direct product, metacyclic, supersoluble, monomial, A-group
Aliases: C3×C13⋊C6, C39⋊2C6, D13⋊C32, C13⋊C3⋊C6, C13⋊(C3×C6), (C3×D13)⋊C3, (C3×C13⋊C3)⋊2C2, SmallGroup(234,7)
Series: Derived ►Chief ►Lower central ►Upper central
| C13 — C3×C13⋊C6 |
Generators and relations for C3×C13⋊C6
G = < a,b,c | a3=b13=c6=1, ab=ba, ac=ca, cbc-1=b10 >
Character table of C3×C13⋊C6
| class | 1 | 2 | 3A | 3B | 3C | 3D | 3E | 3F | 3G | 3H | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 6H | 13A | 13B | 39A | 39B | 39C | 39D | |
| size | 1 | 13 | 1 | 1 | 13 | 13 | 13 | 13 | 13 | 13 | 13 | 13 | 13 | 13 | 13 | 13 | 13 | 13 | 6 | 6 | 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 | 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 | ζ3 | ζ32 | ζ3 | 1 | ζ32 | 1 | ζ32 | ζ3 | ζ6 | ζ6 | ζ65 | ζ6 | ζ65 | -1 | ζ65 | -1 | 1 | 1 | ζ3 | ζ32 | ζ3 | ζ32 | linear of order 6 |
| ρ4 | 1 | -1 | ζ32 | ζ3 | ζ3 | ζ3 | ζ32 | ζ32 | 1 | 1 | -1 | ζ65 | ζ6 | ζ6 | -1 | ζ65 | ζ65 | ζ6 | 1 | 1 | ζ32 | ζ3 | ζ32 | ζ3 | linear of order 6 |
| ρ5 | 1 | -1 | 1 | 1 | ζ3 | ζ32 | ζ32 | ζ3 | ζ3 | ζ32 | ζ65 | -1 | -1 | ζ6 | ζ6 | ζ6 | ζ65 | ζ65 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 6 |
| ρ6 | 1 | 1 | ζ32 | ζ3 | ζ32 | 1 | ζ3 | 1 | ζ3 | ζ32 | ζ3 | ζ3 | ζ32 | ζ3 | ζ32 | 1 | ζ32 | 1 | 1 | 1 | ζ32 | ζ3 | ζ32 | ζ3 | linear of order 3 |
| ρ7 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | ζ32 | ζ3 | ζ3 | ζ32 | ζ3 | 1 | 1 | ζ32 | ζ32 | ζ32 | ζ3 | ζ3 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 3 |
| ρ8 | 1 | 1 | ζ32 | ζ3 | 1 | ζ32 | 1 | ζ3 | ζ32 | ζ3 | ζ32 | ζ3 | ζ32 | 1 | ζ3 | ζ32 | 1 | ζ3 | 1 | 1 | ζ32 | ζ3 | ζ32 | ζ3 | linear of order 3 |
| ρ9 | 1 | -1 | ζ3 | ζ32 | 1 | ζ3 | 1 | ζ32 | ζ3 | ζ32 | ζ65 | ζ6 | ζ65 | -1 | ζ6 | ζ65 | -1 | ζ6 | 1 | 1 | ζ3 | ζ32 | ζ3 | ζ32 | linear of order 6 |
| ρ10 | 1 | -1 | ζ3 | ζ32 | ζ32 | ζ32 | ζ3 | ζ3 | 1 | 1 | -1 | ζ6 | ζ65 | ζ65 | -1 | ζ6 | ζ6 | ζ65 | 1 | 1 | ζ3 | ζ32 | ζ3 | ζ32 | linear of order 6 |
| ρ11 | 1 | -1 | 1 | 1 | ζ32 | ζ3 | ζ3 | ζ32 | ζ32 | ζ3 | ζ6 | -1 | -1 | ζ65 | ζ65 | ζ65 | ζ6 | ζ6 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 6 |
| ρ12 | 1 | -1 | ζ32 | ζ3 | 1 | ζ32 | 1 | ζ3 | ζ32 | ζ3 | ζ6 | ζ65 | ζ6 | -1 | ζ65 | ζ6 | -1 | ζ65 | 1 | 1 | ζ32 | ζ3 | ζ32 | ζ3 | linear of order 6 |
| ρ13 | 1 | 1 | ζ3 | ζ32 | ζ32 | ζ32 | ζ3 | ζ3 | 1 | 1 | 1 | ζ32 | ζ3 | ζ3 | 1 | ζ32 | ζ32 | ζ3 | 1 | 1 | ζ3 | ζ32 | ζ3 | ζ32 | linear of order 3 |
| ρ14 | 1 | -1 | ζ32 | ζ3 | ζ32 | 1 | ζ3 | 1 | ζ3 | ζ32 | ζ65 | ζ65 | ζ6 | ζ65 | ζ6 | -1 | ζ6 | -1 | 1 | 1 | ζ32 | ζ3 | ζ32 | ζ3 | linear of order 6 |
| ρ15 | 1 | 1 | ζ32 | ζ3 | ζ3 | ζ3 | ζ32 | ζ32 | 1 | 1 | 1 | ζ3 | ζ32 | ζ32 | 1 | ζ3 | ζ3 | ζ32 | 1 | 1 | ζ32 | ζ3 | ζ32 | ζ3 | linear of order 3 |
| ρ16 | 1 | 1 | ζ3 | ζ32 | 1 | ζ3 | 1 | ζ32 | ζ3 | ζ32 | ζ3 | ζ32 | ζ3 | 1 | ζ32 | ζ3 | 1 | ζ32 | 1 | 1 | ζ3 | ζ32 | ζ3 | ζ32 | linear of order 3 |
| ρ17 | 1 | 1 | ζ3 | ζ32 | ζ3 | 1 | ζ32 | 1 | ζ32 | ζ3 | ζ32 | ζ32 | ζ3 | ζ32 | ζ3 | 1 | ζ3 | 1 | 1 | 1 | ζ3 | ζ32 | ζ3 | ζ32 | linear of order 3 |
| ρ18 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | ζ3 | ζ32 | ζ32 | ζ3 | ζ32 | 1 | 1 | ζ3 | ζ3 | ζ3 | ζ32 | ζ32 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 3 |
| ρ19 | 6 | 0 | 6 | 6 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1+√13/2 | -1-√13/2 | -1-√13/2 | -1+√13/2 | -1+√13/2 | -1-√13/2 | orthogonal lifted from C13⋊C6 |
| ρ20 | 6 | 0 | 6 | 6 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1-√13/2 | -1+√13/2 | -1+√13/2 | -1-√13/2 | -1-√13/2 | -1+√13/2 | orthogonal lifted from C13⋊C6 |
| ρ21 | 6 | 0 | -3-3√-3 | -3+3√-3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1+√13/2 | -1-√13/2 | ζ32ζ1311+ζ32ζ138+ζ32ζ137+ζ32ζ136+ζ32ζ135+ζ32ζ132 | ζ3ζ1312+ζ3ζ1310+ζ3ζ139+ζ3ζ134+ζ3ζ133+ζ3ζ13 | ζ32ζ1312+ζ32ζ1310+ζ32ζ139+ζ32ζ134+ζ32ζ133+ζ32ζ13 | ζ3ζ1311+ζ3ζ138+ζ3ζ137+ζ3ζ136+ζ3ζ135+ζ3ζ132 | complex faithful |
| ρ22 | 6 | 0 | -3+3√-3 | -3-3√-3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1-√13/2 | -1+√13/2 | ζ3ζ1312+ζ3ζ1310+ζ3ζ139+ζ3ζ134+ζ3ζ133+ζ3ζ13 | ζ32ζ1311+ζ32ζ138+ζ32ζ137+ζ32ζ136+ζ32ζ135+ζ32ζ132 | ζ3ζ1311+ζ3ζ138+ζ3ζ137+ζ3ζ136+ζ3ζ135+ζ3ζ132 | ζ32ζ1312+ζ32ζ1310+ζ32ζ139+ζ32ζ134+ζ32ζ133+ζ32ζ13 | complex faithful |
| ρ23 | 6 | 0 | -3+3√-3 | -3-3√-3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1+√13/2 | -1-√13/2 | ζ3ζ1311+ζ3ζ138+ζ3ζ137+ζ3ζ136+ζ3ζ135+ζ3ζ132 | ζ32ζ1312+ζ32ζ1310+ζ32ζ139+ζ32ζ134+ζ32ζ133+ζ32ζ13 | ζ3ζ1312+ζ3ζ1310+ζ3ζ139+ζ3ζ134+ζ3ζ133+ζ3ζ13 | ζ32ζ1311+ζ32ζ138+ζ32ζ137+ζ32ζ136+ζ32ζ135+ζ32ζ132 | complex faithful |
| ρ24 | 6 | 0 | -3-3√-3 | -3+3√-3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1-√13/2 | -1+√13/2 | ζ32ζ1312+ζ32ζ1310+ζ32ζ139+ζ32ζ134+ζ32ζ133+ζ32ζ13 | ζ3ζ1311+ζ3ζ138+ζ3ζ137+ζ3ζ136+ζ3ζ135+ζ3ζ132 | ζ32ζ1311+ζ32ζ138+ζ32ζ137+ζ32ζ136+ζ32ζ135+ζ32ζ132 | ζ3ζ1312+ζ3ζ1310+ζ3ζ139+ζ3ζ134+ζ3ζ133+ζ3ζ13 | 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)
(1 27 14)(2 31 17 13 36 24)(3 35 20 12 32 21)(4 39 23 11 28 18)(5 30 26 10 37 15)(6 34 16 9 33 25)(7 38 19 8 29 22)
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), (1,27,14)(2,31,17,13,36,24)(3,35,20,12,32,21)(4,39,23,11,28,18)(5,30,26,10,37,15)(6,34,16,9,33,25)(7,38,19,8,29,22)>;
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), (1,27,14)(2,31,17,13,36,24)(3,35,20,12,32,21)(4,39,23,11,28,18)(5,30,26,10,37,15)(6,34,16,9,33,25)(7,38,19,8,29,22) );
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)], [(1,27,14),(2,31,17,13,36,24),(3,35,20,12,32,21),(4,39,23,11,28,18),(5,30,26,10,37,15),(6,34,16,9,33,25),(7,38,19,8,29,22)]])
C3×C13⋊C6 is a maximal subgroup of
C3⋊F13
Matrix representation of C3×C13⋊C6 ►in GL6(𝔽79)
| 55 | 0 | 0 | 0 | 0 | 0 |
| 0 | 55 | 0 | 0 | 0 | 0 |
| 0 | 0 | 55 | 0 | 0 | 0 |
| 0 | 0 | 0 | 55 | 0 | 0 |
| 0 | 0 | 0 | 0 | 55 | 0 |
| 0 | 0 | 0 | 0 | 0 | 55 |
| 0 | 0 | 0 | 0 | 0 | 78 |
| 1 | 0 | 0 | 0 | 0 | 63 |
| 0 | 1 | 0 | 0 | 0 | 77 |
| 0 | 0 | 1 | 0 | 0 | 64 |
| 0 | 0 | 0 | 1 | 0 | 77 |
| 0 | 0 | 0 | 0 | 1 | 63 |
| 13 | 68 | 0 | 11 | 66 | 0 |
| 61 | 48 | 0 | 42 | 66 | 0 |
| 68 | 44 | 0 | 11 | 55 | 55 |
| 48 | 48 | 55 | 55 | 11 | 0 |
| 2 | 68 | 0 | 66 | 42 | 0 |
| 37 | 24 | 0 | 66 | 11 | 0 |
G:=sub<GL(6,GF(79))| [55,0,0,0,0,0,0,55,0,0,0,0,0,0,55,0,0,0,0,0,0,55,0,0,0,0,0,0,55,0,0,0,0,0,0,55],[0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,78,63,77,64,77,63],[13,61,68,48,2,37,68,48,44,48,68,24,0,0,0,55,0,0,11,42,11,55,66,66,66,66,55,11,42,11,0,0,55,0,0,0] >;
C3×C13⋊C6 in GAP, Magma, Sage, TeX
C_3\times C_{13}\rtimes C_6 % in TeX
G:=Group("C3xC13:C6"); // GroupNames label
G:=SmallGroup(234,7);
// by ID
G=gap.SmallGroup(234,7);
# by ID
G:=PCGroup([4,-2,-3,-3,-13,3459,439]);
// Polycyclic
G:=Group<a,b,c|a^3=b^13=c^6=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^10>;
// generators/relations
Export
Subgroup lattice of C3×C13⋊C6 in TeX
Character table of C3×C13⋊C6 in TeX