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G = C3×Dic5order 60 = 22·3·5

Direct product of C3 and Dic5

direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary

Aliases: C3×Dic5, C52C12, C154C4, C10.C6, C6.2D5, C30.2C2, C2.(C3×D5), SmallGroup(60,2)

Series: Derived Chief Lower central Upper central

C1C5 — C3×Dic5
C1C5C10C30 — C3×Dic5
C5 — C3×Dic5
C1C6

Generators and relations for C3×Dic5
 G = < a,b,c | a3=b10=1, c2=b5, ab=ba, ac=ca, cbc-1=b-1 >

5C4
5C12

Character table of C3×Dic5

 class 123A3B4A4B5A5B6A6B10A10B12A12B12C12D15A15B15C15D30A30B30C30D
 size 111155221122555522222222
ρ1111111111111111111111111    trivial
ρ21111-1-1111111-1-1-1-111111111    linear of order 2
ρ311ζ3ζ321111ζ3ζ3211ζ3ζ32ζ3ζ32ζ32ζ32ζ3ζ3ζ3ζ32ζ32ζ3    linear of order 3
ρ411ζ32ζ3-1-111ζ32ζ311ζ6ζ65ζ6ζ65ζ3ζ3ζ32ζ32ζ32ζ3ζ3ζ32    linear of order 6
ρ511ζ32ζ31111ζ32ζ311ζ32ζ3ζ32ζ3ζ3ζ3ζ32ζ32ζ32ζ3ζ3ζ32    linear of order 3
ρ611ζ3ζ32-1-111ζ3ζ3211ζ65ζ6ζ65ζ6ζ32ζ32ζ3ζ3ζ3ζ32ζ32ζ3    linear of order 6
ρ71-111-ii11-1-1-1-1-i-iii1111-1-1-1-1    linear of order 4
ρ81-111i-i11-1-1-1-1ii-i-i1111-1-1-1-1    linear of order 4
ρ91-1ζ62ζ32ζ2ζ211ζ65ζ6-1-1ζ4ζ3ζ4ζ32ζ43ζ3ζ43ζ32ζ32ζ32ζ62ζ62ζ65ζ6ζ6ζ65    linear of order 12
ρ101-1ζ32ζ62ζ2ζ211ζ6ζ65-1-1ζ43ζ32ζ43ζ3ζ4ζ32ζ4ζ3ζ62ζ62ζ32ζ32ζ6ζ65ζ65ζ6    linear of order 12
ρ111-1ζ32ζ62ζ2ζ211ζ6ζ65-1-1ζ4ζ32ζ4ζ3ζ43ζ32ζ43ζ3ζ62ζ62ζ32ζ32ζ6ζ65ζ65ζ6    linear of order 12
ρ121-1ζ62ζ32ζ2ζ211ζ65ζ6-1-1ζ43ζ3ζ43ζ32ζ4ζ3ζ4ζ32ζ32ζ32ζ62ζ62ζ65ζ6ζ6ζ65    linear of order 12
ρ13222200-1+5/2-1-5/222-1-5/2-1+5/20000-1+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
ρ14222200-1-5/2-1+5/222-1+5/2-1-5/20000-1-5/2-1+5/2-1-5/2-1+5/2-1+5/2-1-5/2-1+5/2-1-5/2    orthogonal lifted from D5
ρ152-22200-1-5/2-1+5/2-2-21-5/21+5/20000-1-5/2-1+5/2-1-5/2-1+5/21-5/21+5/21-5/21+5/2    symplectic lifted from Dic5, Schur index 2
ρ162-22200-1+5/2-1-5/2-2-21+5/21-5/20000-1+5/2-1-5/2-1+5/2-1-5/21+5/21-5/21+5/21-5/2    symplectic lifted from Dic5, Schur index 2
ρ1722-1+-3-1--300-1-5/2-1+5/2-1+-3-1--3-1+5/2-1-5/20000ζ32ζ5332ζ52ζ32ζ5432ζ5ζ3ζ533ζ52ζ3ζ543ζ5ζ3ζ543ζ5ζ32ζ5332ζ52ζ32ζ5432ζ5ζ3ζ533ζ52    complex lifted from C3×D5
ρ1822-1--3-1+-300-1+5/2-1-5/2-1--3-1+-3-1-5/2-1+5/20000ζ3ζ543ζ5ζ3ζ533ζ52ζ32ζ5432ζ5ζ32ζ5332ζ52ζ32ζ5332ζ52ζ3ζ543ζ5ζ3ζ533ζ52ζ32ζ5432ζ5    complex lifted from C3×D5
ρ192-2-1--3-1+-300-1-5/2-1+5/21+-31--31-5/21+5/20000ζ3ζ533ζ52ζ3ζ543ζ5ζ32ζ5332ζ52ζ32ζ5432ζ532ζ5432ζ53ζ533ζ523ζ543ζ532ζ5332ζ52    complex faithful
ρ2022-1+-3-1--300-1+5/2-1-5/2-1+-3-1--3-1-5/2-1+5/20000ζ32ζ5432ζ5ζ32ζ5332ζ52ζ3ζ543ζ5ζ3ζ533ζ52ζ3ζ533ζ52ζ32ζ5432ζ5ζ32ζ5332ζ52ζ3ζ543ζ5    complex lifted from C3×D5
ρ212-2-1--3-1+-300-1+5/2-1-5/21+-31--31+5/21-5/20000ζ3ζ543ζ5ζ3ζ533ζ52ζ32ζ5432ζ5ζ32ζ5332ζ5232ζ5332ζ523ζ543ζ53ζ533ζ5232ζ5432ζ5    complex faithful
ρ2222-1--3-1+-300-1-5/2-1+5/2-1--3-1+-3-1+5/2-1-5/20000ζ3ζ533ζ52ζ3ζ543ζ5ζ32ζ5332ζ52ζ32ζ5432ζ5ζ32ζ5432ζ5ζ3ζ533ζ52ζ3ζ543ζ5ζ32ζ5332ζ52    complex lifted from C3×D5
ρ232-2-1+-3-1--300-1-5/2-1+5/21--31+-31-5/21+5/20000ζ32ζ5332ζ52ζ32ζ5432ζ5ζ3ζ533ζ52ζ3ζ543ζ53ζ543ζ532ζ5332ζ5232ζ5432ζ53ζ533ζ52    complex faithful
ρ242-2-1+-3-1--300-1+5/2-1-5/21--31+-31+5/21-5/20000ζ32ζ5432ζ5ζ32ζ5332ζ52ζ3ζ543ζ5ζ3ζ533ζ523ζ533ζ5232ζ5432ζ532ζ5332ζ523ζ543ζ5    complex faithful

Smallest permutation representation of C3×Dic5
Regular action on 60 points
Generators in S60
(1 29 19)(2 30 20)(3 21 11)(4 22 12)(5 23 13)(6 24 14)(7 25 15)(8 26 16)(9 27 17)(10 28 18)(31 51 41)(32 52 42)(33 53 43)(34 54 44)(35 55 45)(36 56 46)(37 57 47)(38 58 48)(39 59 49)(40 60 50)
(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 36 6 31)(2 35 7 40)(3 34 8 39)(4 33 9 38)(5 32 10 37)(11 44 16 49)(12 43 17 48)(13 42 18 47)(14 41 19 46)(15 50 20 45)(21 54 26 59)(22 53 27 58)(23 52 28 57)(24 51 29 56)(25 60 30 55)

G:=sub<Sym(60)| (1,29,19)(2,30,20)(3,21,11)(4,22,12)(5,23,13)(6,24,14)(7,25,15)(8,26,16)(9,27,17)(10,28,18)(31,51,41)(32,52,42)(33,53,43)(34,54,44)(35,55,45)(36,56,46)(37,57,47)(38,58,48)(39,59,49)(40,60,50), (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), (1,36,6,31)(2,35,7,40)(3,34,8,39)(4,33,9,38)(5,32,10,37)(11,44,16,49)(12,43,17,48)(13,42,18,47)(14,41,19,46)(15,50,20,45)(21,54,26,59)(22,53,27,58)(23,52,28,57)(24,51,29,56)(25,60,30,55)>;

G:=Group( (1,29,19)(2,30,20)(3,21,11)(4,22,12)(5,23,13)(6,24,14)(7,25,15)(8,26,16)(9,27,17)(10,28,18)(31,51,41)(32,52,42)(33,53,43)(34,54,44)(35,55,45)(36,56,46)(37,57,47)(38,58,48)(39,59,49)(40,60,50), (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), (1,36,6,31)(2,35,7,40)(3,34,8,39)(4,33,9,38)(5,32,10,37)(11,44,16,49)(12,43,17,48)(13,42,18,47)(14,41,19,46)(15,50,20,45)(21,54,26,59)(22,53,27,58)(23,52,28,57)(24,51,29,56)(25,60,30,55) );

G=PermutationGroup([(1,29,19),(2,30,20),(3,21,11),(4,22,12),(5,23,13),(6,24,14),(7,25,15),(8,26,16),(9,27,17),(10,28,18),(31,51,41),(32,52,42),(33,53,43),(34,54,44),(35,55,45),(36,56,46),(37,57,47),(38,58,48),(39,59,49),(40,60,50)], [(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)], [(1,36,6,31),(2,35,7,40),(3,34,8,39),(4,33,9,38),(5,32,10,37),(11,44,16,49),(12,43,17,48),(13,42,18,47),(14,41,19,46),(15,50,20,45),(21,54,26,59),(22,53,27,58),(23,52,28,57),(24,51,29,56),(25,60,30,55)])

Matrix representation of C3×Dic5 in GL2(𝔽19) generated by

70
07
,
117
314
,
1010
79
G:=sub<GL(2,GF(19))| [7,0,0,7],[1,3,17,14],[10,7,10,9] >;

C3×Dic5 in GAP, Magma, Sage, TeX

C_3\times {\rm Dic}_5
% in TeX

G:=Group("C3xDic5");
// GroupNames label

G:=SmallGroup(60,2);
// by ID

G=gap.SmallGroup(60,2);
# by ID

G:=PCGroup([4,-2,-3,-2,-5,24,771]);
// Polycyclic

G:=Group<a,b,c|a^3=b^10=1,c^2=b^5,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

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