direct product, metacyclic, supersoluble, monomial, Z-group, 3-hyperelementary
Aliases: C2×C13⋊C3, C26⋊C3, C13⋊2C6, SmallGroup(78,2)
Series: Derived ►Chief ►Lower central ►Upper central
| C13 — C2×C13⋊C3 |
Generators and relations for C2×C13⋊C3
G = < a,b,c | a2=b13=c3=1, ab=ba, ac=ca, cbc-1=b9 >
Character table of C2×C13⋊C3
| class | 1 | 2 | 3A | 3B | 6A | 6B | 13A | 13B | 13C | 13D | 26A | 26B | 26C | 26D | |
| size | 1 | 1 | 13 | 13 | 13 | 13 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | |
| ρ1 | 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 | linear of order 2 |
| ρ3 | 1 | 1 | ζ32 | ζ3 | ζ3 | ζ32 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 3 |
| ρ4 | 1 | -1 | ζ3 | ζ32 | ζ6 | ζ65 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | linear of order 6 |
| ρ5 | 1 | -1 | ζ32 | ζ3 | ζ65 | ζ6 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | linear of order 6 |
| ρ6 | 1 | 1 | ζ3 | ζ32 | ζ32 | ζ3 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 3 |
| ρ7 | 3 | -3 | 0 | 0 | 0 | 0 | ζ136+ζ135+ζ132 | ζ139+ζ133+ζ13 | ζ1312+ζ1310+ζ134 | ζ1311+ζ138+ζ137 | -ζ1312-ζ1310-ζ134 | -ζ1311-ζ138-ζ137 | -ζ139-ζ133-ζ13 | -ζ136-ζ135-ζ132 | complex faithful |
| ρ8 | 3 | -3 | 0 | 0 | 0 | 0 | ζ1312+ζ1310+ζ134 | ζ136+ζ135+ζ132 | ζ1311+ζ138+ζ137 | ζ139+ζ133+ζ13 | -ζ1311-ζ138-ζ137 | -ζ139-ζ133-ζ13 | -ζ136-ζ135-ζ132 | -ζ1312-ζ1310-ζ134 | complex faithful |
| ρ9 | 3 | -3 | 0 | 0 | 0 | 0 | ζ139+ζ133+ζ13 | ζ1311+ζ138+ζ137 | ζ136+ζ135+ζ132 | ζ1312+ζ1310+ζ134 | -ζ136-ζ135-ζ132 | -ζ1312-ζ1310-ζ134 | -ζ1311-ζ138-ζ137 | -ζ139-ζ133-ζ13 | complex faithful |
| ρ10 | 3 | 3 | 0 | 0 | 0 | 0 | ζ139+ζ133+ζ13 | ζ1311+ζ138+ζ137 | ζ136+ζ135+ζ132 | ζ1312+ζ1310+ζ134 | ζ136+ζ135+ζ132 | ζ1312+ζ1310+ζ134 | ζ1311+ζ138+ζ137 | ζ139+ζ133+ζ13 | complex lifted from C13⋊C3 |
| ρ11 | 3 | 3 | 0 | 0 | 0 | 0 | ζ1312+ζ1310+ζ134 | ζ136+ζ135+ζ132 | ζ1311+ζ138+ζ137 | ζ139+ζ133+ζ13 | ζ1311+ζ138+ζ137 | ζ139+ζ133+ζ13 | ζ136+ζ135+ζ132 | ζ1312+ζ1310+ζ134 | complex lifted from C13⋊C3 |
| ρ12 | 3 | 3 | 0 | 0 | 0 | 0 | ζ1311+ζ138+ζ137 | ζ1312+ζ1310+ζ134 | ζ139+ζ133+ζ13 | ζ136+ζ135+ζ132 | ζ139+ζ133+ζ13 | ζ136+ζ135+ζ132 | ζ1312+ζ1310+ζ134 | ζ1311+ζ138+ζ137 | complex lifted from C13⋊C3 |
| ρ13 | 3 | -3 | 0 | 0 | 0 | 0 | ζ1311+ζ138+ζ137 | ζ1312+ζ1310+ζ134 | ζ139+ζ133+ζ13 | ζ136+ζ135+ζ132 | -ζ139-ζ133-ζ13 | -ζ136-ζ135-ζ132 | -ζ1312-ζ1310-ζ134 | -ζ1311-ζ138-ζ137 | complex faithful |
| ρ14 | 3 | 3 | 0 | 0 | 0 | 0 | ζ136+ζ135+ζ132 | ζ139+ζ133+ζ13 | ζ1312+ζ1310+ζ134 | ζ1311+ζ138+ζ137 | ζ1312+ζ1310+ζ134 | ζ1311+ζ138+ζ137 | ζ139+ζ133+ζ13 | ζ136+ζ135+ζ132 | complex lifted from C13⋊C3 |
►On 26 points - transitive group 26T5
Generators in S26
(1 14)(2 15)(3 16)(4 17)(5 18)(6 19)(7 20)(8 21)(9 22)(10 23)(11 24)(12 25)(13 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)
(2 4 10)(3 7 6)(5 13 11)(8 9 12)(15 17 23)(16 20 19)(18 26 24)(21 22 25)
G:=sub<Sym(26)| (1,14)(2,15)(3,16)(4,17)(5,18)(6,19)(7,20)(8,21)(9,22)(10,23)(11,24)(12,25)(13,26), (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), (2,4,10)(3,7,6)(5,13,11)(8,9,12)(15,17,23)(16,20,19)(18,26,24)(21,22,25)>;
G:=Group( (1,14)(2,15)(3,16)(4,17)(5,18)(6,19)(7,20)(8,21)(9,22)(10,23)(11,24)(12,25)(13,26), (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), (2,4,10)(3,7,6)(5,13,11)(8,9,12)(15,17,23)(16,20,19)(18,26,24)(21,22,25) );
G=PermutationGroup([[(1,14),(2,15),(3,16),(4,17),(5,18),(6,19),(7,20),(8,21),(9,22),(10,23),(11,24),(12,25),(13,26)], [(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)], [(2,4,10),(3,7,6),(5,13,11),(8,9,12),(15,17,23),(16,20,19),(18,26,24),(21,22,25)]])
G:=TransitiveGroup(26,5);
C2×C13⋊C3 is a maximal subgroup of
C26.C6 C26.A4
Matrix representation of C2×C13⋊C3 ►in GL3(𝔽3) generated by
| 2 | 0 | 0 |
| 0 | 2 | 0 |
| 0 | 0 | 2 |
| 0 | 1 | 2 |
| 0 | 2 | 2 |
| 1 | 2 | 0 |
| 1 | 1 | 2 |
| 0 | 2 | 1 |
| 0 | 2 | 0 |
G:=sub<GL(3,GF(3))| [2,0,0,0,2,0,0,0,2],[0,0,1,1,2,2,2,2,0],[1,0,0,1,2,2,2,1,0] >;
C2×C13⋊C3 in GAP, Magma, Sage, TeX
C_2\times C_{13}\rtimes C_3 % in TeX
G:=Group("C2xC13:C3"); // GroupNames label
G:=SmallGroup(78,2);
// by ID
G=gap.SmallGroup(78,2);
# by ID
G:=PCGroup([3,-2,-3,-13,86]);
// Polycyclic
G:=Group<a,b,c|a^2=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 C2×C13⋊C3 in TeX
Character table of C2×C13⋊C3 in TeX