direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: S3×C13, C3⋊C26, C39⋊3C2, SmallGroup(78,3)
Series: Derived ►Chief ►Lower central ►Upper central
| C3 — S3×C13 |
Generators and relations for S3×C13
G = < a,b,c | a13=b3=c2=1, ab=ba, ac=ca, cbc=b-1 >
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 39 15)(2 27 16)(3 28 17)(4 29 18)(5 30 19)(6 31 20)(7 32 21)(8 33 22)(9 34 23)(10 35 24)(11 36 25)(12 37 26)(13 38 14)
(14 38)(15 39)(16 27)(17 28)(18 29)(19 30)(20 31)(21 32)(22 33)(23 34)(24 35)(25 36)(26 37)
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), (1,39,15)(2,27,16)(3,28,17)(4,29,18)(5,30,19)(6,31,20)(7,32,21)(8,33,22)(9,34,23)(10,35,24)(11,36,25)(12,37,26)(13,38,14), (14,38)(15,39)(16,27)(17,28)(18,29)(19,30)(20,31)(21,32)(22,33)(23,34)(24,35)(25,36)(26,37)>;
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), (1,39,15)(2,27,16)(3,28,17)(4,29,18)(5,30,19)(6,31,20)(7,32,21)(8,33,22)(9,34,23)(10,35,24)(11,36,25)(12,37,26)(13,38,14), (14,38)(15,39)(16,27)(17,28)(18,29)(19,30)(20,31)(21,32)(22,33)(23,34)(24,35)(25,36)(26,37) );
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)], [(1,39,15),(2,27,16),(3,28,17),(4,29,18),(5,30,19),(6,31,20),(7,32,21),(8,33,22),(9,34,23),(10,35,24),(11,36,25),(12,37,26),(13,38,14)], [(14,38),(15,39),(16,27),(17,28),(18,29),(19,30),(20,31),(21,32),(22,33),(23,34),(24,35),(25,36),(26,37)]])
39 conjugacy classes
| class | 1 | 2 | 3 | 13A | ··· | 13L | 26A | ··· | 26L | 39A | ··· | 39L |
| order | 1 | 2 | 3 | 13 | ··· | 13 | 26 | ··· | 26 | 39 | ··· | 39 |
| size | 1 | 3 | 2 | 1 | ··· | 1 | 3 | ··· | 3 | 2 | ··· | 2 |
39 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | |||
| image | C1 | C2 | C13 | C26 | S3 | S3×C13 |
| kernel | S3×C13 | C39 | S3 | C3 | C13 | C1 |
| # reps | 1 | 1 | 12 | 12 | 1 | 12 |
Matrix representation of S3×C13 ►in GL2(𝔽79) generated by
| 8 | 0 |
| 0 | 8 |
| 0 | 78 |
| 1 | 78 |
| 0 | 1 |
| 1 | 0 |
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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