1 | ζ54 | ζ53 | ζ52 | ζ5 | 1 | ζ52 | ζ53 | ζ5 | ζ54 | ζ52 | ζ53 | ζ54 | ζ5 | -ζ5 | -ζ54 | -ζ53 | -ζ53 | -ζ52 | -ζ52 | -ζ5 | -ζ54 | ζ52 | ζ5 | ζ53 | ζ54 | linear of order 10 |
| ρ13 | 1 | -1 | 1 | -i | i | ζ52 | ζ54 | ζ5 | ζ53 | -1 | -ζ5 | -ζ54 | -ζ53 | -ζ52 | ζ5 | ζ54 | ζ52 | ζ53 | ζ43ζ53 | ζ43ζ52 | ζ4ζ54 | ζ43ζ54 | ζ4ζ5 | ζ43ζ5 | ζ4ζ53 | ζ4ζ52 | -ζ5 | -ζ53 | -ζ54 | -ζ52 | linear of order 20 |
| ρ14 | 1 | -1 | 1 | i | -i | ζ5 | ζ52 | ζ53 | ζ54 | -1 | -ζ53 | -ζ52 | -ζ54 | -ζ5 | ζ53 | ζ52 | ζ5 | ζ54 | ζ4ζ54 | ζ4ζ5 | ζ43ζ52 | ζ4ζ52 | ζ43ζ53 | ζ4ζ53 | ζ43ζ54 | ζ43ζ5 | -ζ53 | -ζ54 | -ζ52 | -ζ5 | linear of order 20 |
| ρ15 | 1 | -1 | 1 | -i | i | ζ53 | ζ5 | ζ54 | ζ52 | -1 | -ζ54 | -ζ5 | -ζ52 | -ζ53 | ζ54 | ζ5 | ζ53 | ζ52 | ζ43ζ52 | ζ43ζ53 | ζ4ζ5 | ζ43ζ5 | ζ4ζ54 | ζ43ζ54 | ζ4ζ52 | ζ4ζ53 | -ζ54 | -ζ52 | -ζ5 | -ζ53 | linear of order 20 |
| ρ16 | 1 | -1 | 1 | -i | i | ζ54 | ζ53 | ζ52 | ζ5 | -1 | -ζ52 | -ζ53 | -ζ5 | -ζ54 | ζ52 | ζ53 | ζ54 | ζ5 | ζ43ζ5 | ζ43ζ54 | ζ4ζ53 | ζ43ζ53 | ζ4ζ52 | ζ43ζ52 | ζ4ζ5 | ζ4ζ54 | -ζ52 | -ζ5 | -ζ53 | -ζ54 | linear of order 20 |
| ρ17 | 1 | -1 | 1 | i | -i | ζ52 | ζ54 | ζ5 | ζ53 | -1 | -ζ5 | -ζ54 | -ζ53 | -ζ52 | ζ5 | ζ54 | ζ52 | ζ53 | ζ4ζ53 | ζ4ζ52 | ζ43ζ54 | ζ4ζ54 | ζ43ζ5 | ζ4ζ5 | ζ43ζ53 | ζ43ζ52 | -ζ5 | -ζ53 | -ζ54 | -ζ52 | linear of order 20 |
| ρ18 | 1 | -1 | 1 | i | -i | ζ53 | ζ5 | ζ54 | ζ52 | -1 | -ζ54 | -ζ5 | -ζ52 | -ζ53 | ζ54 | ζ5 | ζ53 | ζ52 | ζ4ζ52 | ζ4ζ53 | ζ43ζ5 | ζ4ζ5 | ζ43ζ54 | ζ4ζ54 | ζ43ζ52 | ζ43ζ53 | -ζ54 | -ζ52 | -ζ5 | -ζ53 | linear of order 20 |
| ρ19 | 1 | -1 | 1 | -i | i | ζ5 | ζ52 | ζ53 | ζ54 | -1 | -ζ53 | -ζ52 | -ζ54 | -ζ5 | ζ53 | ζ52 | ζ5 | ζ54 | ζ43ζ54 | ζ43ζ5 | ζ4ζ52 | ζ43ζ52 | ζ4ζ53 | ζ43ζ53 | ζ4ζ54 | ζ4ζ5 | -ζ53 | -ζ54 | -ζ52 | -ζ5 | linear of order 20 |
| ρ20 | 1 | -1 | 1 | i | -i | ζ54 | ζ53 | ζ52 | ζ5 | -1 | -ζ52 | -ζ53 | -ζ5 | -ζ54 | ζ52 | ζ53 | ζ54 | ζ5 | ζ4ζ5 | ζ4ζ54 | ζ43ζ53 | ζ4ζ53 | ζ43ζ52 | ζ4ζ52 | ζ43ζ5 | ζ43ζ54 | -ζ52 | -ζ5 | -ζ53 | -ζ54 | linear of order 20 |
| ρ21 | 2 | 2 | -1 | 0 | 0 | 2 | 2 | 2 | 2 | -1 | 2 | 2 | 2 | 2 | -1 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ22 | 2 | -2 | -1 | 0 | 0 | 2 | 2 | 2 | 2 | 1 | -2 | -2 | -2 | -2 | -1 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | symplectic lifted from Dic3, Schur index 2 |
| ρ23 | 2 | -2 | -1 | 0 | 0 | 2ζ52 | 2ζ54 | 2ζ5 | 2ζ53 | 1 | -2ζ5 | -2ζ54 | -2ζ53 | -2ζ52 | -ζ5 | -ζ54 | -ζ52 | -ζ53 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ5 | ζ53 | ζ54 | ζ52 | complex faithful |
| ρ24 | 2 | -2 | -1 | 0 | 0 | 2ζ53 | 2ζ5 | 2ζ54 | 2ζ52 | 1 | -2ζ54 | -2ζ5 | -2ζ52 | -2ζ53 | -ζ54 | -ζ5 | -ζ53 | -ζ52 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ54 | ζ52 | ζ5 | ζ53 | complex faithful |
| ρ25 | 2 | 2 | -1 | 0 | 0 | 2ζ53 | 2ζ5 | 2ζ54 | 2ζ52 | -1 | 2ζ54 | 2ζ5 | 2ζ52 | 2ζ53 | -ζ54 | -ζ5 | -ζ53 | -ζ52 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -ζ54 | -ζ52 | -ζ5 | -ζ53 | complex lifted from C5×S3 |
| ρ26 | 2 | -2 | -1 | 0 | 0 | 2ζ54 | 2ζ53 | 2ζ52 | 2ζ5 | 1 | -2ζ52 | -2ζ53 | -2ζ5 | -2ζ54 | -ζ52 | -ζ53 | -ζ54 | -ζ5 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ52 | ζ5 | ζ53 | ζ54 | complex faithful |
| ρ27 | 2 | -2 | -1 | 0 | 0 | 2ζ5 | 2ζ52 | 2ζ53 | 2ζ54 | 1 | -2ζ53 | -2ζ52 | -2ζ54 | -2ζ5 | -ζ53 | -ζ52 | -ζ5 | -ζ54 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ53 | ζ54 | ζ52 | ζ5 | complex faithful |
| ρ28 | 2 | 2 | -1 | 0 | 0 | 2ζ5 | 2ζ52 | 2ζ53 | 2ζ54 | -1 | 2ζ53 | 2ζ52 | 2ζ54 | 2ζ5 | -ζ53 | -ζ52 | -ζ5 | -ζ54 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -ζ53 | -ζ54 | -ζ52 | -ζ5 | complex lifted from C5×S3 |
| ρ29 | 2 | 2 | -1 | 0 | 0 | 2ζ52 | 2ζ54 | 2ζ5 | 2ζ53 | -1 | 2ζ5 | 2ζ54 | 2ζ53 | 2ζ52 | -ζ5 | -ζ54 | -ζ52 | -ζ53 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -ζ5 | -ζ53 | -ζ54 | -ζ52 | complex lifted from C5×S3 |
| ρ30 | 2 | 2 | -1 | 0 | 0 | 2ζ54 | 2ζ53 | 2ζ52 | 2ζ5 | -1 | 2ζ52 | 2ζ53 | 2ζ5 | 2ζ54 | -ζ52 | -ζ53 | -ζ54 | -ζ5 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -ζ52 | -ζ5 | -ζ53 | -ζ54 | complex lifted from C5×S3 |
►Regular action on 60 points
Generators in S60
(1 29 23 17 11)(2 30 24 18 12)(3 25 19 13 7)(4 26 20 14 8)(5 27 21 15 9)(6 28 22 16 10)(31 55 49 43 37)(32 56 50 44 38)(33 57 51 45 39)(34 58 52 46 40)(35 59 53 47 41)(36 60 54 48 42)
(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 34 4 31)(2 33 5 36)(3 32 6 35)(7 38 10 41)(8 37 11 40)(9 42 12 39)(13 44 16 47)(14 43 17 46)(15 48 18 45)(19 50 22 53)(20 49 23 52)(21 54 24 51)(25 56 28 59)(26 55 29 58)(27 60 30 57)
G:=sub<Sym(60)| (1,29,23,17,11)(2,30,24,18,12)(3,25,19,13,7)(4,26,20,14,8)(5,27,21,15,9)(6,28,22,16,10)(31,55,49,43,37)(32,56,50,44,38)(33,57,51,45,39)(34,58,52,46,40)(35,59,53,47,41)(36,60,54,48,42), (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), (1,34,4,31)(2,33,5,36)(3,32,6,35)(7,38,10,41)(8,37,11,40)(9,42,12,39)(13,44,16,47)(14,43,17,46)(15,48,18,45)(19,50,22,53)(20,49,23,52)(21,54,24,51)(25,56,28,59)(26,55,29,58)(27,60,30,57)>;
G:=Group( (1,29,23,17,11)(2,30,24,18,12)(3,25,19,13,7)(4,26,20,14,8)(5,27,21,15,9)(6,28,22,16,10)(31,55,49,43,37)(32,56,50,44,38)(33,57,51,45,39)(34,58,52,46,40)(35,59,53,47,41)(36,60,54,48,42), (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), (1,34,4,31)(2,33,5,36)(3,32,6,35)(7,38,10,41)(8,37,11,40)(9,42,12,39)(13,44,16,47)(14,43,17,46)(15,48,18,45)(19,50,22,53)(20,49,23,52)(21,54,24,51)(25,56,28,59)(26,55,29,58)(27,60,30,57) );
G=PermutationGroup([[(1,29,23,17,11),(2,30,24,18,12),(3,25,19,13,7),(4,26,20,14,8),(5,27,21,15,9),(6,28,22,16,10),(31,55,49,43,37),(32,56,50,44,38),(33,57,51,45,39),(34,58,52,46,40),(35,59,53,47,41),(36,60,54,48,42)], [(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)], [(1,34,4,31),(2,33,5,36),(3,32,6,35),(7,38,10,41),(8,37,11,40),(9,42,12,39),(13,44,16,47),(14,43,17,46),(15,48,18,45),(19,50,22,53),(20,49,23,52),(21,54,24,51),(25,56,28,59),(26,55,29,58),(27,60,30,57)]])
C5×Dic3 is a maximal subgroup of
D30.C2 C3⋊D20 C15⋊Q8 S3×C20
Matrix representation of C5×Dic3 ►in GL2(𝔽11) generated by
| 5 | 0 |
| 0 | 5 |
| 1 | 5 |
| 2 | 0 |
| 7 | 8 |
| 2 | 4 |
G:=sub<GL(2,GF(11))| [5,0,0,5],[1,2,5,0],[7,2,8,4] >;
C5×Dic3 in GAP, Magma, Sage, TeX
C_5\times {\rm Dic}_3 % in TeX
G:=Group("C5xDic3"); // GroupNames label
G:=SmallGroup(60,1);
// by ID
G=gap.SmallGroup(60,1);
# by ID
G:=PCGroup([4,-2,-5,-2,-3,40,643]);
// Polycyclic
G:=Group<a,b,c|a^5=b^6=1,c^2=b^3,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
Export
Subgroup lattice of C5×Dic3 in TeX
Character table of C5×Dic3 in TeX