direct product, metacyclic, supersoluble, monomial, A-group, 3-hyperelementary
Aliases: C3×C13⋊C3, C39⋊C3, C13⋊C32, SmallGroup(117,3)
Series: Derived ►Chief ►Lower central ►Upper central
| C13 — C3×C13⋊C3 |
Generators and relations for C3×C13⋊C3
G = < a,b,c | a3=b13=c3=1, ab=ba, ac=ca, cbc-1=b9 >
Character table of C3×C13⋊C3
| class | 1 | 3A | 3B | 3C | 3D | 3E | 3F | 3G | 3H | 13A | 13B | 13C | 13D | 39A | 39B | 39C | 39D | 39E | 39F | 39G | 39H | |
| size | 1 | 1 | 1 | 13 | 13 | 13 | 13 | 13 | 13 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | |
| 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 | ζ32 | ζ3 | ζ3 | ζ3 | ζ32 | 1 | ζ32 | 1 | 1 | 1 | 1 | 1 | ζ32 | ζ32 | ζ3 | ζ3 | ζ3 | ζ3 | ζ32 | ζ32 | linear of order 3 |
| ρ3 | 1 | ζ3 | ζ32 | 1 | ζ3 | 1 | ζ32 | ζ32 | ζ3 | 1 | 1 | 1 | 1 | ζ3 | ζ3 | ζ32 | ζ32 | ζ32 | ζ32 | ζ3 | ζ3 | linear of order 3 |
| ρ4 | 1 | ζ32 | ζ3 | ζ32 | 1 | ζ3 | ζ32 | 1 | ζ3 | 1 | 1 | 1 | 1 | ζ32 | ζ32 | ζ3 | ζ3 | ζ3 | ζ3 | ζ32 | ζ32 | linear of order 3 |
| ρ5 | 1 | ζ3 | ζ32 | ζ3 | 1 | ζ32 | ζ3 | 1 | ζ32 | 1 | 1 | 1 | 1 | ζ3 | ζ3 | ζ32 | ζ32 | ζ32 | ζ32 | ζ3 | ζ3 | linear of order 3 |
| ρ6 | 1 | ζ3 | ζ32 | ζ32 | ζ32 | ζ3 | 1 | ζ3 | 1 | 1 | 1 | 1 | 1 | ζ3 | ζ3 | ζ32 | ζ32 | ζ32 | ζ32 | ζ3 | ζ3 | linear of order 3 |
| ρ7 | 1 | ζ32 | ζ3 | 1 | ζ32 | 1 | ζ3 | ζ3 | ζ32 | 1 | 1 | 1 | 1 | ζ32 | ζ32 | ζ3 | ζ3 | ζ3 | ζ3 | ζ32 | ζ32 | linear of order 3 |
| ρ8 | 1 | 1 | 1 | ζ3 | ζ32 | ζ32 | ζ32 | ζ3 | ζ3 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 3 |
| ρ9 | 1 | 1 | 1 | ζ32 | ζ3 | ζ3 | ζ3 | ζ32 | ζ32 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 3 |
| ρ10 | 3 | 3 | 3 | 0 | 0 | 0 | 0 | 0 | 0 | ζ1311+ζ138+ζ137 | ζ139+ζ133+ζ13 | ζ136+ζ135+ζ132 | ζ1312+ζ1310+ζ134 | ζ1311+ζ138+ζ137 | ζ139+ζ133+ζ13 | ζ1312+ζ1310+ζ134 | ζ1311+ζ138+ζ137 | ζ139+ζ133+ζ13 | ζ136+ζ135+ζ132 | ζ136+ζ135+ζ132 | ζ1312+ζ1310+ζ134 | complex lifted from 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 | ζ1312+ζ1310+ζ134 | ζ139+ζ133+ζ13 | ζ136+ζ135+ζ132 | ζ1312+ζ1310+ζ134 | ζ1311+ζ138+ζ137 | ζ1311+ζ138+ζ137 | ζ139+ζ133+ζ13 | complex lifted from C13⋊C3 |
| ρ12 | 3 | 3 | 3 | 0 | 0 | 0 | 0 | 0 | 0 | ζ139+ζ133+ζ13 | ζ136+ζ135+ζ132 | ζ1312+ζ1310+ζ134 | ζ1311+ζ138+ζ137 | ζ139+ζ133+ζ13 | ζ136+ζ135+ζ132 | ζ1311+ζ138+ζ137 | ζ139+ζ133+ζ13 | ζ136+ζ135+ζ132 | ζ1312+ζ1310+ζ134 | ζ1312+ζ1310+ζ134 | ζ1311+ζ138+ζ137 | complex lifted from C13⋊C3 |
| ρ13 | 3 | 3 | 3 | 0 | 0 | 0 | 0 | 0 | 0 | ζ1312+ζ1310+ζ134 | ζ1311+ζ138+ζ137 | ζ139+ζ133+ζ13 | ζ136+ζ135+ζ132 | ζ1312+ζ1310+ζ134 | ζ1311+ζ138+ζ137 | ζ136+ζ135+ζ132 | ζ1312+ζ1310+ζ134 | ζ1311+ζ138+ζ137 | ζ139+ζ133+ζ13 | ζ139+ζ133+ζ13 | ζ136+ζ135+ζ132 | complex lifted from C13⋊C3 |
| ρ14 | 3 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | ζ136+ζ135+ζ132 | ζ1312+ζ1310+ζ134 | ζ1311+ζ138+ζ137 | ζ139+ζ133+ζ13 | ζ32ζ136+ζ32ζ135+ζ32ζ132 | ζ32ζ1312+ζ32ζ1310+ζ32ζ134 | ζ3ζ139+ζ3ζ133+ζ3ζ13 | ζ3ζ136+ζ3ζ135+ζ3ζ132 | ζ3ζ1312+ζ3ζ1310+ζ3ζ134 | ζ3ζ1311+ζ3ζ138+ζ3ζ137 | ζ32ζ1311+ζ32ζ138+ζ32ζ137 | ζ32ζ139+ζ32ζ133+ζ32ζ13 | complex faithful |
| ρ15 | 3 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | ζ1312+ζ1310+ζ134 | ζ1311+ζ138+ζ137 | ζ139+ζ133+ζ13 | ζ136+ζ135+ζ132 | ζ32ζ1312+ζ32ζ1310+ζ32ζ134 | ζ32ζ1311+ζ32ζ138+ζ32ζ137 | ζ3ζ136+ζ3ζ135+ζ3ζ132 | ζ3ζ1312+ζ3ζ1310+ζ3ζ134 | ζ3ζ1311+ζ3ζ138+ζ3ζ137 | ζ3ζ139+ζ3ζ133+ζ3ζ13 | ζ32ζ139+ζ32ζ133+ζ32ζ13 | ζ32ζ136+ζ32ζ135+ζ32ζ132 | complex faithful |
| ρ16 | 3 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | ζ1311+ζ138+ζ137 | ζ139+ζ133+ζ13 | ζ136+ζ135+ζ132 | ζ1312+ζ1310+ζ134 | ζ32ζ1311+ζ32ζ138+ζ32ζ137 | ζ32ζ139+ζ32ζ133+ζ32ζ13 | ζ3ζ1312+ζ3ζ1310+ζ3ζ134 | ζ3ζ1311+ζ3ζ138+ζ3ζ137 | ζ3ζ139+ζ3ζ133+ζ3ζ13 | ζ3ζ136+ζ3ζ135+ζ3ζ132 | ζ32ζ136+ζ32ζ135+ζ32ζ132 | ζ32ζ1312+ζ32ζ1310+ζ32ζ134 | complex faithful |
| ρ17 | 3 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | ζ1312+ζ1310+ζ134 | ζ1311+ζ138+ζ137 | ζ139+ζ133+ζ13 | ζ136+ζ135+ζ132 | ζ3ζ1312+ζ3ζ1310+ζ3ζ134 | ζ3ζ1311+ζ3ζ138+ζ3ζ137 | ζ32ζ136+ζ32ζ135+ζ32ζ132 | ζ32ζ1312+ζ32ζ1310+ζ32ζ134 | ζ32ζ1311+ζ32ζ138+ζ32ζ137 | ζ32ζ139+ζ32ζ133+ζ32ζ13 | ζ3ζ139+ζ3ζ133+ζ3ζ13 | ζ3ζ136+ζ3ζ135+ζ3ζ132 | complex faithful |
| ρ18 | 3 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | ζ136+ζ135+ζ132 | ζ1312+ζ1310+ζ134 | ζ1311+ζ138+ζ137 | ζ139+ζ133+ζ13 | ζ3ζ136+ζ3ζ135+ζ3ζ132 | ζ3ζ1312+ζ3ζ1310+ζ3ζ134 | ζ32ζ139+ζ32ζ133+ζ32ζ13 | ζ32ζ136+ζ32ζ135+ζ32ζ132 | ζ32ζ1312+ζ32ζ1310+ζ32ζ134 | ζ32ζ1311+ζ32ζ138+ζ32ζ137 | ζ3ζ1311+ζ3ζ138+ζ3ζ137 | ζ3ζ139+ζ3ζ133+ζ3ζ13 | complex faithful |
| ρ19 | 3 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | ζ139+ζ133+ζ13 | ζ136+ζ135+ζ132 | ζ1312+ζ1310+ζ134 | ζ1311+ζ138+ζ137 | ζ3ζ139+ζ3ζ133+ζ3ζ13 | ζ3ζ136+ζ3ζ135+ζ3ζ132 | ζ32ζ1311+ζ32ζ138+ζ32ζ137 | ζ32ζ139+ζ32ζ133+ζ32ζ13 | ζ32ζ136+ζ32ζ135+ζ32ζ132 | ζ32ζ1312+ζ32ζ1310+ζ32ζ134 | ζ3ζ1312+ζ3ζ1310+ζ3ζ134 | ζ3ζ1311+ζ3ζ138+ζ3ζ137 | complex faithful |
| ρ20 | 3 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | ζ1311+ζ138+ζ137 | ζ139+ζ133+ζ13 | ζ136+ζ135+ζ132 | ζ1312+ζ1310+ζ134 | ζ3ζ1311+ζ3ζ138+ζ3ζ137 | ζ3ζ139+ζ3ζ133+ζ3ζ13 | ζ32ζ1312+ζ32ζ1310+ζ32ζ134 | ζ32ζ1311+ζ32ζ138+ζ32ζ137 | ζ32ζ139+ζ32ζ133+ζ32ζ13 | ζ32ζ136+ζ32ζ135+ζ32ζ132 | ζ3ζ136+ζ3ζ135+ζ3ζ132 | ζ3ζ1312+ζ3ζ1310+ζ3ζ134 | complex faithful |
| ρ21 | 3 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | ζ139+ζ133+ζ13 | ζ136+ζ135+ζ132 | ζ1312+ζ1310+ζ134 | ζ1311+ζ138+ζ137 | ζ32ζ139+ζ32ζ133+ζ32ζ13 | ζ32ζ136+ζ32ζ135+ζ32ζ132 | ζ3ζ1311+ζ3ζ138+ζ3ζ137 | ζ3ζ139+ζ3ζ133+ζ3ζ13 | ζ3ζ136+ζ3ζ135+ζ3ζ132 | ζ3ζ1312+ζ3ζ1310+ζ3ζ134 | ζ32ζ1312+ζ32ζ1310+ζ32ζ134 | ζ32ζ1311+ζ32ζ138+ζ32ζ137 | complex faithful |
►On 39 points
Generators in S39
(1 27 14)(2 28 15)(3 29 16)(4 30 17)(5 31 18)(6 32 19)(7 33 20)(8 34 21)(9 35 22)(10 36 23)(11 37 24)(12 38 25)(13 39 26)
(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)| (1,27,14)(2,28,15)(3,29,16)(4,30,17)(5,31,18)(6,32,19)(7,33,20)(8,34,21)(9,35,22)(10,36,23)(11,37,24)(12,38,25)(13,39,26), (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( (1,27,14)(2,28,15)(3,29,16)(4,30,17)(5,31,18)(6,32,19)(7,33,20)(8,34,21)(9,35,22)(10,36,23)(11,37,24)(12,38,25)(13,39,26), (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([[(1,27,14),(2,28,15),(3,29,16),(4,30,17),(5,31,18),(6,32,19),(7,33,20),(8,34,21),(9,35,22),(10,36,23),(11,37,24),(12,38,25),(13,39,26)], [(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)]])
C3×C13⋊C3 is a maximal subgroup of
D39⋊C3 C117⋊C3 C117⋊3C3 C13⋊He3
C3×C13⋊C3 is a maximal quotient of C117⋊C3 C117⋊3C3 C39.C32 C13⋊He3
Matrix representation of C3×C13⋊C3 ►in GL3(𝔽79) generated by
| 23 | 0 | 0 |
| 0 | 23 | 0 |
| 0 | 0 | 23 |
| 69 | 54 | 1 |
| 1 | 0 | 0 |
| 0 | 1 | 0 |
| 1 | 0 | 0 |
| 24 | 68 | 54 |
| 3 | 55 | 10 |
G:=sub<GL(3,GF(79))| [23,0,0,0,23,0,0,0,23],[69,1,0,54,0,1,1,0,0],[1,24,3,0,68,55,0,54,10] >;
C3×C13⋊C3 in GAP, Magma, Sage, TeX
C_3\times C_{13}\rtimes C_3 % in TeX
G:=Group("C3xC13:C3"); // GroupNames label
G:=SmallGroup(117,3);
// by ID
G=gap.SmallGroup(117,3);
# by ID
G:=PCGroup([3,-3,-3,-13,245]);
// Polycyclic
G:=Group<a,b,c|a^3=b^13=c^3=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^9>;
// generators/relations
Export
Subgroup lattice of C3×C13⋊C3 in TeX
Character table of C3×C13⋊C3 in TeX