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G = C5×Dic3order 60 = 22·3·5

Direct product of C5 and Dic3

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

Aliases: C5×Dic3, C3⋊C20, C155C4, C6.C10, C30.3C2, C10.2S3, C2.(C5×S3), SmallGroup(60,1)

Series: Derived Chief Lower central Upper central

C1C3 — C5×Dic3
C1C3C6C30 — C5×Dic3
C3 — C5×Dic3
C1C10

Generators and relations for C5×Dic3
 G = < a,b,c | a5=b6=1, c2=b3, ab=ba, ac=ca, cbc-1=b-1 >

3C4
3C20

Character table of C5×Dic3

 class 1234A4B5A5B5C5D610A10B10C10D15A15B15C15D20A20B20C20D20E20F20G20H30A30B30C30D
 size 112331111211112222333333332222
ρ1111111111111111111111111111111    trivial
ρ2111-1-11111111111111-1-1-1-1-1-1-1-11111    linear of order 2
ρ31-11-ii1111-1-1-1-1-11111-i-ii-ii-iii-1-1-1-1    linear of order 4
ρ41-11i-i1111-1-1-1-1-11111ii-ii-ii-i-i-1-1-1-1    linear of order 4
ρ5111-1-1ζ53ζ5ζ54ζ521ζ54ζ5ζ52ζ53ζ54ζ5ζ53ζ5252535554545253ζ54ζ52ζ5ζ53    linear of order 10
ρ6111-1-1ζ5ζ52ζ53ζ541ζ53ζ52ζ54ζ5ζ53ζ52ζ5ζ5454552525353545ζ53ζ54ζ52ζ5    linear of order 10
ρ711111ζ52ζ54ζ5ζ531ζ5ζ54ζ53ζ52ζ5ζ54ζ52ζ53ζ53ζ52ζ54ζ54ζ5ζ5ζ53ζ52ζ5ζ53ζ54ζ52    linear of order 5
ρ811111ζ5ζ52ζ53ζ541ζ53ζ52ζ54ζ5ζ53ζ52ζ5ζ54ζ54ζ5ζ52ζ52ζ53ζ53ζ54ζ5ζ53ζ54ζ52ζ5    linear of order 5
ρ911111ζ54ζ53ζ52ζ51ζ52ζ53ζ5ζ54ζ52ζ53ζ54ζ5ζ5ζ54ζ53ζ53ζ52ζ52ζ5ζ54ζ52ζ5ζ53ζ54    linear of order 5
ρ1011111ζ53ζ5ζ54ζ521ζ54ζ5ζ52ζ53ζ54ζ5ζ53ζ52ζ52ζ53ζ5ζ5ζ54ζ54ζ52ζ53ζ54ζ52ζ5ζ53    linear of order 5
ρ11111-1-1ζ52ζ54ζ5ζ531ζ5ζ54ζ53ζ52ζ5ζ54ζ52ζ5353525454555352ζ5ζ53ζ54ζ52    linear of order 10
ρ12111-1-1ζ54ζ53ζ52ζ51ζ52ζ53ζ5ζ54ζ52ζ53ζ54ζ555453535252554ζ52ζ5ζ53ζ54    linear of order 10
ρ131-11ζ2ζ2ζ52ζ54ζ5ζ53-15545352ζ5ζ54ζ52ζ53ζ43ζ53ζ43ζ52ζ4ζ54ζ43ζ54ζ4ζ5ζ43ζ5ζ4ζ53ζ4ζ525535452    linear of order 20
ρ141-11ζ2ζ2ζ5ζ52ζ53ζ54-15352545ζ53ζ52ζ5ζ54ζ4ζ54ζ4ζ5ζ43ζ52ζ4ζ52ζ43ζ53ζ4ζ53ζ43ζ54ζ43ζ55354525    linear of order 20
ρ151-11ζ2ζ2ζ53ζ5ζ54ζ52-15455253ζ54ζ5ζ53ζ52ζ43ζ52ζ43ζ53ζ4ζ5ζ43ζ5ζ4ζ54ζ43ζ54ζ4ζ52ζ4ζ535452553    linear of order 20
ρ161-11ζ2ζ2ζ54ζ53ζ52ζ5-15253554ζ52ζ53ζ54ζ5ζ43ζ5ζ43ζ54ζ4ζ53ζ43ζ53ζ4ζ52ζ43ζ52ζ4ζ5ζ4ζ545255354    linear of order 20
ρ171-11ζ2ζ2ζ52ζ54ζ5ζ53-15545352ζ5ζ54ζ52ζ53ζ4ζ53ζ4ζ52ζ43ζ54ζ4ζ54ζ43ζ5ζ4ζ5ζ43ζ53ζ43ζ525535452    linear of order 20
ρ181-11ζ2ζ2ζ53ζ5ζ54ζ52-15455253ζ54ζ5ζ53ζ52ζ4ζ52ζ4ζ53ζ43ζ5ζ4ζ5ζ43ζ54ζ4ζ54ζ43ζ52ζ43ζ535452553    linear of order 20
ρ191-11ζ2ζ2ζ5ζ52ζ53ζ54-15352545ζ53ζ52ζ5ζ54ζ43ζ54ζ43ζ5ζ4ζ52ζ43ζ52ζ4ζ53ζ43ζ53ζ4ζ54ζ4ζ55354525    linear of order 20
ρ201-11ζ2ζ2ζ54ζ53ζ52ζ5-15253554ζ52ζ53ζ54ζ5ζ4ζ5ζ4ζ54ζ43ζ53ζ4ζ53ζ43ζ52ζ4ζ52ζ43ζ5ζ43ζ545255354    linear of order 20
ρ2122-1002222-12222-1-1-1-100000000-1-1-1-1    orthogonal lifted from S3
ρ222-2-10022221-2-2-2-2-1-1-1-1000000001111    symplectic lifted from Dic3, Schur index 2
ρ232-2-10052545531-2ζ5-2ζ54-2ζ53-2ζ52554525300000000ζ5ζ53ζ54ζ52    complex faithful
ρ242-2-10053554521-2ζ54-2ζ5-2ζ52-2ζ53545535200000000ζ54ζ52ζ5ζ53    complex faithful
ρ2522-1005355452-154552535455352000000005452553    complex lifted from C5×S3
ρ262-2-10054535251-2ζ52-2ζ53-2ζ5-2ζ54525354500000000ζ52ζ5ζ53ζ54    complex faithful
ρ272-2-10055253541-2ζ53-2ζ52-2ζ54-2ζ5535255400000000ζ53ζ54ζ52ζ5    complex faithful
ρ2822-1005525354-153525455352554000000005354525    complex lifted from C5×S3
ρ2922-1005254553-155453525545253000000005535452    complex lifted from C5×S3
ρ3022-1005453525-152535545253545000000005255354    complex lifted from C5×S3

Smallest permutation representation of C5×Dic3
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)])

Matrix representation of C5×Dic3 in GL2(𝔽11) generated by

50
05
,
15
20
,
78
24
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

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