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
C1 — C13 — C39 — C3×C13⋊C3 — S3×C13⋊C3 |
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 |
(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