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## G = S3×C10order 60 = 22·3·5

### Direct product of C10 and S3

Aliases: S3×C10, C6⋊C10, C303C2, C154C22, C3⋊(C2×C10), SmallGroup(60,11)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — S3×C10
 Chief series C1 — C3 — C15 — C5×S3 — S3×C10
 Lower central C3 — S3×C10
 Upper central C1 — C10

Generators and relations for S3×C10
G = < a,b,c | a10=b3=c2=1, ab=ba, ac=ca, cbc=b-1 >

Character table of S3×C10

 class 1 2A 2B 2C 3 5A 5B 5C 5D 6 10A 10B 10C 10D 10E 10F 10G 10H 10I 10J 10K 10L 15A 15B 15C 15D 30A 30B 30C 30D size 1 1 3 3 2 1 1 1 1 2 1 1 1 1 3 3 3 3 3 3 3 3 2 2 2 2 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 -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 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 1 1 -1 -1 -1 -1 linear of order 2 ρ5 1 -1 -1 1 1 ζ5 ζ53 ζ52 ζ54 -1 -ζ5 -ζ53 -ζ54 -ζ52 -ζ53 ζ5 ζ53 ζ52 -ζ52 -ζ54 -ζ5 ζ54 ζ53 ζ52 ζ5 ζ54 -ζ54 -ζ53 -ζ5 -ζ52 linear of order 10 ρ6 1 1 1 1 1 ζ5 ζ53 ζ52 ζ54 1 ζ5 ζ53 ζ54 ζ52 ζ53 ζ5 ζ53 ζ52 ζ52 ζ54 ζ5 ζ54 ζ53 ζ52 ζ5 ζ54 ζ54 ζ53 ζ5 ζ52 linear of order 5 ρ7 1 1 -1 -1 1 ζ53 ζ54 ζ5 ζ52 1 ζ53 ζ54 ζ52 ζ5 -ζ54 -ζ53 -ζ54 -ζ5 -ζ5 -ζ52 -ζ53 -ζ52 ζ54 ζ5 ζ53 ζ52 ζ52 ζ54 ζ53 ζ5 linear of order 10 ρ8 1 1 1 1 1 ζ54 ζ52 ζ53 ζ5 1 ζ54 ζ52 ζ5 ζ53 ζ52 ζ54 ζ52 ζ53 ζ53 ζ5 ζ54 ζ5 ζ52 ζ53 ζ54 ζ5 ζ5 ζ52 ζ54 ζ53 linear of order 5 ρ9 1 -1 1 -1 1 ζ53 ζ54 ζ5 ζ52 -1 -ζ53 -ζ54 -ζ52 -ζ5 ζ54 -ζ53 -ζ54 -ζ5 ζ5 ζ52 ζ53 -ζ52 ζ54 ζ5 ζ53 ζ52 -ζ52 -ζ54 -ζ53 -ζ5 linear of order 10 ρ10 1 -1 -1 1 1 ζ53 ζ54 ζ5 ζ52 -1 -ζ53 -ζ54 -ζ52 -ζ5 -ζ54 ζ53 ζ54 ζ5 -ζ5 -ζ52 -ζ53 ζ52 ζ54 ζ5 ζ53 ζ52 -ζ52 -ζ54 -ζ53 -ζ5 linear of order 10 ρ11 1 1 -1 -1 1 ζ54 ζ52 ζ53 ζ5 1 ζ54 ζ52 ζ5 ζ53 -ζ52 -ζ54 -ζ52 -ζ53 -ζ53 -ζ5 -ζ54 -ζ5 ζ52 ζ53 ζ54 ζ5 ζ5 ζ52 ζ54 ζ53 linear of order 10 ρ12 1 1 1 1 1 ζ53 ζ54 ζ5 ζ52 1 ζ53 ζ54 ζ52 ζ5 ζ54 ζ53 ζ54 ζ5 ζ5 ζ52 ζ53 ζ52 ζ54 ζ5 ζ53 ζ52 ζ52 ζ54 ζ53 ζ5 linear of order 5 ρ13 1 -1 1 -1 1 ζ5 ζ53 ζ52 ζ54 -1 -ζ5 -ζ53 -ζ54 -ζ52 ζ53 -ζ5 -ζ53 -ζ52 ζ52 ζ54 ζ5 -ζ54 ζ53 ζ52 ζ5 ζ54 -ζ54 -ζ53 -ζ5 -ζ52 linear of order 10 ρ14 1 -1 1 -1 1 ζ54 ζ52 ζ53 ζ5 -1 -ζ54 -ζ52 -ζ5 -ζ53 ζ52 -ζ54 -ζ52 -ζ53 ζ53 ζ5 ζ54 -ζ5 ζ52 ζ53 ζ54 ζ5 -ζ5 -ζ52 -ζ54 -ζ53 linear of order 10 ρ15 1 -1 -1 1 1 ζ54 ζ52 ζ53 ζ5 -1 -ζ54 -ζ52 -ζ5 -ζ53 -ζ52 ζ54 ζ52 ζ53 -ζ53 -ζ5 -ζ54 ζ5 ζ52 ζ53 ζ54 ζ5 -ζ5 -ζ52 -ζ54 -ζ53 linear of order 10 ρ16 1 -1 1 -1 1 ζ52 ζ5 ζ54 ζ53 -1 -ζ52 -ζ5 -ζ53 -ζ54 ζ5 -ζ52 -ζ5 -ζ54 ζ54 ζ53 ζ52 -ζ53 ζ5 ζ54 ζ52 ζ53 -ζ53 -ζ5 -ζ52 -ζ54 linear of order 10 ρ17 1 1 1 1 1 ζ52 ζ5 ζ54 ζ53 1 ζ52 ζ5 ζ53 ζ54 ζ5 ζ52 ζ5 ζ54 ζ54 ζ53 ζ52 ζ53 ζ5 ζ54 ζ52 ζ53 ζ53 ζ5 ζ52 ζ54 linear of order 5 ρ18 1 1 -1 -1 1 ζ52 ζ5 ζ54 ζ53 1 ζ52 ζ5 ζ53 ζ54 -ζ5 -ζ52 -ζ5 -ζ54 -ζ54 -ζ53 -ζ52 -ζ53 ζ5 ζ54 ζ52 ζ53 ζ53 ζ5 ζ52 ζ54 linear of order 10 ρ19 1 1 -1 -1 1 ζ5 ζ53 ζ52 ζ54 1 ζ5 ζ53 ζ54 ζ52 -ζ53 -ζ5 -ζ53 -ζ52 -ζ52 -ζ54 -ζ5 -ζ54 ζ53 ζ52 ζ5 ζ54 ζ54 ζ53 ζ5 ζ52 linear of order 10 ρ20 1 -1 -1 1 1 ζ52 ζ5 ζ54 ζ53 -1 -ζ52 -ζ5 -ζ53 -ζ54 -ζ5 ζ52 ζ5 ζ54 -ζ54 -ζ53 -ζ52 ζ53 ζ5 ζ54 ζ52 ζ53 -ζ53 -ζ5 -ζ52 -ζ54 linear of order 10 ρ21 2 -2 0 0 -1 2 2 2 2 1 -2 -2 -2 -2 0 0 0 0 0 0 0 0 -1 -1 -1 -1 1 1 1 1 orthogonal lifted from D6 ρ22 2 2 0 0 -1 2 2 2 2 -1 2 2 2 2 0 0 0 0 0 0 0 0 -1 -1 -1 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ23 2 2 0 0 -1 2ζ5 2ζ53 2ζ52 2ζ54 -1 2ζ5 2ζ53 2ζ54 2ζ52 0 0 0 0 0 0 0 0 -ζ53 -ζ52 -ζ5 -ζ54 -ζ54 -ζ53 -ζ5 -ζ52 complex lifted from C5×S3 ρ24 2 2 0 0 -1 2ζ52 2ζ5 2ζ54 2ζ53 -1 2ζ52 2ζ5 2ζ53 2ζ54 0 0 0 0 0 0 0 0 -ζ5 -ζ54 -ζ52 -ζ53 -ζ53 -ζ5 -ζ52 -ζ54 complex lifted from C5×S3 ρ25 2 -2 0 0 -1 2ζ54 2ζ52 2ζ53 2ζ5 1 -2ζ54 -2ζ52 -2ζ5 -2ζ53 0 0 0 0 0 0 0 0 -ζ52 -ζ53 -ζ54 -ζ5 ζ5 ζ52 ζ54 ζ53 complex faithful ρ26 2 -2 0 0 -1 2ζ52 2ζ5 2ζ54 2ζ53 1 -2ζ52 -2ζ5 -2ζ53 -2ζ54 0 0 0 0 0 0 0 0 -ζ5 -ζ54 -ζ52 -ζ53 ζ53 ζ5 ζ52 ζ54 complex faithful ρ27 2 2 0 0 -1 2ζ53 2ζ54 2ζ5 2ζ52 -1 2ζ53 2ζ54 2ζ52 2ζ5 0 0 0 0 0 0 0 0 -ζ54 -ζ5 -ζ53 -ζ52 -ζ52 -ζ54 -ζ53 -ζ5 complex lifted from C5×S3 ρ28 2 -2 0 0 -1 2ζ53 2ζ54 2ζ5 2ζ52 1 -2ζ53 -2ζ54 -2ζ52 -2ζ5 0 0 0 0 0 0 0 0 -ζ54 -ζ5 -ζ53 -ζ52 ζ52 ζ54 ζ53 ζ5 complex faithful ρ29 2 2 0 0 -1 2ζ54 2ζ52 2ζ53 2ζ5 -1 2ζ54 2ζ52 2ζ5 2ζ53 0 0 0 0 0 0 0 0 -ζ52 -ζ53 -ζ54 -ζ5 -ζ5 -ζ52 -ζ54 -ζ53 complex lifted from C5×S3 ρ30 2 -2 0 0 -1 2ζ5 2ζ53 2ζ52 2ζ54 1 -2ζ5 -2ζ53 -2ζ54 -2ζ52 0 0 0 0 0 0 0 0 -ζ53 -ζ52 -ζ5 -ζ54 ζ54 ζ53 ζ5 ζ52 complex faithful

Permutation representations of S3×C10
On 30 points - transitive group 30T12
Generators in S30
(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)
(1 13 25)(2 14 26)(3 15 27)(4 16 28)(5 17 29)(6 18 30)(7 19 21)(8 20 22)(9 11 23)(10 12 24)
(11 23)(12 24)(13 25)(14 26)(15 27)(16 28)(17 29)(18 30)(19 21)(20 22)

G:=sub<Sym(30)| (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), (1,13,25)(2,14,26)(3,15,27)(4,16,28)(5,17,29)(6,18,30)(7,19,21)(8,20,22)(9,11,23)(10,12,24), (11,23)(12,24)(13,25)(14,26)(15,27)(16,28)(17,29)(18,30)(19,21)(20,22)>;

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), (1,13,25)(2,14,26)(3,15,27)(4,16,28)(5,17,29)(6,18,30)(7,19,21)(8,20,22)(9,11,23)(10,12,24), (11,23)(12,24)(13,25)(14,26)(15,27)(16,28)(17,29)(18,30)(19,21)(20,22) );

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)], [(1,13,25),(2,14,26),(3,15,27),(4,16,28),(5,17,29),(6,18,30),(7,19,21),(8,20,22),(9,11,23),(10,12,24)], [(11,23),(12,24),(13,25),(14,26),(15,27),(16,28),(17,29),(18,30),(19,21),(20,22)]])

G:=TransitiveGroup(30,12);

S3×C10 is a maximal subgroup of   C15⋊D4  C5⋊D12

Matrix representation of S3×C10 in GL2(𝔽11) generated by

 2 0 0 2
,
 0 5 2 10
,
 10 0 9 1
G:=sub<GL(2,GF(11))| [2,0,0,2],[0,2,5,10],[10,9,0,1] >;

S3×C10 in GAP, Magma, Sage, TeX

S_3\times C_{10}
% in TeX

G:=Group("S3xC10");
// GroupNames label

G:=SmallGroup(60,11);
// by ID

G=gap.SmallGroup(60,11);
# by ID

G:=PCGroup([4,-2,-2,-5,-3,643]);
// Polycyclic

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

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