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G = S3×C10  order 60 = 22·3·5

Direct product of C10 and S3

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

Aliases: S3×C10, C6⋊C10, C30⋊3C2, C15⋊4C22, 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 >

C1
C2
3C2
3C2
C3
C5
3C22
C6
S3
S3
C10
3C10
3C10
C15
D6
3C2×C10
C5×S3
C5×S3
C30
S3×C10

Character table of S3×C10

 class 12A2B2C35A5B5C5D610A10B10C10D10E10F10G10H10I10J10K10L15A15B15C15D30A30B30C30D
 size 113321111211113333333322222222
ρ1111111111111111111111111111111    trivial
ρ211-1-11111111111-1-1-1-1-1-1-1-111111111    linear of order 2
ρ31-11-111111-1-1-1-1-11-1-1-1111-11111-1-1-1-1    linear of order 2
ρ41-1-1111111-1-1-1-1-1-1111-1-1-111111-1-1-1-1    linear of order 2
ρ51-1-111ζ5ζ53ζ52ζ54-1-ζ5-ζ53-ζ54-ζ52-ζ53ζ5ζ53ζ52-ζ52-ζ54-ζ5ζ54ζ53ζ52ζ5ζ54-ζ54-ζ53-ζ5-ζ52    linear of order 10
ρ611111ζ5ζ53ζ52ζ541ζ5ζ53ζ54ζ52ζ53ζ5ζ53ζ52ζ52ζ54ζ5ζ54ζ53ζ52ζ5ζ54ζ54ζ53ζ5ζ52    linear of order 5
ρ711-1-11ζ53ζ54ζ5ζ521ζ53ζ54ζ52ζ5-ζ54-ζ53-ζ54-ζ5-ζ5-ζ52-ζ53-ζ52ζ54ζ5ζ53ζ52ζ52ζ54ζ53ζ5    linear of order 10
ρ811111ζ54ζ52ζ53ζ51ζ54ζ52ζ5ζ53ζ52ζ54ζ52ζ53ζ53ζ5ζ54ζ5ζ52ζ53ζ54ζ5ζ5ζ52ζ54ζ53    linear of order 5
ρ91-11-11ζ53ζ54ζ5ζ52-1-ζ53-ζ54-ζ52-ζ5ζ54-ζ53-ζ54-ζ5ζ5ζ52ζ53-ζ52ζ54ζ5ζ53ζ52-ζ52-ζ54-ζ53-ζ5    linear of order 10
ρ101-1-111ζ53ζ54ζ5ζ52-1-ζ53-ζ54-ζ52-ζ5-ζ54ζ53ζ54ζ5-ζ5-ζ52-ζ53ζ52ζ54ζ5ζ53ζ52-ζ52-ζ54-ζ53-ζ5    linear of order 10
ρ1111-1-11ζ54ζ52ζ53ζ51ζ54ζ52ζ5ζ53-ζ52-ζ54-ζ52-ζ53-ζ53-ζ5-ζ54-ζ5ζ52ζ53ζ54ζ5ζ5ζ52ζ54ζ53    linear of order 10
ρ1211111ζ53ζ54ζ5ζ521ζ53ζ54ζ52ζ5ζ54ζ53ζ54ζ5ζ5ζ52ζ53ζ52ζ54ζ5ζ53ζ52ζ52ζ54ζ53ζ5    linear of order 5
ρ131-11-11ζ5ζ53ζ52ζ54-1-ζ5-ζ53-ζ54-ζ52ζ53-ζ5-ζ53-ζ52ζ52ζ54ζ5-ζ54ζ53ζ52ζ5ζ54-ζ54-ζ53-ζ5-ζ52    linear of order 10
ρ141-11-11ζ54ζ52ζ53ζ5-1-ζ54-ζ52-ζ5-ζ53ζ52-ζ54-ζ52-ζ53ζ53ζ5ζ54-ζ5ζ52ζ53ζ54ζ5-ζ5-ζ52-ζ54-ζ53    linear of order 10
ρ151-1-111ζ54ζ52ζ53ζ5-1-ζ54-ζ52-ζ5-ζ53-ζ52ζ54ζ52ζ53-ζ53-ζ5-ζ54ζ5ζ52ζ53ζ54ζ5-ζ5-ζ52-ζ54-ζ53    linear of order 10
ρ161-11-11ζ52ζ5ζ54ζ53-1-ζ52-ζ5-ζ53-ζ54ζ5-ζ52-ζ5-ζ54ζ54ζ53ζ52-ζ53ζ5ζ54ζ52ζ53-ζ53-ζ5-ζ52-ζ54    linear of order 10
ρ1711111ζ52ζ5ζ54ζ531ζ52ζ5ζ53ζ54ζ5ζ52ζ5ζ54ζ54ζ53ζ52ζ53ζ5ζ54ζ52ζ53ζ53ζ5ζ52ζ54    linear of order 5
ρ1811-1-11ζ52ζ5ζ54ζ531ζ52ζ5ζ53ζ54-ζ5-ζ52-ζ5-ζ54-ζ54-ζ53-ζ52-ζ53ζ5ζ54ζ52ζ53ζ53ζ5ζ52ζ54    linear of order 10
ρ1911-1-11ζ5ζ53ζ52ζ541ζ5ζ53ζ54ζ52-ζ53-ζ5-ζ53-ζ52-ζ52-ζ54-ζ5-ζ54ζ53ζ52ζ5ζ54ζ54ζ53ζ5ζ52    linear of order 10
ρ201-1-111ζ52ζ5ζ54ζ53-1-ζ52-ζ5-ζ53-ζ54-ζ5ζ52ζ5ζ54-ζ54-ζ53-ζ52ζ53ζ5ζ54ζ52ζ53-ζ53-ζ5-ζ52-ζ54    linear of order 10
ρ212-200-122221-2-2-2-200000000-1-1-1-11111    orthogonal lifted from D6
ρ222200-12222-1222200000000-1-1-1-1-1-1-1-1    orthogonal lifted from S3
ρ232200-12ζ52ζ532ζ522ζ54-12ζ52ζ532ζ542ζ5200000000-ζ53-ζ52-ζ5-ζ54-ζ54-ζ53-ζ5-ζ52    complex lifted from C5×S3
ρ242200-12ζ522ζ52ζ542ζ53-12ζ522ζ52ζ532ζ5400000000-ζ5-ζ54-ζ52-ζ53-ζ53-ζ5-ζ52-ζ54    complex lifted from C5×S3
ρ252-200-12ζ542ζ522ζ532ζ51-2ζ54-2ζ52-2ζ5-2ζ5300000000-ζ52-ζ53-ζ54-ζ5ζ5ζ52ζ54ζ53    complex faithful
ρ262-200-12ζ522ζ52ζ542ζ531-2ζ52-2ζ5-2ζ53-2ζ5400000000-ζ5-ζ54-ζ52-ζ53ζ53ζ5ζ52ζ54    complex faithful
ρ272200-12ζ532ζ542ζ52ζ52-12ζ532ζ542ζ522ζ500000000-ζ54-ζ5-ζ53-ζ52-ζ52-ζ54-ζ53-ζ5    complex lifted from C5×S3
ρ282-200-12ζ532ζ542ζ52ζ521-2ζ53-2ζ54-2ζ52-2ζ500000000-ζ54-ζ5-ζ53-ζ52ζ52ζ54ζ53ζ5    complex faithful
ρ292200-12ζ542ζ522ζ532ζ5-12ζ542ζ522ζ52ζ5300000000-ζ52-ζ53-ζ54-ζ5-ζ5-ζ52-ζ54-ζ53    complex lifted from C5×S3
ρ302-200-12ζ52ζ532ζ522ζ541-2ζ5-2ζ53-2ζ54-2ζ5200000000-ζ53-ζ52-ζ5-ζ54ζ54ζ53ζ5ζ52    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

20
02
,
05
210
,
100
91
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
 

Export

Subgroup lattice of S3×C10 in TeX
Character table of S3×C10 in TeX

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