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G = S3×C11order 66 = 2·3·11

Direct product of C11 and S3

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

Aliases: S3×C11, C3⋊C22, C333C2, SmallGroup(66,1)

Series: Derived Chief Lower central Upper central

C1C3 — S3×C11
C1C3C33 — S3×C11
C3 — S3×C11
C1C11

Generators and relations for S3×C11
 G = < a,b,c | a11=b3=c2=1, ab=ba, ac=ca, cbc=b-1 >

3C2
3C22

Smallest permutation representation of S3×C11
On 33 points
Generators in S33
(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)
(1 21 30)(2 22 31)(3 12 32)(4 13 33)(5 14 23)(6 15 24)(7 16 25)(8 17 26)(9 18 27)(10 19 28)(11 20 29)
(12 32)(13 33)(14 23)(15 24)(16 25)(17 26)(18 27)(19 28)(20 29)(21 30)(22 31)

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

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

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

33 conjugacy classes

class 1  2  3 11A···11J22A···22J33A···33J
order12311···1122···2233···33
size1321···13···32···2

33 irreducible representations

dim111122
type+++
imageC1C2C11C22S3S3×C11
kernelS3×C11C33S3C3C11C1
# reps111010110

Matrix representation of S3×C11 in GL2(𝔽23) generated by

60
06
,
021
1222
,
121
022
G:=sub<GL(2,GF(23))| [6,0,0,6],[0,12,21,22],[1,0,21,22] >;

S3×C11 in GAP, Magma, Sage, TeX

S_3\times C_{11}
% in TeX

G:=Group("S3xC11");
// GroupNames label

G:=SmallGroup(66,1);
// by ID

G=gap.SmallGroup(66,1);
# by ID

G:=PCGroup([3,-2,-11,-3,398]);
// Polycyclic

G:=Group<a,b,c|a^11=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×C11 in TeX

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