metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C13⋊C4, D13.C2, SmallGroup(52,3)
Series: Derived ►Chief ►Lower central ►Upper central
| C13 — C13⋊C4 |
Generators and relations for C13⋊C4
G = < a,b | a13=b4=1, bab-1=a5 >
Character table of C13⋊C4
| class | 1 | 2 | 4A | 4B | 13A | 13B | 13C | |
| size | 1 | 13 | 13 | 13 | 4 | 4 | 4 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | linear of order 2 |
| ρ3 | 1 | -1 | -i | i | 1 | 1 | 1 | linear of order 4 |
| ρ4 | 1 | -1 | i | -i | 1 | 1 | 1 | linear of order 4 |
| ρ5 | 4 | 0 | 0 | 0 | ζ139+ζ137+ζ136+ζ134 | ζ1312+ζ138+ζ135+ζ13 | ζ1311+ζ1310+ζ133+ζ132 | orthogonal faithful |
| ρ6 | 4 | 0 | 0 | 0 | ζ1312+ζ138+ζ135+ζ13 | ζ1311+ζ1310+ζ133+ζ132 | ζ139+ζ137+ζ136+ζ134 | orthogonal faithful |
| ρ7 | 4 | 0 | 0 | 0 | ζ1311+ζ1310+ζ133+ζ132 | ζ139+ζ137+ζ136+ζ134 | ζ1312+ζ138+ζ135+ζ13 | orthogonal faithful |
►On 13 points: primitive - transitive group 13T4
Generators in S13
(1 2 3 4 5 6 7 8 9 10 11 12 13)
(2 9 13 6)(3 4 12 11)(5 7 10 8)
G:=sub<Sym(13)| (1,2,3,4,5,6,7,8,9,10,11,12,13), (2,9,13,6)(3,4,12,11)(5,7,10,8)>;
G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13), (2,9,13,6)(3,4,12,11)(5,7,10,8) );
G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10,11,12,13)], [(2,9,13,6),(3,4,12,11),(5,7,10,8)]])
G:=TransitiveGroup(13,4);
►On 26 points - transitive group 26T4
Generators in S26
(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 26)(2 21 13 18)(3 16 12 23)(4 24 11 15)(5 19 10 20)(6 14 9 25)(7 22 8 17)
G:=sub<Sym(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), (1,26)(2,21,13,18)(3,16,12,23)(4,24,11,15)(5,19,10,20)(6,14,9,25)(7,22,8,17)>;
G:=Group( (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), (1,26)(2,21,13,18)(3,16,12,23)(4,24,11,15)(5,19,10,20)(6,14,9,25)(7,22,8,17) );
G=PermutationGroup([[(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)], [(1,26),(2,21,13,18),(3,16,12,23),(4,24,11,15),(5,19,10,20),(6,14,9,25),(7,22,8,17)]])
G:=TransitiveGroup(26,4);
C13⋊C4 is a maximal subgroup of
F13 C39⋊C4 C65⋊C4 C13⋊F5 C65⋊2C4 C91⋊C4 (C3×C39)⋊C4
C13⋊C4 is a maximal quotient of C13⋊C8 C39⋊C4 C65⋊C4 C13⋊F5 C65⋊2C4 C91⋊C4 (C3×C39)⋊C4
| action | f(x) | Disc(f) |
|---|---|---|
| 13T4 | x13-455x11+1300x10+76895x9-393900x8-5556850x7+39456950x6+152496175x5-1553609850x4-557344775x3+21836871550x2-10720383375x-99005292450 | 248·318·524·1321·184612 |
Matrix representation of C13⋊C4 ►in GL4(𝔽5) generated by
| 0 | 4 | 0 | 0 |
| 0 | 4 | 0 | 1 |
| 1 | 4 | 0 | 0 |
| 0 | 1 | 1 | 0 |
| 1 | 2 | 4 | 0 |
| 0 | 1 | 4 | 1 |
| 0 | 3 | 3 | 0 |
| 0 | 0 | 3 | 0 |
G:=sub<GL(4,GF(5))| [0,0,1,0,4,4,4,1,0,0,0,1,0,1,0,0],[1,0,0,0,2,1,3,0,4,4,3,3,0,1,0,0] >;
C13⋊C4 in GAP, Magma, Sage, TeX
C_{13}\rtimes C_4 % in TeX
G:=Group("C13:C4"); // GroupNames label
G:=SmallGroup(52,3);
// by ID
G=gap.SmallGroup(52,3);
# by ID
G:=PCGroup([3,-2,-2,-13,6,290,221]);
// Polycyclic
G:=Group<a,b|a^13=b^4=1,b*a*b^-1=a^5>;
// generators/relations
Export
Subgroup lattice of C13⋊C4 in TeX
Character table of C13⋊C4 in TeX