direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C2×C13⋊C4, C26⋊C4, D13⋊C4, D26.C2, D13.C22, C13⋊(C2×C4), SmallGroup(104,12)
Series: Derived ►Chief ►Lower central ►Upper central
| C13 — C2×C13⋊C4 |
Generators and relations for C2×C13⋊C4
G = < a,b,c | a2=b13=c4=1, ab=ba, ac=ca, cbc-1=b5 >
Character table of C2×C13⋊C4
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 13A | 13B | 13C | 26A | 26B | 26C | |
| size | 1 | 1 | 13 | 13 | 13 | 13 | 13 | 13 | 4 | 4 | 4 | 4 | 4 | 4 | |
| 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 | 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 | linear of order 2 |
| ρ5 | 1 | 1 | -1 | -1 | -i | -i | i | i | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 4 |
| ρ6 | 1 | 1 | -1 | -1 | i | i | -i | -i | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 4 |
| ρ7 | 1 | -1 | -1 | 1 | i | -i | -i | i | 1 | 1 | 1 | -1 | -1 | -1 | linear of order 4 |
| ρ8 | 1 | -1 | -1 | 1 | -i | i | i | -i | 1 | 1 | 1 | -1 | -1 | -1 | linear of order 4 |
| ρ9 | 4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | ζ1312+ζ138+ζ135+ζ13 | ζ1311+ζ1310+ζ133+ζ132 | ζ139+ζ137+ζ136+ζ134 | ζ1311+ζ1310+ζ133+ζ132 | ζ139+ζ137+ζ136+ζ134 | ζ1312+ζ138+ζ135+ζ13 | orthogonal lifted from C13⋊C4 |
| ρ10 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | ζ1311+ζ1310+ζ133+ζ132 | ζ139+ζ137+ζ136+ζ134 | ζ1312+ζ138+ζ135+ζ13 | -ζ139-ζ137-ζ136-ζ134 | -ζ1312-ζ138-ζ135-ζ13 | -ζ1311-ζ1310-ζ133-ζ132 | orthogonal faithful |
| ρ11 | 4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | ζ1311+ζ1310+ζ133+ζ132 | ζ139+ζ137+ζ136+ζ134 | ζ1312+ζ138+ζ135+ζ13 | ζ139+ζ137+ζ136+ζ134 | ζ1312+ζ138+ζ135+ζ13 | ζ1311+ζ1310+ζ133+ζ132 | orthogonal lifted from C13⋊C4 |
| ρ12 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | ζ1312+ζ138+ζ135+ζ13 | ζ1311+ζ1310+ζ133+ζ132 | ζ139+ζ137+ζ136+ζ134 | -ζ1311-ζ1310-ζ133-ζ132 | -ζ139-ζ137-ζ136-ζ134 | -ζ1312-ζ138-ζ135-ζ13 | orthogonal faithful |
| ρ13 | 4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | ζ139+ζ137+ζ136+ζ134 | ζ1312+ζ138+ζ135+ζ13 | ζ1311+ζ1310+ζ133+ζ132 | ζ1312+ζ138+ζ135+ζ13 | ζ1311+ζ1310+ζ133+ζ132 | ζ139+ζ137+ζ136+ζ134 | orthogonal lifted from C13⋊C4 |
| ρ14 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | ζ139+ζ137+ζ136+ζ134 | ζ1312+ζ138+ζ135+ζ13 | ζ1311+ζ1310+ζ133+ζ132 | -ζ1312-ζ138-ζ135-ζ13 | -ζ1311-ζ1310-ζ133-ζ132 | -ζ139-ζ137-ζ136-ζ134 | orthogonal faithful |
►On 26 points - transitive group 26T7
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)
(1 14)(2 22 13 19)(3 17 12 24)(4 25 11 16)(5 20 10 21)(6 15 9 26)(7 23 8 18)
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), (1,14)(2,22,13,19)(3,17,12,24)(4,25,11,16)(5,20,10,21)(6,15,9,26)(7,23,8,18)>;
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), (1,14)(2,22,13,19)(3,17,12,24)(4,25,11,16)(5,20,10,21)(6,15,9,26)(7,23,8,18) );
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)], [(1,14),(2,22,13,19),(3,17,12,24),(4,25,11,16),(5,20,10,21),(6,15,9,26),(7,23,8,18)]])
G:=TransitiveGroup(26,7);
C2×C13⋊C4 is a maximal subgroup of
C52⋊C4 D13.D4
C2×C13⋊C4 is a maximal quotient of D13⋊C8 C52.C4 C52⋊C4 C13⋊M4(2) D13.D4
Matrix representation of C2×C13⋊C4 ►in GL4(𝔽5) generated by
| 4 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 4 |
| 3 | 0 | 1 | 0 |
| 1 | 0 | 0 | 0 |
| 4 | 0 | 0 | 4 |
| 2 | 1 | 0 | 0 |
| 1 | 2 | 3 | 0 |
| 0 | 2 | 2 | 1 |
| 0 | 0 | 2 | 0 |
| 0 | 3 | 3 | 0 |
G:=sub<GL(4,GF(5))| [4,0,0,0,0,4,0,0,0,0,4,0,0,0,0,4],[3,1,4,2,0,0,0,1,1,0,0,0,0,0,4,0],[1,0,0,0,2,2,0,3,3,2,2,3,0,1,0,0] >;
C2×C13⋊C4 in GAP, Magma, Sage, TeX
C_2\times C_{13}\rtimes C_4 % in TeX
G:=Group("C2xC13:C4"); // GroupNames label
G:=SmallGroup(104,12);
// by ID
G=gap.SmallGroup(104,12);
# by ID
G:=PCGroup([4,-2,-2,-2,-13,16,1027,395]);
// Polycyclic
G:=Group<a,b,c|a^2=b^13=c^4=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^5>;
// generators/relations
Export
Subgroup lattice of C2×C13⋊C4 in TeX
Character table of C2×C13⋊C4 in TeX