metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C65⋊3C4, C13⋊Dic5, D13.D5, C5⋊3(C13⋊C4), (C5×D13).1C2, SmallGroup(260,6)
Series: Derived ►Chief ►Lower central ►Upper central
| C65 — C65⋊C4 |
Generators and relations for C65⋊C4
G = < a,b | a65=b4=1, bab-1=a34 >
Character table of C65⋊C4
| class | 1 | 2 | 4A | 4B | 5A | 5B | 10A | 10B | 13A | 13B | 13C | 65A | 65B | 65C | 65D | 65E | 65F | 65G | 65H | 65I | 65J | 65K | 65L | |
| size | 1 | 13 | 65 | 65 | 2 | 2 | 26 | 26 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | |
| ρ1 | 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 | linear of order 2 |
| ρ3 | 1 | -1 | -i | i | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 4 |
| ρ4 | 1 | -1 | i | -i | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 4 |
| ρ5 | 2 | 2 | 0 | 0 | -1+√5/2 | -1-√5/2 | -1+√5/2 | -1-√5/2 | 2 | 2 | 2 | -1+√5/2 | -1-√5/2 | -1-√5/2 | -1-√5/2 | -1-√5/2 | -1-√5/2 | -1-√5/2 | -1+√5/2 | -1+√5/2 | -1+√5/2 | -1+√5/2 | -1+√5/2 | orthogonal lifted from D5 |
| ρ6 | 2 | 2 | 0 | 0 | -1-√5/2 | -1+√5/2 | -1-√5/2 | -1+√5/2 | 2 | 2 | 2 | -1-√5/2 | -1+√5/2 | -1+√5/2 | -1+√5/2 | -1+√5/2 | -1+√5/2 | -1+√5/2 | -1-√5/2 | -1-√5/2 | -1-√5/2 | -1-√5/2 | -1-√5/2 | orthogonal lifted from D5 |
| ρ7 | 2 | -2 | 0 | 0 | -1+√5/2 | -1-√5/2 | 1-√5/2 | 1+√5/2 | 2 | 2 | 2 | -1+√5/2 | -1-√5/2 | -1-√5/2 | -1-√5/2 | -1-√5/2 | -1-√5/2 | -1-√5/2 | -1+√5/2 | -1+√5/2 | -1+√5/2 | -1+√5/2 | -1+√5/2 | symplectic lifted from Dic5, Schur index 2 |
| ρ8 | 2 | -2 | 0 | 0 | -1-√5/2 | -1+√5/2 | 1+√5/2 | 1-√5/2 | 2 | 2 | 2 | -1-√5/2 | -1+√5/2 | -1+√5/2 | -1+√5/2 | -1+√5/2 | -1+√5/2 | -1+√5/2 | -1-√5/2 | -1-√5/2 | -1-√5/2 | -1-√5/2 | -1-√5/2 | symplectic lifted from Dic5, Schur index 2 |
| ρ9 | 4 | 0 | 0 | 0 | 4 | 4 | 0 | 0 | ζ1312+ζ138+ζ135+ζ13 | ζ139+ζ137+ζ136+ζ134 | ζ1311+ζ1310+ζ133+ζ132 | ζ1311+ζ1310+ζ133+ζ132 | ζ1311+ζ1310+ζ133+ζ132 | ζ139+ζ137+ζ136+ζ134 | ζ139+ζ137+ζ136+ζ134 | ζ1312+ζ138+ζ135+ζ13 | ζ1311+ζ1310+ζ133+ζ132 | ζ1312+ζ138+ζ135+ζ13 | ζ139+ζ137+ζ136+ζ134 | ζ139+ζ137+ζ136+ζ134 | ζ1312+ζ138+ζ135+ζ13 | ζ1311+ζ1310+ζ133+ζ132 | ζ1312+ζ138+ζ135+ζ13 | orthogonal lifted from C13⋊C4 |
| ρ10 | 4 | 0 | 0 | 0 | 4 | 4 | 0 | 0 | ζ139+ζ137+ζ136+ζ134 | ζ1311+ζ1310+ζ133+ζ132 | ζ1312+ζ138+ζ135+ζ13 | ζ1312+ζ138+ζ135+ζ13 | ζ1312+ζ138+ζ135+ζ13 | ζ1311+ζ1310+ζ133+ζ132 | ζ1311+ζ1310+ζ133+ζ132 | ζ139+ζ137+ζ136+ζ134 | ζ1312+ζ138+ζ135+ζ13 | ζ139+ζ137+ζ136+ζ134 | ζ1311+ζ1310+ζ133+ζ132 | ζ1311+ζ1310+ζ133+ζ132 | ζ139+ζ137+ζ136+ζ134 | ζ1312+ζ138+ζ135+ζ13 | ζ139+ζ137+ζ136+ζ134 | orthogonal lifted from C13⋊C4 |
| ρ11 | 4 | 0 | 0 | 0 | 4 | 4 | 0 | 0 | ζ1311+ζ1310+ζ133+ζ132 | ζ1312+ζ138+ζ135+ζ13 | ζ139+ζ137+ζ136+ζ134 | ζ139+ζ137+ζ136+ζ134 | ζ139+ζ137+ζ136+ζ134 | ζ1312+ζ138+ζ135+ζ13 | ζ1312+ζ138+ζ135+ζ13 | ζ1311+ζ1310+ζ133+ζ132 | ζ139+ζ137+ζ136+ζ134 | ζ1311+ζ1310+ζ133+ζ132 | ζ1312+ζ138+ζ135+ζ13 | ζ1312+ζ138+ζ135+ζ13 | ζ1311+ζ1310+ζ133+ζ132 | ζ139+ζ137+ζ136+ζ134 | ζ1311+ζ1310+ζ133+ζ132 | orthogonal lifted from C13⋊C4 |
| ρ12 | 4 | 0 | 0 | 0 | -1-√5 | -1+√5 | 0 | 0 | ζ1312+ζ138+ζ135+ζ13 | ζ139+ζ137+ζ136+ζ134 | ζ1311+ζ1310+ζ133+ζ132 | ζ53ζ1311+ζ53ζ132+ζ52ζ1310+ζ52ζ133 | ζ54ζ1310+ζ54ζ133+ζ5ζ1311+ζ5ζ132 | ζ54ζ137+ζ54ζ136+ζ5ζ139+ζ5ζ134 | ζ54ζ139+ζ54ζ134+ζ5ζ137+ζ5ζ136 | ζ54ζ1312+ζ54ζ13+ζ5ζ138+ζ5ζ135 | ζ54ζ1311+ζ54ζ132+ζ5ζ1310+ζ5ζ133 | ζ54ζ138+ζ54ζ135+ζ5ζ1312+ζ5ζ13 | ζ53ζ139+ζ53ζ134+ζ52ζ137+ζ52ζ136 | ζ53ζ137+ζ53ζ136+ζ52ζ139+ζ52ζ134 | ζ53ζ138+ζ53ζ135+ζ52ζ1312+ζ52ζ13 | ζ53ζ1310+ζ53ζ133+ζ52ζ1311+ζ52ζ132 | ζ53ζ1312+ζ53ζ13+ζ52ζ138+ζ52ζ135 | complex faithful |
| ρ13 | 4 | 0 | 0 | 0 | -1-√5 | -1+√5 | 0 | 0 | ζ1312+ζ138+ζ135+ζ13 | ζ139+ζ137+ζ136+ζ134 | ζ1311+ζ1310+ζ133+ζ132 | ζ53ζ1310+ζ53ζ133+ζ52ζ1311+ζ52ζ132 | ζ54ζ1311+ζ54ζ132+ζ5ζ1310+ζ5ζ133 | ζ54ζ139+ζ54ζ134+ζ5ζ137+ζ5ζ136 | ζ54ζ137+ζ54ζ136+ζ5ζ139+ζ5ζ134 | ζ54ζ138+ζ54ζ135+ζ5ζ1312+ζ5ζ13 | ζ54ζ1310+ζ54ζ133+ζ5ζ1311+ζ5ζ132 | ζ54ζ1312+ζ54ζ13+ζ5ζ138+ζ5ζ135 | ζ53ζ137+ζ53ζ136+ζ52ζ139+ζ52ζ134 | ζ53ζ139+ζ53ζ134+ζ52ζ137+ζ52ζ136 | ζ53ζ1312+ζ53ζ13+ζ52ζ138+ζ52ζ135 | ζ53ζ1311+ζ53ζ132+ζ52ζ1310+ζ52ζ133 | ζ53ζ138+ζ53ζ135+ζ52ζ1312+ζ52ζ13 | complex faithful |
| ρ14 | 4 | 0 | 0 | 0 | -1+√5 | -1-√5 | 0 | 0 | ζ139+ζ137+ζ136+ζ134 | ζ1311+ζ1310+ζ133+ζ132 | ζ1312+ζ138+ζ135+ζ13 | ζ54ζ1312+ζ54ζ13+ζ5ζ138+ζ5ζ135 | ζ53ζ1312+ζ53ζ13+ζ52ζ138+ζ52ζ135 | ζ53ζ1311+ζ53ζ132+ζ52ζ1310+ζ52ζ133 | ζ53ζ1310+ζ53ζ133+ζ52ζ1311+ζ52ζ132 | ζ53ζ139+ζ53ζ134+ζ52ζ137+ζ52ζ136 | ζ53ζ138+ζ53ζ135+ζ52ζ1312+ζ52ζ13 | ζ53ζ137+ζ53ζ136+ζ52ζ139+ζ52ζ134 | ζ54ζ1311+ζ54ζ132+ζ5ζ1310+ζ5ζ133 | ζ54ζ1310+ζ54ζ133+ζ5ζ1311+ζ5ζ132 | ζ54ζ139+ζ54ζ134+ζ5ζ137+ζ5ζ136 | ζ54ζ138+ζ54ζ135+ζ5ζ1312+ζ5ζ13 | ζ54ζ137+ζ54ζ136+ζ5ζ139+ζ5ζ134 | complex faithful |
| ρ15 | 4 | 0 | 0 | 0 | -1+√5 | -1-√5 | 0 | 0 | ζ1311+ζ1310+ζ133+ζ132 | ζ1312+ζ138+ζ135+ζ13 | ζ139+ζ137+ζ136+ζ134 | ζ54ζ139+ζ54ζ134+ζ5ζ137+ζ5ζ136 | ζ53ζ139+ζ53ζ134+ζ52ζ137+ζ52ζ136 | ζ53ζ138+ζ53ζ135+ζ52ζ1312+ζ52ζ13 | ζ53ζ1312+ζ53ζ13+ζ52ζ138+ζ52ζ135 | ζ53ζ1310+ζ53ζ133+ζ52ζ1311+ζ52ζ132 | ζ53ζ137+ζ53ζ136+ζ52ζ139+ζ52ζ134 | ζ53ζ1311+ζ53ζ132+ζ52ζ1310+ζ52ζ133 | ζ54ζ138+ζ54ζ135+ζ5ζ1312+ζ5ζ13 | ζ54ζ1312+ζ54ζ13+ζ5ζ138+ζ5ζ135 | ζ54ζ1310+ζ54ζ133+ζ5ζ1311+ζ5ζ132 | ζ54ζ137+ζ54ζ136+ζ5ζ139+ζ5ζ134 | ζ54ζ1311+ζ54ζ132+ζ5ζ1310+ζ5ζ133 | complex faithful |
| ρ16 | 4 | 0 | 0 | 0 | -1-√5 | -1+√5 | 0 | 0 | ζ1311+ζ1310+ζ133+ζ132 | ζ1312+ζ138+ζ135+ζ13 | ζ139+ζ137+ζ136+ζ134 | ζ53ζ139+ζ53ζ134+ζ52ζ137+ζ52ζ136 | ζ54ζ137+ζ54ζ136+ζ5ζ139+ζ5ζ134 | ζ54ζ1312+ζ54ζ13+ζ5ζ138+ζ5ζ135 | ζ54ζ138+ζ54ζ135+ζ5ζ1312+ζ5ζ13 | ζ54ζ1311+ζ54ζ132+ζ5ζ1310+ζ5ζ133 | ζ54ζ139+ζ54ζ134+ζ5ζ137+ζ5ζ136 | ζ54ζ1310+ζ54ζ133+ζ5ζ1311+ζ5ζ132 | ζ53ζ138+ζ53ζ135+ζ52ζ1312+ζ52ζ13 | ζ53ζ1312+ζ53ζ13+ζ52ζ138+ζ52ζ135 | ζ53ζ1310+ζ53ζ133+ζ52ζ1311+ζ52ζ132 | ζ53ζ137+ζ53ζ136+ζ52ζ139+ζ52ζ134 | ζ53ζ1311+ζ53ζ132+ζ52ζ1310+ζ52ζ133 | complex faithful |
| ρ17 | 4 | 0 | 0 | 0 | -1-√5 | -1+√5 | 0 | 0 | ζ139+ζ137+ζ136+ζ134 | ζ1311+ζ1310+ζ133+ζ132 | ζ1312+ζ138+ζ135+ζ13 | ζ53ζ1312+ζ53ζ13+ζ52ζ138+ζ52ζ135 | ζ54ζ138+ζ54ζ135+ζ5ζ1312+ζ5ζ13 | ζ54ζ1310+ζ54ζ133+ζ5ζ1311+ζ5ζ132 | ζ54ζ1311+ζ54ζ132+ζ5ζ1310+ζ5ζ133 | ζ54ζ137+ζ54ζ136+ζ5ζ139+ζ5ζ134 | ζ54ζ1312+ζ54ζ13+ζ5ζ138+ζ5ζ135 | ζ54ζ139+ζ54ζ134+ζ5ζ137+ζ5ζ136 | ζ53ζ1311+ζ53ζ132+ζ52ζ1310+ζ52ζ133 | ζ53ζ1310+ζ53ζ133+ζ52ζ1311+ζ52ζ132 | ζ53ζ139+ζ53ζ134+ζ52ζ137+ζ52ζ136 | ζ53ζ138+ζ53ζ135+ζ52ζ1312+ζ52ζ13 | ζ53ζ137+ζ53ζ136+ζ52ζ139+ζ52ζ134 | complex faithful |
| ρ18 | 4 | 0 | 0 | 0 | -1+√5 | -1-√5 | 0 | 0 | ζ139+ζ137+ζ136+ζ134 | ζ1311+ζ1310+ζ133+ζ132 | ζ1312+ζ138+ζ135+ζ13 | ζ54ζ138+ζ54ζ135+ζ5ζ1312+ζ5ζ13 | ζ53ζ138+ζ53ζ135+ζ52ζ1312+ζ52ζ13 | ζ53ζ1310+ζ53ζ133+ζ52ζ1311+ζ52ζ132 | ζ53ζ1311+ζ53ζ132+ζ52ζ1310+ζ52ζ133 | ζ53ζ137+ζ53ζ136+ζ52ζ139+ζ52ζ134 | ζ53ζ1312+ζ53ζ13+ζ52ζ138+ζ52ζ135 | ζ53ζ139+ζ53ζ134+ζ52ζ137+ζ52ζ136 | ζ54ζ1310+ζ54ζ133+ζ5ζ1311+ζ5ζ132 | ζ54ζ1311+ζ54ζ132+ζ5ζ1310+ζ5ζ133 | ζ54ζ137+ζ54ζ136+ζ5ζ139+ζ5ζ134 | ζ54ζ1312+ζ54ζ13+ζ5ζ138+ζ5ζ135 | ζ54ζ139+ζ54ζ134+ζ5ζ137+ζ5ζ136 | complex faithful |
| ρ19 | 4 | 0 | 0 | 0 | -1+√5 | -1-√5 | 0 | 0 | ζ1312+ζ138+ζ135+ζ13 | ζ139+ζ137+ζ136+ζ134 | ζ1311+ζ1310+ζ133+ζ132 | ζ54ζ1311+ζ54ζ132+ζ5ζ1310+ζ5ζ133 | ζ53ζ1311+ζ53ζ132+ζ52ζ1310+ζ52ζ133 | ζ53ζ139+ζ53ζ134+ζ52ζ137+ζ52ζ136 | ζ53ζ137+ζ53ζ136+ζ52ζ139+ζ52ζ134 | ζ53ζ138+ζ53ζ135+ζ52ζ1312+ζ52ζ13 | ζ53ζ1310+ζ53ζ133+ζ52ζ1311+ζ52ζ132 | ζ53ζ1312+ζ53ζ13+ζ52ζ138+ζ52ζ135 | ζ54ζ139+ζ54ζ134+ζ5ζ137+ζ5ζ136 | ζ54ζ137+ζ54ζ136+ζ5ζ139+ζ5ζ134 | ζ54ζ138+ζ54ζ135+ζ5ζ1312+ζ5ζ13 | ζ54ζ1310+ζ54ζ133+ζ5ζ1311+ζ5ζ132 | ζ54ζ1312+ζ54ζ13+ζ5ζ138+ζ5ζ135 | complex faithful |
| ρ20 | 4 | 0 | 0 | 0 | -1+√5 | -1-√5 | 0 | 0 | ζ1311+ζ1310+ζ133+ζ132 | ζ1312+ζ138+ζ135+ζ13 | ζ139+ζ137+ζ136+ζ134 | ζ54ζ137+ζ54ζ136+ζ5ζ139+ζ5ζ134 | ζ53ζ137+ζ53ζ136+ζ52ζ139+ζ52ζ134 | ζ53ζ1312+ζ53ζ13+ζ52ζ138+ζ52ζ135 | ζ53ζ138+ζ53ζ135+ζ52ζ1312+ζ52ζ13 | ζ53ζ1311+ζ53ζ132+ζ52ζ1310+ζ52ζ133 | ζ53ζ139+ζ53ζ134+ζ52ζ137+ζ52ζ136 | ζ53ζ1310+ζ53ζ133+ζ52ζ1311+ζ52ζ132 | ζ54ζ1312+ζ54ζ13+ζ5ζ138+ζ5ζ135 | ζ54ζ138+ζ54ζ135+ζ5ζ1312+ζ5ζ13 | ζ54ζ1311+ζ54ζ132+ζ5ζ1310+ζ5ζ133 | ζ54ζ139+ζ54ζ134+ζ5ζ137+ζ5ζ136 | ζ54ζ1310+ζ54ζ133+ζ5ζ1311+ζ5ζ132 | complex faithful |
| ρ21 | 4 | 0 | 0 | 0 | -1-√5 | -1+√5 | 0 | 0 | ζ1311+ζ1310+ζ133+ζ132 | ζ1312+ζ138+ζ135+ζ13 | ζ139+ζ137+ζ136+ζ134 | ζ53ζ137+ζ53ζ136+ζ52ζ139+ζ52ζ134 | ζ54ζ139+ζ54ζ134+ζ5ζ137+ζ5ζ136 | ζ54ζ138+ζ54ζ135+ζ5ζ1312+ζ5ζ13 | ζ54ζ1312+ζ54ζ13+ζ5ζ138+ζ5ζ135 | ζ54ζ1310+ζ54ζ133+ζ5ζ1311+ζ5ζ132 | ζ54ζ137+ζ54ζ136+ζ5ζ139+ζ5ζ134 | ζ54ζ1311+ζ54ζ132+ζ5ζ1310+ζ5ζ133 | ζ53ζ1312+ζ53ζ13+ζ52ζ138+ζ52ζ135 | ζ53ζ138+ζ53ζ135+ζ52ζ1312+ζ52ζ13 | ζ53ζ1311+ζ53ζ132+ζ52ζ1310+ζ52ζ133 | ζ53ζ139+ζ53ζ134+ζ52ζ137+ζ52ζ136 | ζ53ζ1310+ζ53ζ133+ζ52ζ1311+ζ52ζ132 | complex faithful |
| ρ22 | 4 | 0 | 0 | 0 | -1+√5 | -1-√5 | 0 | 0 | ζ1312+ζ138+ζ135+ζ13 | ζ139+ζ137+ζ136+ζ134 | ζ1311+ζ1310+ζ133+ζ132 | ζ54ζ1310+ζ54ζ133+ζ5ζ1311+ζ5ζ132 | ζ53ζ1310+ζ53ζ133+ζ52ζ1311+ζ52ζ132 | ζ53ζ137+ζ53ζ136+ζ52ζ139+ζ52ζ134 | ζ53ζ139+ζ53ζ134+ζ52ζ137+ζ52ζ136 | ζ53ζ1312+ζ53ζ13+ζ52ζ138+ζ52ζ135 | ζ53ζ1311+ζ53ζ132+ζ52ζ1310+ζ52ζ133 | ζ53ζ138+ζ53ζ135+ζ52ζ1312+ζ52ζ13 | ζ54ζ137+ζ54ζ136+ζ5ζ139+ζ5ζ134 | ζ54ζ139+ζ54ζ134+ζ5ζ137+ζ5ζ136 | ζ54ζ1312+ζ54ζ13+ζ5ζ138+ζ5ζ135 | ζ54ζ1311+ζ54ζ132+ζ5ζ1310+ζ5ζ133 | ζ54ζ138+ζ54ζ135+ζ5ζ1312+ζ5ζ13 | complex faithful |
| ρ23 | 4 | 0 | 0 | 0 | -1-√5 | -1+√5 | 0 | 0 | ζ139+ζ137+ζ136+ζ134 | ζ1311+ζ1310+ζ133+ζ132 | ζ1312+ζ138+ζ135+ζ13 | ζ53ζ138+ζ53ζ135+ζ52ζ1312+ζ52ζ13 | ζ54ζ1312+ζ54ζ13+ζ5ζ138+ζ5ζ135 | ζ54ζ1311+ζ54ζ132+ζ5ζ1310+ζ5ζ133 | ζ54ζ1310+ζ54ζ133+ζ5ζ1311+ζ5ζ132 | ζ54ζ139+ζ54ζ134+ζ5ζ137+ζ5ζ136 | ζ54ζ138+ζ54ζ135+ζ5ζ1312+ζ5ζ13 | ζ54ζ137+ζ54ζ136+ζ5ζ139+ζ5ζ134 | ζ53ζ1310+ζ53ζ133+ζ52ζ1311+ζ52ζ132 | ζ53ζ1311+ζ53ζ132+ζ52ζ1310+ζ52ζ133 | ζ53ζ137+ζ53ζ136+ζ52ζ139+ζ52ζ134 | ζ53ζ1312+ζ53ζ13+ζ52ζ138+ζ52ζ135 | ζ53ζ139+ζ53ζ134+ζ52ζ137+ζ52ζ136 | complex faithful |
►On 65 points
Generators in S65
(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 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65)
(2 45 52 35)(3 24 38 4)(5 47 10 7)(6 26 61 41)(8 49 33 44)(9 28 19 13)(11 51 56 16)(12 30 42 50)(14 53)(15 32 65 22)(17 55 37 25)(18 34 23 59)(20 57 60 62)(21 36 46 31)(27 40)(29 63 64 43)(39 48 54 58)
G:=sub<Sym(65)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65), (2,45,52,35)(3,24,38,4)(5,47,10,7)(6,26,61,41)(8,49,33,44)(9,28,19,13)(11,51,56,16)(12,30,42,50)(14,53)(15,32,65,22)(17,55,37,25)(18,34,23,59)(20,57,60,62)(21,36,46,31)(27,40)(29,63,64,43)(39,48,54,58)>;
G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65), (2,45,52,35)(3,24,38,4)(5,47,10,7)(6,26,61,41)(8,49,33,44)(9,28,19,13)(11,51,56,16)(12,30,42,50)(14,53)(15,32,65,22)(17,55,37,25)(18,34,23,59)(20,57,60,62)(21,36,46,31)(27,40)(29,63,64,43)(39,48,54,58) );
G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65)], [(2,45,52,35),(3,24,38,4),(5,47,10,7),(6,26,61,41),(8,49,33,44),(9,28,19,13),(11,51,56,16),(12,30,42,50),(14,53),(15,32,65,22),(17,55,37,25),(18,34,23,59),(20,57,60,62),(21,36,46,31),(27,40),(29,63,64,43),(39,48,54,58)]])
Matrix representation of C65⋊C4 ►in GL4(𝔽521) generated by
| 393 | 419 | 419 | 393 |
| 128 | 319 | 131 | 345 |
| 176 | 417 | 444 | 420 |
| 101 | 256 | 320 | 3 |
| 1 | 0 | 0 | 0 |
| 25 | 24 | 24 | 25 |
| 0 | 0 | 0 | 1 |
| 497 | 472 | 103 | 496 |
G:=sub<GL(4,GF(521))| [393,128,176,101,419,319,417,256,419,131,444,320,393,345,420,3],[1,25,0,497,0,24,0,472,0,24,0,103,0,25,1,496] >;
C65⋊C4 in GAP, Magma, Sage, TeX
C_{65}\rtimes C_4 % in TeX
G:=Group("C65:C4"); // GroupNames label
G:=SmallGroup(260,6);
// by ID
G=gap.SmallGroup(260,6);
# by ID
G:=PCGroup([4,-2,-2,-5,-13,8,194,1603,1927]);
// Polycyclic
G:=Group<a,b|a^65=b^4=1,b*a*b^-1=a^34>;
// generators/relations
Export
Subgroup lattice of C65⋊C4 in TeX
Character table of C65⋊C4 in TeX