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## G = S3×C11order 66 = 2·3·11

### Direct product of C11 and S3

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — S3×C11
 Chief series C1 — C3 — C33 — S3×C11
 Lower central C3 — S3×C11
 Upper central C1 — C11

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

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

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

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

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

33 conjugacy classes

 class 1 2 3 11A ··· 11J 22A ··· 22J 33A ··· 33J order 1 2 3 11 ··· 11 22 ··· 22 33 ··· 33 size 1 3 2 1 ··· 1 3 ··· 3 2 ··· 2

33 irreducible representations

 dim 1 1 1 1 2 2 type + + + image C1 C2 C11 C22 S3 S3×C11 kernel S3×C11 C33 S3 C3 C11 C1 # reps 1 1 10 10 1 10

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

 6 0 0 6
,
 0 21 12 22
,
 1 21 0 22
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

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