direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: S3×C13⋊C4, D39⋊C4, D13.1D6, C13⋊(C4×S3), C39⋊(C2×C4), C39⋊C4⋊C2, (S3×C13)⋊C4, (S3×D13).C2, (C3×D13).C22, (C3×C13⋊C4)⋊C2, C3⋊1(C2×C13⋊C4), SmallGroup(312,46)
Series: Derived ►Chief ►Lower central ►Upper central
| C39 — S3×C13⋊C4 |
Generators and relations for S3×C13⋊C4
G = < a,b,c,d | a3=b2=c13=d4=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c5 >
Character table of S3×C13⋊C4
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 6 | 12A | 12B | 13A | 13B | 13C | 26A | 26B | 26C | 39A | 39B | 39C | |
| size | 1 | 3 | 13 | 39 | 2 | 13 | 13 | 39 | 39 | 26 | 26 | 26 | 4 | 4 | 4 | 12 | 12 | 12 | 8 | 8 | 8 | |
| 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 | -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 | linear of order 2 |
| ρ5 | 1 | -1 | -1 | 1 | 1 | -i | i | i | -i | -1 | -i | i | 1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | 1 | linear of order 4 |
| ρ6 | 1 | -1 | -1 | 1 | 1 | i | -i | -i | i | -1 | i | -i | 1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | 1 | linear of order 4 |
| ρ7 | 1 | 1 | -1 | -1 | 1 | -i | i | -i | i | -1 | -i | i | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 4 |
| ρ8 | 1 | 1 | -1 | -1 | 1 | i | -i | i | -i | -1 | i | -i | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 4 |
| ρ9 | 2 | 0 | 2 | 0 | -1 | 2 | 2 | 0 | 0 | -1 | -1 | -1 | 2 | 2 | 2 | 0 | 0 | 0 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ10 | 2 | 0 | 2 | 0 | -1 | -2 | -2 | 0 | 0 | -1 | 1 | 1 | 2 | 2 | 2 | 0 | 0 | 0 | -1 | -1 | -1 | orthogonal lifted from D6 |
| ρ11 | 2 | 0 | -2 | 0 | -1 | 2i | -2i | 0 | 0 | 1 | -i | i | 2 | 2 | 2 | 0 | 0 | 0 | -1 | -1 | -1 | complex lifted from C4×S3 |
| ρ12 | 2 | 0 | -2 | 0 | -1 | -2i | 2i | 0 | 0 | 1 | i | -i | 2 | 2 | 2 | 0 | 0 | 0 | -1 | -1 | -1 | complex lifted from C4×S3 |
| ρ13 | 4 | 4 | 0 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ1312+ζ138+ζ135+ζ13 | ζ139+ζ137+ζ136+ζ134 | ζ1311+ζ1310+ζ133+ζ132 | ζ1311+ζ1310+ζ133+ζ132 | ζ139+ζ137+ζ136+ζ134 | ζ1312+ζ138+ζ135+ζ13 | ζ1312+ζ138+ζ135+ζ13 | ζ1311+ζ1310+ζ133+ζ132 | ζ139+ζ137+ζ136+ζ134 | orthogonal lifted from C13⋊C4 |
| ρ14 | 4 | -4 | 0 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ1312+ζ138+ζ135+ζ13 | ζ139+ζ137+ζ136+ζ134 | ζ1311+ζ1310+ζ133+ζ132 | -ζ1311-ζ1310-ζ133-ζ132 | -ζ139-ζ137-ζ136-ζ134 | -ζ1312-ζ138-ζ135-ζ13 | ζ1312+ζ138+ζ135+ζ13 | ζ1311+ζ1310+ζ133+ζ132 | ζ139+ζ137+ζ136+ζ134 | orthogonal lifted from C2×C13⋊C4 |
| ρ15 | 4 | -4 | 0 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ1311+ζ1310+ζ133+ζ132 | ζ1312+ζ138+ζ135+ζ13 | ζ139+ζ137+ζ136+ζ134 | -ζ139-ζ137-ζ136-ζ134 | -ζ1312-ζ138-ζ135-ζ13 | -ζ1311-ζ1310-ζ133-ζ132 | ζ1311+ζ1310+ζ133+ζ132 | ζ139+ζ137+ζ136+ζ134 | ζ1312+ζ138+ζ135+ζ13 | orthogonal lifted from C2×C13⋊C4 |
| ρ16 | 4 | 4 | 0 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ1311+ζ1310+ζ133+ζ132 | ζ1312+ζ138+ζ135+ζ13 | ζ139+ζ137+ζ136+ζ134 | ζ139+ζ137+ζ136+ζ134 | ζ1312+ζ138+ζ135+ζ13 | ζ1311+ζ1310+ζ133+ζ132 | ζ1311+ζ1310+ζ133+ζ132 | ζ139+ζ137+ζ136+ζ134 | ζ1312+ζ138+ζ135+ζ13 | orthogonal lifted from C13⋊C4 |
| ρ17 | 4 | 4 | 0 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ139+ζ137+ζ136+ζ134 | ζ1311+ζ1310+ζ133+ζ132 | ζ1312+ζ138+ζ135+ζ13 | ζ1312+ζ138+ζ135+ζ13 | ζ1311+ζ1310+ζ133+ζ132 | ζ139+ζ137+ζ136+ζ134 | ζ139+ζ137+ζ136+ζ134 | ζ1312+ζ138+ζ135+ζ13 | ζ1311+ζ1310+ζ133+ζ132 | orthogonal lifted from C13⋊C4 |
| ρ18 | 4 | -4 | 0 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ139+ζ137+ζ136+ζ134 | ζ1311+ζ1310+ζ133+ζ132 | ζ1312+ζ138+ζ135+ζ13 | -ζ1312-ζ138-ζ135-ζ13 | -ζ1311-ζ1310-ζ133-ζ132 | -ζ139-ζ137-ζ136-ζ134 | ζ139+ζ137+ζ136+ζ134 | ζ1312+ζ138+ζ135+ζ13 | ζ1311+ζ1310+ζ133+ζ132 | orthogonal lifted from C2×C13⋊C4 |
| ρ19 | 8 | 0 | 0 | 0 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2ζ1311+2ζ1310+2ζ133+2ζ132 | 2ζ1312+2ζ138+2ζ135+2ζ13 | 2ζ139+2ζ137+2ζ136+2ζ134 | 0 | 0 | 0 | -ζ1311-ζ1310-ζ133-ζ132 | -ζ139-ζ137-ζ136-ζ134 | -ζ1312-ζ138-ζ135-ζ13 | orthogonal faithful |
| ρ20 | 8 | 0 | 0 | 0 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2ζ139+2ζ137+2ζ136+2ζ134 | 2ζ1311+2ζ1310+2ζ133+2ζ132 | 2ζ1312+2ζ138+2ζ135+2ζ13 | 0 | 0 | 0 | -ζ139-ζ137-ζ136-ζ134 | -ζ1312-ζ138-ζ135-ζ13 | -ζ1311-ζ1310-ζ133-ζ132 | orthogonal faithful |
| ρ21 | 8 | 0 | 0 | 0 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2ζ1312+2ζ138+2ζ135+2ζ13 | 2ζ139+2ζ137+2ζ136+2ζ134 | 2ζ1311+2ζ1310+2ζ133+2ζ132 | 0 | 0 | 0 | -ζ1312-ζ138-ζ135-ζ13 | -ζ1311-ζ1310-ζ133-ζ132 | -ζ139-ζ137-ζ136-ζ134 | orthogonal 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 9 13 6)(3 4 12 11)(5 7 10 8)(15 22 26 19)(16 17 25 24)(18 20 23 21)(28 35 39 32)(29 30 38 37)(31 33 36 34)
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), (2,9,13,6)(3,4,12,11)(5,7,10,8)(15,22,26,19)(16,17,25,24)(18,20,23,21)(28,35,39,32)(29,30,38,37)(31,33,36,34)>;
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), (2,9,13,6)(3,4,12,11)(5,7,10,8)(15,22,26,19)(16,17,25,24)(18,20,23,21)(28,35,39,32)(29,30,38,37)(31,33,36,34) );
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)], [(2,9,13,6),(3,4,12,11),(5,7,10,8),(15,22,26,19),(16,17,25,24),(18,20,23,21),(28,35,39,32),(29,30,38,37),(31,33,36,34)]])
Matrix representation of S3×C13⋊C4 ►in GL6(𝔽157)
| 1 | 19 | 0 | 0 | 0 | 0 |
| 99 | 155 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 156 | 138 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 33 | 135 | 33 | 156 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 129 | 0 | 0 | 0 | 0 | 0 |
| 0 | 129 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 124 | 146 | 92 | 125 |
| 0 | 0 | 135 | 65 | 10 | 32 |
| 0 | 0 | 0 | 0 | 1 | 0 |
G:=sub<GL(6,GF(157))| [1,99,0,0,0,0,19,155,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[156,0,0,0,0,0,138,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,33,1,0,0,0,0,135,0,1,0,0,0,33,0,0,1,0,0,156,0,0,0],[129,0,0,0,0,0,0,129,0,0,0,0,0,0,1,124,135,0,0,0,0,146,65,0,0,0,0,92,10,1,0,0,0,125,32,0] >;
S3×C13⋊C4 in GAP, Magma, Sage, TeX
S_3\times C_{13}\rtimes C_4 % in TeX
G:=Group("S3xC13:C4"); // GroupNames label
G:=SmallGroup(312,46);
// by ID
G=gap.SmallGroup(312,46);
# by ID
G:=PCGroup([5,-2,-2,-2,-3,-13,20,168,4804,1814]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^2=c^13=d^4=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^5>;
// generators/relations
Export
Subgroup lattice of S3×C13⋊C4 in TeX
Character table of S3×C13⋊C4 in TeX