direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: S3×D13, D39⋊C2, C3⋊1D26, C13⋊1D6, C39⋊C22, (S3×C13)⋊C2, (C3×D13)⋊C2, SmallGroup(156,11)
Series: Derived ►Chief ►Lower central ►Upper central
| C39 — S3×D13 |
Generators and relations for S3×D13
G = < a,b,c,d | a3=b2=c13=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >
Character table of S3×D13
| class | 1 | 2A | 2B | 2C | 3 | 6 | 13A | 13B | 13C | 13D | 13E | 13F | 26A | 26B | 26C | 26D | 26E | 26F | 39A | 39B | 39C | 39D | 39E | 39F | |
| size | 1 | 3 | 13 | 39 | 2 | 26 | 2 | 2 | 2 | 2 | 2 | 2 | 6 | 6 | 6 | 6 | 6 | 6 | 4 | 4 | 4 | 4 | 4 | 4 | |
| ρ1 | 1 | 1 | 1 | 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 | 1 | 1 | 1 | linear of order 2 |
| ρ3 | 1 | -1 | 1 | -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 |
| ρ4 | 1 | 1 | -1 | -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 |
| ρ5 | 2 | 0 | -2 | 0 | -1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from D6 |
| ρ6 | 2 | 0 | 2 | 0 | -1 | -1 | 2 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ7 | 2 | 2 | 0 | 0 | 2 | 0 | ζ137+ζ136 | ζ1312+ζ13 | ζ138+ζ135 | ζ1310+ζ133 | ζ139+ζ134 | ζ1311+ζ132 | ζ139+ζ134 | ζ138+ζ135 | ζ1310+ζ133 | ζ1311+ζ132 | ζ137+ζ136 | ζ1312+ζ13 | ζ1312+ζ13 | ζ139+ζ134 | ζ138+ζ135 | ζ1310+ζ133 | ζ1311+ζ132 | ζ137+ζ136 | orthogonal lifted from D13 |
| ρ8 | 2 | -2 | 0 | 0 | 2 | 0 | ζ137+ζ136 | ζ1312+ζ13 | ζ138+ζ135 | ζ1310+ζ133 | ζ139+ζ134 | ζ1311+ζ132 | -ζ139-ζ134 | -ζ138-ζ135 | -ζ1310-ζ133 | -ζ1311-ζ132 | -ζ137-ζ136 | -ζ1312-ζ13 | ζ1312+ζ13 | ζ139+ζ134 | ζ138+ζ135 | ζ1310+ζ133 | ζ1311+ζ132 | ζ137+ζ136 | orthogonal lifted from D26 |
| ρ9 | 2 | 2 | 0 | 0 | 2 | 0 | ζ1310+ζ133 | ζ137+ζ136 | ζ139+ζ134 | ζ138+ζ135 | ζ1311+ζ132 | ζ1312+ζ13 | ζ1311+ζ132 | ζ139+ζ134 | ζ138+ζ135 | ζ1312+ζ13 | ζ1310+ζ133 | ζ137+ζ136 | ζ137+ζ136 | ζ1311+ζ132 | ζ139+ζ134 | ζ138+ζ135 | ζ1312+ζ13 | ζ1310+ζ133 | orthogonal lifted from D13 |
| ρ10 | 2 | 2 | 0 | 0 | 2 | 0 | ζ138+ζ135 | ζ1310+ζ133 | ζ1311+ζ132 | ζ139+ζ134 | ζ1312+ζ13 | ζ137+ζ136 | ζ1312+ζ13 | ζ1311+ζ132 | ζ139+ζ134 | ζ137+ζ136 | ζ138+ζ135 | ζ1310+ζ133 | ζ1310+ζ133 | ζ1312+ζ13 | ζ1311+ζ132 | ζ139+ζ134 | ζ137+ζ136 | ζ138+ζ135 | orthogonal lifted from D13 |
| ρ11 | 2 | -2 | 0 | 0 | 2 | 0 | ζ139+ζ134 | ζ138+ζ135 | ζ1312+ζ13 | ζ1311+ζ132 | ζ137+ζ136 | ζ1310+ζ133 | -ζ137-ζ136 | -ζ1312-ζ13 | -ζ1311-ζ132 | -ζ1310-ζ133 | -ζ139-ζ134 | -ζ138-ζ135 | ζ138+ζ135 | ζ137+ζ136 | ζ1312+ζ13 | ζ1311+ζ132 | ζ1310+ζ133 | ζ139+ζ134 | orthogonal lifted from D26 |
| ρ12 | 2 | -2 | 0 | 0 | 2 | 0 | ζ138+ζ135 | ζ1310+ζ133 | ζ1311+ζ132 | ζ139+ζ134 | ζ1312+ζ13 | ζ137+ζ136 | -ζ1312-ζ13 | -ζ1311-ζ132 | -ζ139-ζ134 | -ζ137-ζ136 | -ζ138-ζ135 | -ζ1310-ζ133 | ζ1310+ζ133 | ζ1312+ζ13 | ζ1311+ζ132 | ζ139+ζ134 | ζ137+ζ136 | ζ138+ζ135 | orthogonal lifted from D26 |
| ρ13 | 2 | 2 | 0 | 0 | 2 | 0 | ζ1312+ζ13 | ζ1311+ζ132 | ζ1310+ζ133 | ζ137+ζ136 | ζ138+ζ135 | ζ139+ζ134 | ζ138+ζ135 | ζ1310+ζ133 | ζ137+ζ136 | ζ139+ζ134 | ζ1312+ζ13 | ζ1311+ζ132 | ζ1311+ζ132 | ζ138+ζ135 | ζ1310+ζ133 | ζ137+ζ136 | ζ139+ζ134 | ζ1312+ζ13 | orthogonal lifted from D13 |
| ρ14 | 2 | -2 | 0 | 0 | 2 | 0 | ζ1311+ζ132 | ζ139+ζ134 | ζ137+ζ136 | ζ1312+ζ13 | ζ1310+ζ133 | ζ138+ζ135 | -ζ1310-ζ133 | -ζ137-ζ136 | -ζ1312-ζ13 | -ζ138-ζ135 | -ζ1311-ζ132 | -ζ139-ζ134 | ζ139+ζ134 | ζ1310+ζ133 | ζ137+ζ136 | ζ1312+ζ13 | ζ138+ζ135 | ζ1311+ζ132 | orthogonal lifted from D26 |
| ρ15 | 2 | 2 | 0 | 0 | 2 | 0 | ζ139+ζ134 | ζ138+ζ135 | ζ1312+ζ13 | ζ1311+ζ132 | ζ137+ζ136 | ζ1310+ζ133 | ζ137+ζ136 | ζ1312+ζ13 | ζ1311+ζ132 | ζ1310+ζ133 | ζ139+ζ134 | ζ138+ζ135 | ζ138+ζ135 | ζ137+ζ136 | ζ1312+ζ13 | ζ1311+ζ132 | ζ1310+ζ133 | ζ139+ζ134 | orthogonal lifted from D13 |
| ρ16 | 2 | 2 | 0 | 0 | 2 | 0 | ζ1311+ζ132 | ζ139+ζ134 | ζ137+ζ136 | ζ1312+ζ13 | ζ1310+ζ133 | ζ138+ζ135 | ζ1310+ζ133 | ζ137+ζ136 | ζ1312+ζ13 | ζ138+ζ135 | ζ1311+ζ132 | ζ139+ζ134 | ζ139+ζ134 | ζ1310+ζ133 | ζ137+ζ136 | ζ1312+ζ13 | ζ138+ζ135 | ζ1311+ζ132 | orthogonal lifted from D13 |
| ρ17 | 2 | -2 | 0 | 0 | 2 | 0 | ζ1310+ζ133 | ζ137+ζ136 | ζ139+ζ134 | ζ138+ζ135 | ζ1311+ζ132 | ζ1312+ζ13 | -ζ1311-ζ132 | -ζ139-ζ134 | -ζ138-ζ135 | -ζ1312-ζ13 | -ζ1310-ζ133 | -ζ137-ζ136 | ζ137+ζ136 | ζ1311+ζ132 | ζ139+ζ134 | ζ138+ζ135 | ζ1312+ζ13 | ζ1310+ζ133 | orthogonal lifted from D26 |
| ρ18 | 2 | -2 | 0 | 0 | 2 | 0 | ζ1312+ζ13 | ζ1311+ζ132 | ζ1310+ζ133 | ζ137+ζ136 | ζ138+ζ135 | ζ139+ζ134 | -ζ138-ζ135 | -ζ1310-ζ133 | -ζ137-ζ136 | -ζ139-ζ134 | -ζ1312-ζ13 | -ζ1311-ζ132 | ζ1311+ζ132 | ζ138+ζ135 | ζ1310+ζ133 | ζ137+ζ136 | ζ139+ζ134 | ζ1312+ζ13 | orthogonal lifted from D26 |
| ρ19 | 4 | 0 | 0 | 0 | -2 | 0 | 2ζ137+2ζ136 | 2ζ1312+2ζ13 | 2ζ138+2ζ135 | 2ζ1310+2ζ133 | 2ζ139+2ζ134 | 2ζ1311+2ζ132 | 0 | 0 | 0 | 0 | 0 | 0 | -ζ1312-ζ13 | -ζ139-ζ134 | -ζ138-ζ135 | -ζ1310-ζ133 | -ζ1311-ζ132 | -ζ137-ζ136 | orthogonal faithful |
| ρ20 | 4 | 0 | 0 | 0 | -2 | 0 | 2ζ1311+2ζ132 | 2ζ139+2ζ134 | 2ζ137+2ζ136 | 2ζ1312+2ζ13 | 2ζ1310+2ζ133 | 2ζ138+2ζ135 | 0 | 0 | 0 | 0 | 0 | 0 | -ζ139-ζ134 | -ζ1310-ζ133 | -ζ137-ζ136 | -ζ1312-ζ13 | -ζ138-ζ135 | -ζ1311-ζ132 | orthogonal faithful |
| ρ21 | 4 | 0 | 0 | 0 | -2 | 0 | 2ζ1310+2ζ133 | 2ζ137+2ζ136 | 2ζ139+2ζ134 | 2ζ138+2ζ135 | 2ζ1311+2ζ132 | 2ζ1312+2ζ13 | 0 | 0 | 0 | 0 | 0 | 0 | -ζ137-ζ136 | -ζ1311-ζ132 | -ζ139-ζ134 | -ζ138-ζ135 | -ζ1312-ζ13 | -ζ1310-ζ133 | orthogonal faithful |
| ρ22 | 4 | 0 | 0 | 0 | -2 | 0 | 2ζ139+2ζ134 | 2ζ138+2ζ135 | 2ζ1312+2ζ13 | 2ζ1311+2ζ132 | 2ζ137+2ζ136 | 2ζ1310+2ζ133 | 0 | 0 | 0 | 0 | 0 | 0 | -ζ138-ζ135 | -ζ137-ζ136 | -ζ1312-ζ13 | -ζ1311-ζ132 | -ζ1310-ζ133 | -ζ139-ζ134 | orthogonal faithful |
| ρ23 | 4 | 0 | 0 | 0 | -2 | 0 | 2ζ138+2ζ135 | 2ζ1310+2ζ133 | 2ζ1311+2ζ132 | 2ζ139+2ζ134 | 2ζ1312+2ζ13 | 2ζ137+2ζ136 | 0 | 0 | 0 | 0 | 0 | 0 | -ζ1310-ζ133 | -ζ1312-ζ13 | -ζ1311-ζ132 | -ζ139-ζ134 | -ζ137-ζ136 | -ζ138-ζ135 | orthogonal faithful |
| ρ24 | 4 | 0 | 0 | 0 | -2 | 0 | 2ζ1312+2ζ13 | 2ζ1311+2ζ132 | 2ζ1310+2ζ133 | 2ζ137+2ζ136 | 2ζ138+2ζ135 | 2ζ139+2ζ134 | 0 | 0 | 0 | 0 | 0 | 0 | -ζ1311-ζ132 | -ζ138-ζ135 | -ζ1310-ζ133 | -ζ137-ζ136 | -ζ139-ζ134 | -ζ1312-ζ13 | orthogonal faithful |
►On 39 points
Generators in S39
(1 21 27)(2 22 28)(3 23 29)(4 24 30)(5 25 31)(6 26 32)(7 14 33)(8 15 34)(9 16 35)(10 17 36)(11 18 37)(12 19 38)(13 20 39)
(14 33)(15 34)(16 35)(17 36)(18 37)(19 38)(20 39)(21 27)(22 28)(23 29)(24 30)(25 31)(26 32)
(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 13)(2 12)(3 11)(4 10)(5 9)(6 8)(15 26)(16 25)(17 24)(18 23)(19 22)(20 21)(27 39)(28 38)(29 37)(30 36)(31 35)(32 34)
G:=sub<Sym(39)| (1,21,27)(2,22,28)(3,23,29)(4,24,30)(5,25,31)(6,26,32)(7,14,33)(8,15,34)(9,16,35)(10,17,36)(11,18,37)(12,19,38)(13,20,39), (14,33)(15,34)(16,35)(17,36)(18,37)(19,38)(20,39)(21,27)(22,28)(23,29)(24,30)(25,31)(26,32), (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), (1,13)(2,12)(3,11)(4,10)(5,9)(6,8)(15,26)(16,25)(17,24)(18,23)(19,22)(20,21)(27,39)(28,38)(29,37)(30,36)(31,35)(32,34)>;
G:=Group( (1,21,27)(2,22,28)(3,23,29)(4,24,30)(5,25,31)(6,26,32)(7,14,33)(8,15,34)(9,16,35)(10,17,36)(11,18,37)(12,19,38)(13,20,39), (14,33)(15,34)(16,35)(17,36)(18,37)(19,38)(20,39)(21,27)(22,28)(23,29)(24,30)(25,31)(26,32), (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), (1,13)(2,12)(3,11)(4,10)(5,9)(6,8)(15,26)(16,25)(17,24)(18,23)(19,22)(20,21)(27,39)(28,38)(29,37)(30,36)(31,35)(32,34) );
G=PermutationGroup([[(1,21,27),(2,22,28),(3,23,29),(4,24,30),(5,25,31),(6,26,32),(7,14,33),(8,15,34),(9,16,35),(10,17,36),(11,18,37),(12,19,38),(13,20,39)], [(14,33),(15,34),(16,35),(17,36),(18,37),(19,38),(20,39),(21,27),(22,28),(23,29),(24,30),(25,31),(26,32)], [(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)], [(1,13),(2,12),(3,11),(4,10),(5,9),(6,8),(15,26),(16,25),(17,24),(18,23),(19,22),(20,21),(27,39),(28,38),(29,37),(30,36),(31,35),(32,34)]])
S3×D13 is a maximal subgroup of
D39⋊S3
S3×D13 is a maximal quotient of D78.C2 C39⋊D4 C3⋊D52 C13⋊D12 C39⋊Q8 D39⋊S3
Matrix representation of S3×D13 ►in GL4(𝔽79) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 77 | 9 |
| 0 | 0 | 26 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 53 | 78 |
| 39 | 1 | 0 | 0 |
| 13 | 51 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 26 | 38 | 0 | 0 |
| 55 | 53 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(79))| [1,0,0,0,0,1,0,0,0,0,77,26,0,0,9,1],[1,0,0,0,0,1,0,0,0,0,1,53,0,0,0,78],[39,13,0,0,1,51,0,0,0,0,1,0,0,0,0,1],[26,55,0,0,38,53,0,0,0,0,1,0,0,0,0,1] >;
S3×D13 in GAP, Magma, Sage, TeX
S_3\times D_{13} % in TeX
G:=Group("S3xD13"); // GroupNames label
G:=SmallGroup(156,11);
// by ID
G=gap.SmallGroup(156,11);
# by ID
G:=PCGroup([4,-2,-2,-3,-13,54,2307]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^2=c^13=d^2=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=c^-1>;
// generators/relations
Export
Subgroup lattice of S3×D13 in TeX
Character table of S3×D13 in TeX