direct product, metacyclic, supersoluble, monomial, A-group
Aliases: S3×C13⋊C3, C39⋊3C6, (S3×C13)⋊C3, C13⋊2(C3×S3), C3⋊(C2×C13⋊C3), (C3×C13⋊C3)⋊3C2, SmallGroup(234,8)
Series: Derived ►Chief ►Lower central ►Upper central
| C39 — S3×C13⋊C3 |
Generators and relations for S3×C13⋊C3
G = < a,b,c,d | a3=b2=c13=d3=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c9 >
Character table of S3×C13⋊C3
| class | 1 | 2 | 3A | 3B | 3C | 3D | 3E | 6A | 6B | 13A | 13B | 13C | 13D | 26A | 26B | 26C | 26D | 39A | 39B | 39C | 39D | |
| size | 1 | 3 | 2 | 13 | 13 | 26 | 26 | 39 | 39 | 3 | 3 | 3 | 3 | 9 | 9 | 9 | 9 | 6 | 6 | 6 | 6 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 1 | -1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | linear of order 2 |
| ρ3 | 1 | -1 | 1 | ζ3 | ζ32 | ζ32 | ζ3 | ζ6 | ζ65 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | linear of order 6 |
| ρ4 | 1 | -1 | 1 | ζ32 | ζ3 | ζ3 | ζ32 | ζ65 | ζ6 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | linear of order 6 |
| ρ5 | 1 | 1 | 1 | ζ3 | ζ32 | ζ32 | ζ3 | ζ32 | ζ3 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 3 |
| ρ6 | 1 | 1 | 1 | ζ32 | ζ3 | ζ3 | ζ32 | ζ3 | ζ32 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 3 |
| ρ7 | 2 | 0 | -1 | 2 | 2 | -1 | -1 | 0 | 0 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ8 | 2 | 0 | -1 | -1-√-3 | -1+√-3 | ζ65 | ζ6 | 0 | 0 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | complex lifted from C3×S3 |
| ρ9 | 2 | 0 | -1 | -1+√-3 | -1-√-3 | ζ6 | ζ65 | 0 | 0 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | complex lifted from C3×S3 |
| ρ10 | 3 | -3 | 3 | 0 | 0 | 0 | 0 | 0 | 0 | ζ139+ζ133+ζ13 | ζ136+ζ135+ζ132 | ζ1312+ζ1310+ζ134 | ζ1311+ζ138+ζ137 | -ζ139-ζ133-ζ13 | -ζ1312-ζ1310-ζ134 | -ζ1311-ζ138-ζ137 | -ζ136-ζ135-ζ132 | ζ139+ζ133+ζ13 | ζ1311+ζ138+ζ137 | ζ1312+ζ1310+ζ134 | ζ136+ζ135+ζ132 | complex lifted from C2×C13⋊C3 |
| ρ11 | 3 | 3 | 3 | 0 | 0 | 0 | 0 | 0 | 0 | ζ136+ζ135+ζ132 | ζ1312+ζ1310+ζ134 | ζ1311+ζ138+ζ137 | ζ139+ζ133+ζ13 | ζ136+ζ135+ζ132 | ζ1311+ζ138+ζ137 | ζ139+ζ133+ζ13 | ζ1312+ζ1310+ζ134 | ζ136+ζ135+ζ132 | ζ139+ζ133+ζ13 | ζ1311+ζ138+ζ137 | ζ1312+ζ1310+ζ134 | complex lifted from C13⋊C3 |
| ρ12 | 3 | 3 | 3 | 0 | 0 | 0 | 0 | 0 | 0 | ζ1312+ζ1310+ζ134 | ζ1311+ζ138+ζ137 | ζ139+ζ133+ζ13 | ζ136+ζ135+ζ132 | ζ1312+ζ1310+ζ134 | ζ139+ζ133+ζ13 | ζ136+ζ135+ζ132 | ζ1311+ζ138+ζ137 | ζ1312+ζ1310+ζ134 | ζ136+ζ135+ζ132 | ζ139+ζ133+ζ13 | ζ1311+ζ138+ζ137 | complex lifted from C13⋊C3 |
| ρ13 | 3 | 3 | 3 | 0 | 0 | 0 | 0 | 0 | 0 | ζ1311+ζ138+ζ137 | ζ139+ζ133+ζ13 | ζ136+ζ135+ζ132 | ζ1312+ζ1310+ζ134 | ζ1311+ζ138+ζ137 | ζ136+ζ135+ζ132 | ζ1312+ζ1310+ζ134 | ζ139+ζ133+ζ13 | ζ1311+ζ138+ζ137 | ζ1312+ζ1310+ζ134 | ζ136+ζ135+ζ132 | ζ139+ζ133+ζ13 | complex lifted from C13⋊C3 |
| ρ14 | 3 | -3 | 3 | 0 | 0 | 0 | 0 | 0 | 0 | ζ136+ζ135+ζ132 | ζ1312+ζ1310+ζ134 | ζ1311+ζ138+ζ137 | ζ139+ζ133+ζ13 | -ζ136-ζ135-ζ132 | -ζ1311-ζ138-ζ137 | -ζ139-ζ133-ζ13 | -ζ1312-ζ1310-ζ134 | ζ136+ζ135+ζ132 | ζ139+ζ133+ζ13 | ζ1311+ζ138+ζ137 | ζ1312+ζ1310+ζ134 | complex lifted from C2×C13⋊C3 |
| ρ15 | 3 | 3 | 3 | 0 | 0 | 0 | 0 | 0 | 0 | ζ139+ζ133+ζ13 | ζ136+ζ135+ζ132 | ζ1312+ζ1310+ζ134 | ζ1311+ζ138+ζ137 | ζ139+ζ133+ζ13 | ζ1312+ζ1310+ζ134 | ζ1311+ζ138+ζ137 | ζ136+ζ135+ζ132 | ζ139+ζ133+ζ13 | ζ1311+ζ138+ζ137 | ζ1312+ζ1310+ζ134 | ζ136+ζ135+ζ132 | complex lifted from C13⋊C3 |
| ρ16 | 3 | -3 | 3 | 0 | 0 | 0 | 0 | 0 | 0 | ζ1312+ζ1310+ζ134 | ζ1311+ζ138+ζ137 | ζ139+ζ133+ζ13 | ζ136+ζ135+ζ132 | -ζ1312-ζ1310-ζ134 | -ζ139-ζ133-ζ13 | -ζ136-ζ135-ζ132 | -ζ1311-ζ138-ζ137 | ζ1312+ζ1310+ζ134 | ζ136+ζ135+ζ132 | ζ139+ζ133+ζ13 | ζ1311+ζ138+ζ137 | complex lifted from C2×C13⋊C3 |
| ρ17 | 3 | -3 | 3 | 0 | 0 | 0 | 0 | 0 | 0 | ζ1311+ζ138+ζ137 | ζ139+ζ133+ζ13 | ζ136+ζ135+ζ132 | ζ1312+ζ1310+ζ134 | -ζ1311-ζ138-ζ137 | -ζ136-ζ135-ζ132 | -ζ1312-ζ1310-ζ134 | -ζ139-ζ133-ζ13 | ζ1311+ζ138+ζ137 | ζ1312+ζ1310+ζ134 | ζ136+ζ135+ζ132 | ζ139+ζ133+ζ13 | complex lifted from C2×C13⋊C3 |
| ρ18 | 6 | 0 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | 2ζ139+2ζ133+2ζ13 | 2ζ136+2ζ135+2ζ132 | 2ζ1312+2ζ1310+2ζ134 | 2ζ1311+2ζ138+2ζ137 | 0 | 0 | 0 | 0 | -ζ139-ζ133-ζ13 | -ζ1311-ζ138-ζ137 | -ζ1312-ζ1310-ζ134 | -ζ136-ζ135-ζ132 | complex faithful |
| ρ19 | 6 | 0 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | 2ζ1311+2ζ138+2ζ137 | 2ζ139+2ζ133+2ζ13 | 2ζ136+2ζ135+2ζ132 | 2ζ1312+2ζ1310+2ζ134 | 0 | 0 | 0 | 0 | -ζ1311-ζ138-ζ137 | -ζ1312-ζ1310-ζ134 | -ζ136-ζ135-ζ132 | -ζ139-ζ133-ζ13 | complex faithful |
| ρ20 | 6 | 0 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | 2ζ1312+2ζ1310+2ζ134 | 2ζ1311+2ζ138+2ζ137 | 2ζ139+2ζ133+2ζ13 | 2ζ136+2ζ135+2ζ132 | 0 | 0 | 0 | 0 | -ζ1312-ζ1310-ζ134 | -ζ136-ζ135-ζ132 | -ζ139-ζ133-ζ13 | -ζ1311-ζ138-ζ137 | complex faithful |
| ρ21 | 6 | 0 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | 2ζ136+2ζ135+2ζ132 | 2ζ1312+2ζ1310+2ζ134 | 2ζ1311+2ζ138+2ζ137 | 2ζ139+2ζ133+2ζ13 | 0 | 0 | 0 | 0 | -ζ136-ζ135-ζ132 | -ζ139-ζ133-ζ13 | -ζ1311-ζ138-ζ137 | -ζ1312-ζ1310-ζ134 | complex faithful |
►On 39 points
Generators in S39
(1 14 27)(2 15 28)(3 16 29)(4 17 30)(5 18 31)(6 19 32)(7 20 33)(8 21 34)(9 22 35)(10 23 36)(11 24 37)(12 25 38)(13 26 39)
(14 27)(15 28)(16 29)(17 30)(18 31)(19 32)(20 33)(21 34)(22 35)(23 36)(24 37)(25 38)(26 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)
(2 4 10)(3 7 6)(5 13 11)(8 9 12)(15 17 23)(16 20 19)(18 26 24)(21 22 25)(28 30 36)(29 33 32)(31 39 37)(34 35 38)
G:=sub<Sym(39)| (1,14,27)(2,15,28)(3,16,29)(4,17,30)(5,18,31)(6,19,32)(7,20,33)(8,21,34)(9,22,35)(10,23,36)(11,24,37)(12,25,38)(13,26,39), (14,27)(15,28)(16,29)(17,30)(18,31)(19,32)(20,33)(21,34)(22,35)(23,36)(24,37)(25,38)(26,39), (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), (2,4,10)(3,7,6)(5,13,11)(8,9,12)(15,17,23)(16,20,19)(18,26,24)(21,22,25)(28,30,36)(29,33,32)(31,39,37)(34,35,38)>;
G:=Group( (1,14,27)(2,15,28)(3,16,29)(4,17,30)(5,18,31)(6,19,32)(7,20,33)(8,21,34)(9,22,35)(10,23,36)(11,24,37)(12,25,38)(13,26,39), (14,27)(15,28)(16,29)(17,30)(18,31)(19,32)(20,33)(21,34)(22,35)(23,36)(24,37)(25,38)(26,39), (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), (2,4,10)(3,7,6)(5,13,11)(8,9,12)(15,17,23)(16,20,19)(18,26,24)(21,22,25)(28,30,36)(29,33,32)(31,39,37)(34,35,38) );
G=PermutationGroup([[(1,14,27),(2,15,28),(3,16,29),(4,17,30),(5,18,31),(6,19,32),(7,20,33),(8,21,34),(9,22,35),(10,23,36),(11,24,37),(12,25,38),(13,26,39)], [(14,27),(15,28),(16,29),(17,30),(18,31),(19,32),(20,33),(21,34),(22,35),(23,36),(24,37),(25,38),(26,39)], [(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)], [(2,4,10),(3,7,6),(5,13,11),(8,9,12),(15,17,23),(16,20,19),(18,26,24),(21,22,25),(28,30,36),(29,33,32),(31,39,37),(34,35,38)]])
Matrix representation of S3×C13⋊C3 ►in GL5(𝔽79)
| 78 | 78 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 78 | 78 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 9 | 13 | 47 |
| 0 | 0 | 1 | 0 | 52 |
| 0 | 0 | 0 | 1 | 4 |
| 55 | 0 | 0 | 0 | 0 |
| 0 | 55 | 0 | 0 | 0 |
| 0 | 0 | 16 | 28 | 15 |
| 0 | 0 | 67 | 6 | 55 |
| 0 | 0 | 72 | 34 | 57 |
G:=sub<GL(5,GF(79))| [78,1,0,0,0,78,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,78,0,0,0,0,78,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,9,1,0,0,0,13,0,1,0,0,47,52,4],[55,0,0,0,0,0,55,0,0,0,0,0,16,67,72,0,0,28,6,34,0,0,15,55,57] >;
S3×C13⋊C3 in GAP, Magma, Sage, TeX
S_3\times C_{13}\rtimes C_3 % in TeX
G:=Group("S3xC13:C3"); // GroupNames label
G:=SmallGroup(234,8);
// by ID
G=gap.SmallGroup(234,8);
# by ID
G:=PCGroup([4,-2,-3,-3,-13,146,439]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^2=c^13=d^3=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^9>;
// generators/relations
Export
Subgroup lattice of S3×C13⋊C3 in TeX
Character table of S3×C13⋊C3 in TeX