direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: S3×C17, C3⋊C34, C51⋊3C2, SmallGroup(102,1)
Series: Derived ►Chief ►Lower central ►Upper central
| C3 — S3×C17 |
Generators and relations for S3×C17
G = < a,b,c | a17=b3=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of S3×C17
►On 51 points
Generators in S51
(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 40 41 42 43 44 45 46 47 48 49 50 51)
(1 51 21)(2 35 22)(3 36 23)(4 37 24)(5 38 25)(6 39 26)(7 40 27)(8 41 28)(9 42 29)(10 43 30)(11 44 31)(12 45 32)(13 46 33)(14 47 34)(15 48 18)(16 49 19)(17 50 20)
(18 48)(19 49)(20 50)(21 51)(22 35)(23 36)(24 37)(25 38)(26 39)(27 40)(28 41)(29 42)(30 43)(31 44)(32 45)(33 46)(34 47)
G:=sub<Sym(51)| (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)(35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51), (1,51,21)(2,35,22)(3,36,23)(4,37,24)(5,38,25)(6,39,26)(7,40,27)(8,41,28)(9,42,29)(10,43,30)(11,44,31)(12,45,32)(13,46,33)(14,47,34)(15,48,18)(16,49,19)(17,50,20), (18,48)(19,49)(20,50)(21,51)(22,35)(23,36)(24,37)(25,38)(26,39)(27,40)(28,41)(29,42)(30,43)(31,44)(32,45)(33,46)(34,47)>;
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)(35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51), (1,51,21)(2,35,22)(3,36,23)(4,37,24)(5,38,25)(6,39,26)(7,40,27)(8,41,28)(9,42,29)(10,43,30)(11,44,31)(12,45,32)(13,46,33)(14,47,34)(15,48,18)(16,49,19)(17,50,20), (18,48)(19,49)(20,50)(21,51)(22,35)(23,36)(24,37)(25,38)(26,39)(27,40)(28,41)(29,42)(30,43)(31,44)(32,45)(33,46)(34,47) );
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),(35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51)], [(1,51,21),(2,35,22),(3,36,23),(4,37,24),(5,38,25),(6,39,26),(7,40,27),(8,41,28),(9,42,29),(10,43,30),(11,44,31),(12,45,32),(13,46,33),(14,47,34),(15,48,18),(16,49,19),(17,50,20)], [(18,48),(19,49),(20,50),(21,51),(22,35),(23,36),(24,37),(25,38),(26,39),(27,40),(28,41),(29,42),(30,43),(31,44),(32,45),(33,46),(34,47)]])
51 conjugacy classes
| class | 1 | 2 | 3 | 17A | ··· | 17P | 34A | ··· | 34P | 51A | ··· | 51P |
| order | 1 | 2 | 3 | 17 | ··· | 17 | 34 | ··· | 34 | 51 | ··· | 51 |
| size | 1 | 3 | 2 | 1 | ··· | 1 | 3 | ··· | 3 | 2 | ··· | 2 |
51 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | |||
| image | C1 | C2 | C17 | C34 | S3 | S3×C17 |
| kernel | S3×C17 | C51 | S3 | C3 | C17 | C1 |
| # reps | 1 | 1 | 16 | 16 | 1 | 16 |
Matrix representation of S3×C17 ►in GL2(𝔽103) generated by
| 34 | 0 |
| 0 | 34 |
| 0 | 102 |
| 1 | 102 |
| 0 | 1 |
| 1 | 0 |
G:=sub<GL(2,GF(103))| [34,0,0,34],[0,1,102,102],[0,1,1,0] >;
S3×C17 in GAP, Magma, Sage, TeX
S_3\times C_{17} % in TeX
G:=Group("S3xC17"); // GroupNames label
G:=SmallGroup(102,1);
// by ID
G=gap.SmallGroup(102,1);
# by ID
G:=PCGroup([3,-2,-17,-3,614]);
// Polycyclic
G:=Group<a,b,c|a^17=b^3=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export