direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: S3×C11, C3⋊C22, C33⋊3C2, SmallGroup(66,1)
Series: Derived ►Chief ►Lower central ►Upper central
| C3 — S3×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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