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G = S3×C13order 78 = 2·3·13

Direct product of C13 and S3

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

Aliases: S3×C13, C3⋊C26, C393C2, SmallGroup(78,3)

Series: Derived Chief Lower central Upper central

C1C3 — S3×C13
C1C3C39 — S3×C13
C3 — S3×C13
C1C13

Generators and relations for S3×C13
 G = < a,b,c | a13=b3=c2=1, ab=ba, ac=ca, cbc=b-1 >

3C2
3C26

Smallest permutation representation of S3×C13
On 39 points
Generators in S39
(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)
(1 29 20)(2 30 21)(3 31 22)(4 32 23)(5 33 24)(6 34 25)(7 35 26)(8 36 14)(9 37 15)(10 38 16)(11 39 17)(12 27 18)(13 28 19)
(14 36)(15 37)(16 38)(17 39)(18 27)(19 28)(20 29)(21 30)(22 31)(23 32)(24 33)(25 34)(26 35)

G:=sub<Sym(39)| (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), (1,29,20)(2,30,21)(3,31,22)(4,32,23)(5,33,24)(6,34,25)(7,35,26)(8,36,14)(9,37,15)(10,38,16)(11,39,17)(12,27,18)(13,28,19), (14,36)(15,37)(16,38)(17,39)(18,27)(19,28)(20,29)(21,30)(22,31)(23,32)(24,33)(25,34)(26,35)>;

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)(27,28,29,30,31,32,33,34,35,36,37,38,39), (1,29,20)(2,30,21)(3,31,22)(4,32,23)(5,33,24)(6,34,25)(7,35,26)(8,36,14)(9,37,15)(10,38,16)(11,39,17)(12,27,18)(13,28,19), (14,36)(15,37)(16,38)(17,39)(18,27)(19,28)(20,29)(21,30)(22,31)(23,32)(24,33)(25,34)(26,35) );

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),(27,28,29,30,31,32,33,34,35,36,37,38,39)], [(1,29,20),(2,30,21),(3,31,22),(4,32,23),(5,33,24),(6,34,25),(7,35,26),(8,36,14),(9,37,15),(10,38,16),(11,39,17),(12,27,18),(13,28,19)], [(14,36),(15,37),(16,38),(17,39),(18,27),(19,28),(20,29),(21,30),(22,31),(23,32),(24,33),(25,34),(26,35)])

39 conjugacy classes

class 1  2  3 13A···13L26A···26L39A···39L
order12313···1326···2639···39
size1321···13···32···2

39 irreducible representations

dim111122
type+++
imageC1C2C13C26S3S3×C13
kernelS3×C13C39S3C3C13C1
# reps111212112

Matrix representation of S3×C13 in GL2(𝔽79) generated by

80
08
,
078
178
,
01
10
G:=sub<GL(2,GF(79))| [8,0,0,8],[0,1,78,78],[0,1,1,0] >;

S3×C13 in GAP, Magma, Sage, TeX

S_3\times C_{13}
% in TeX

G:=Group("S3xC13");
// GroupNames label

G:=SmallGroup(78,3);
// by ID

G=gap.SmallGroup(78,3);
# by ID

G:=PCGroup([3,-2,-13,-3,470]);
// Polycyclic

G:=Group<a,b,c|a^13=b^3=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

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Subgroup lattice of S3×C13 in TeX

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