metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C65⋊1C4, C13⋊1F5, D65.1C2, C5⋊1(C13⋊C4), SmallGroup(260,9)
Series: Derived ►Chief ►Lower central ►Upper central
| C65 — C13⋊F5 |
Generators and relations for C13⋊F5
G = < a,b,c | a13=b5=c4=1, ab=ba, cac-1=a5, cbc-1=b3 >
Character table of C13⋊F5
| class | 1 | 2 | 4A | 4B | 5 | 13A | 13B | 13C | 65A | 65B | 65C | 65D | 65E | 65F | 65G | 65H | 65I | 65J | 65K | 65L | |
| size | 1 | 65 | 65 | 65 | 4 | 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 | trivial |
| ρ2 | 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 | 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 | linear of order 4 |
| ρ5 | 4 | 0 | 0 | 0 | -1 | 4 | 4 | 4 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from F5 |
| ρ6 | 4 | 0 | 0 | 0 | 4 | ζ1312+ζ138+ζ135+ζ13 | ζ139+ζ137+ζ136+ζ134 | ζ1311+ζ1310+ζ133+ζ132 | ζ1312+ζ138+ζ135+ζ13 | ζ1311+ζ1310+ζ133+ζ132 | ζ1312+ζ138+ζ135+ζ13 | ζ1312+ζ138+ζ135+ζ13 | ζ1311+ζ1310+ζ133+ζ132 | ζ1312+ζ138+ζ135+ζ13 | ζ139+ζ137+ζ136+ζ134 | ζ139+ζ137+ζ136+ζ134 | ζ1311+ζ1310+ζ133+ζ132 | ζ1311+ζ1310+ζ133+ζ132 | ζ139+ζ137+ζ136+ζ134 | ζ139+ζ137+ζ136+ζ134 | orthogonal lifted from C13⋊C4 |
| ρ7 | 4 | 0 | 0 | 0 | 4 | ζ139+ζ137+ζ136+ζ134 | ζ1311+ζ1310+ζ133+ζ132 | ζ1312+ζ138+ζ135+ζ13 | ζ139+ζ137+ζ136+ζ134 | ζ1312+ζ138+ζ135+ζ13 | ζ139+ζ137+ζ136+ζ134 | ζ139+ζ137+ζ136+ζ134 | ζ1312+ζ138+ζ135+ζ13 | ζ139+ζ137+ζ136+ζ134 | ζ1311+ζ1310+ζ133+ζ132 | ζ1311+ζ1310+ζ133+ζ132 | ζ1312+ζ138+ζ135+ζ13 | ζ1312+ζ138+ζ135+ζ13 | ζ1311+ζ1310+ζ133+ζ132 | ζ1311+ζ1310+ζ133+ζ132 | orthogonal lifted from C13⋊C4 |
| ρ8 | 4 | 0 | 0 | 0 | 4 | ζ1311+ζ1310+ζ133+ζ132 | ζ1312+ζ138+ζ135+ζ13 | ζ139+ζ137+ζ136+ζ134 | ζ1311+ζ1310+ζ133+ζ132 | ζ139+ζ137+ζ136+ζ134 | ζ1311+ζ1310+ζ133+ζ132 | ζ1311+ζ1310+ζ133+ζ132 | ζ139+ζ137+ζ136+ζ134 | ζ1311+ζ1310+ζ133+ζ132 | ζ1312+ζ138+ζ135+ζ13 | ζ1312+ζ138+ζ135+ζ13 | ζ139+ζ137+ζ136+ζ134 | ζ139+ζ137+ζ136+ζ134 | ζ1312+ζ138+ζ135+ζ13 | ζ1312+ζ138+ζ135+ζ13 | orthogonal lifted from C13⋊C4 |
| ρ9 | 4 | 0 | 0 | 0 | -1 | ζ1312+ζ138+ζ135+ζ13 | ζ139+ζ137+ζ136+ζ134 | ζ1311+ζ1310+ζ133+ζ132 | -ζ54ζ138+ζ54ζ13-ζ52ζ138+ζ52ζ135+ζ5ζ1312-ζ5ζ138-ζ138 | ζ54ζ1311-ζ54ζ133+ζ53ζ1310-ζ53ζ133-ζ5ζ133+ζ5ζ132-ζ133 | ζ54ζ138-ζ54ζ135+ζ53ζ1312-ζ53ζ135-ζ52ζ135+ζ52ζ13-ζ135 | -ζ54ζ138+ζ54ζ135-ζ53ζ138+ζ53ζ13+ζ52ζ1312-ζ52ζ138-ζ138 | ζ53ζ133-ζ53ζ132+ζ52ζ1310-ζ52ζ132+ζ5ζ1311-ζ5ζ132-ζ132 | -ζ53ζ1312+ζ53ζ135-ζ52ζ1312+ζ52ζ138-ζ5ζ1312+ζ5ζ13-ζ1312 | ζ54ζ139-ζ54ζ136+ζ53ζ137-ζ53ζ136-ζ5ζ136+ζ5ζ134-ζ136 | ζ53ζ139-ζ53ζ136-ζ52ζ136+ζ52ζ134+ζ5ζ137-ζ5ζ136-ζ136 | -ζ54ζ1311+ζ54ζ133-ζ52ζ1311+ζ52ζ132-ζ5ζ1311+ζ5ζ1310-ζ1311 | ζ54ζ1310-ζ54ζ132+ζ52ζ1311-ζ52ζ132+ζ5ζ133-ζ5ζ132-ζ132 | ζ54ζ137-ζ54ζ134+ζ52ζ139-ζ52ζ134+ζ5ζ136-ζ5ζ134-ζ134 | -ζ54ζ137+ζ54ζ134-ζ53ζ137+ζ53ζ136+ζ5ζ139-ζ5ζ137-ζ137 | orthogonal faithful |
| ρ10 | 4 | 0 | 0 | 0 | -1 | ζ1312+ζ138+ζ135+ζ13 | ζ139+ζ137+ζ136+ζ134 | ζ1311+ζ1310+ζ133+ζ132 | -ζ54ζ138+ζ54ζ135-ζ53ζ138+ζ53ζ13+ζ52ζ1312-ζ52ζ138-ζ138 | -ζ54ζ1311+ζ54ζ133-ζ52ζ1311+ζ52ζ132-ζ5ζ1311+ζ5ζ1310-ζ1311 | -ζ54ζ138+ζ54ζ13-ζ52ζ138+ζ52ζ135+ζ5ζ1312-ζ5ζ138-ζ138 | -ζ53ζ1312+ζ53ζ135-ζ52ζ1312+ζ52ζ138-ζ5ζ1312+ζ5ζ13-ζ1312 | ζ54ζ1310-ζ54ζ132+ζ52ζ1311-ζ52ζ132+ζ5ζ133-ζ5ζ132-ζ132 | ζ54ζ138-ζ54ζ135+ζ53ζ1312-ζ53ζ135-ζ52ζ135+ζ52ζ13-ζ135 | ζ53ζ139-ζ53ζ136-ζ52ζ136+ζ52ζ134+ζ5ζ137-ζ5ζ136-ζ136 | -ζ54ζ137+ζ54ζ134-ζ53ζ137+ζ53ζ136+ζ5ζ139-ζ5ζ137-ζ137 | ζ53ζ133-ζ53ζ132+ζ52ζ1310-ζ52ζ132+ζ5ζ1311-ζ5ζ132-ζ132 | ζ54ζ1311-ζ54ζ133+ζ53ζ1310-ζ53ζ133-ζ5ζ133+ζ5ζ132-ζ133 | ζ54ζ139-ζ54ζ136+ζ53ζ137-ζ53ζ136-ζ5ζ136+ζ5ζ134-ζ136 | ζ54ζ137-ζ54ζ134+ζ52ζ139-ζ52ζ134+ζ5ζ136-ζ5ζ134-ζ134 | orthogonal faithful |
| ρ11 | 4 | 0 | 0 | 0 | -1 | ζ139+ζ137+ζ136+ζ134 | ζ1311+ζ1310+ζ133+ζ132 | ζ1312+ζ138+ζ135+ζ13 | ζ54ζ139-ζ54ζ136+ζ53ζ137-ζ53ζ136-ζ5ζ136+ζ5ζ134-ζ136 | ζ54ζ138-ζ54ζ135+ζ53ζ1312-ζ53ζ135-ζ52ζ135+ζ52ζ13-ζ135 | ζ54ζ137-ζ54ζ134+ζ52ζ139-ζ52ζ134+ζ5ζ136-ζ5ζ134-ζ134 | ζ53ζ139-ζ53ζ136-ζ52ζ136+ζ52ζ134+ζ5ζ137-ζ5ζ136-ζ136 | -ζ54ζ138+ζ54ζ135-ζ53ζ138+ζ53ζ13+ζ52ζ1312-ζ52ζ138-ζ138 | -ζ54ζ137+ζ54ζ134-ζ53ζ137+ζ53ζ136+ζ5ζ139-ζ5ζ137-ζ137 | -ζ54ζ1311+ζ54ζ133-ζ52ζ1311+ζ52ζ132-ζ5ζ1311+ζ5ζ1310-ζ1311 | ζ53ζ133-ζ53ζ132+ζ52ζ1310-ζ52ζ132+ζ5ζ1311-ζ5ζ132-ζ132 | -ζ54ζ138+ζ54ζ13-ζ52ζ138+ζ52ζ135+ζ5ζ1312-ζ5ζ138-ζ138 | -ζ53ζ1312+ζ53ζ135-ζ52ζ1312+ζ52ζ138-ζ5ζ1312+ζ5ζ13-ζ1312 | ζ54ζ1311-ζ54ζ133+ζ53ζ1310-ζ53ζ133-ζ5ζ133+ζ5ζ132-ζ133 | ζ54ζ1310-ζ54ζ132+ζ52ζ1311-ζ52ζ132+ζ5ζ133-ζ5ζ132-ζ132 | orthogonal faithful |
| ρ12 | 4 | 0 | 0 | 0 | -1 | ζ1311+ζ1310+ζ133+ζ132 | ζ1312+ζ138+ζ135+ζ13 | ζ139+ζ137+ζ136+ζ134 | ζ54ζ1310-ζ54ζ132+ζ52ζ1311-ζ52ζ132+ζ5ζ133-ζ5ζ132-ζ132 | ζ53ζ139-ζ53ζ136-ζ52ζ136+ζ52ζ134+ζ5ζ137-ζ5ζ136-ζ136 | ζ53ζ133-ζ53ζ132+ζ52ζ1310-ζ52ζ132+ζ5ζ1311-ζ5ζ132-ζ132 | ζ54ζ1311-ζ54ζ133+ζ53ζ1310-ζ53ζ133-ζ5ζ133+ζ5ζ132-ζ133 | ζ54ζ137-ζ54ζ134+ζ52ζ139-ζ52ζ134+ζ5ζ136-ζ5ζ134-ζ134 | -ζ54ζ1311+ζ54ζ133-ζ52ζ1311+ζ52ζ132-ζ5ζ1311+ζ5ζ1310-ζ1311 | -ζ53ζ1312+ζ53ζ135-ζ52ζ1312+ζ52ζ138-ζ5ζ1312+ζ5ζ13-ζ1312 | ζ54ζ138-ζ54ζ135+ζ53ζ1312-ζ53ζ135-ζ52ζ135+ζ52ζ13-ζ135 | -ζ54ζ137+ζ54ζ134-ζ53ζ137+ζ53ζ136+ζ5ζ139-ζ5ζ137-ζ137 | ζ54ζ139-ζ54ζ136+ζ53ζ137-ζ53ζ136-ζ5ζ136+ζ5ζ134-ζ136 | -ζ54ζ138+ζ54ζ135-ζ53ζ138+ζ53ζ13+ζ52ζ1312-ζ52ζ138-ζ138 | -ζ54ζ138+ζ54ζ13-ζ52ζ138+ζ52ζ135+ζ5ζ1312-ζ5ζ138-ζ138 | orthogonal faithful |
| ρ13 | 4 | 0 | 0 | 0 | -1 | ζ1311+ζ1310+ζ133+ζ132 | ζ1312+ζ138+ζ135+ζ13 | ζ139+ζ137+ζ136+ζ134 | ζ53ζ133-ζ53ζ132+ζ52ζ1310-ζ52ζ132+ζ5ζ1311-ζ5ζ132-ζ132 | ζ54ζ139-ζ54ζ136+ζ53ζ137-ζ53ζ136-ζ5ζ136+ζ5ζ134-ζ136 | -ζ54ζ1311+ζ54ζ133-ζ52ζ1311+ζ52ζ132-ζ5ζ1311+ζ5ζ1310-ζ1311 | ζ54ζ1310-ζ54ζ132+ζ52ζ1311-ζ52ζ132+ζ5ζ133-ζ5ζ132-ζ132 | -ζ54ζ137+ζ54ζ134-ζ53ζ137+ζ53ζ136+ζ5ζ139-ζ5ζ137-ζ137 | ζ54ζ1311-ζ54ζ133+ζ53ζ1310-ζ53ζ133-ζ5ζ133+ζ5ζ132-ζ133 | -ζ54ζ138+ζ54ζ135-ζ53ζ138+ζ53ζ13+ζ52ζ1312-ζ52ζ138-ζ138 | -ζ53ζ1312+ζ53ζ135-ζ52ζ1312+ζ52ζ138-ζ5ζ1312+ζ5ζ13-ζ1312 | ζ53ζ139-ζ53ζ136-ζ52ζ136+ζ52ζ134+ζ5ζ137-ζ5ζ136-ζ136 | ζ54ζ137-ζ54ζ134+ζ52ζ139-ζ52ζ134+ζ5ζ136-ζ5ζ134-ζ134 | -ζ54ζ138+ζ54ζ13-ζ52ζ138+ζ52ζ135+ζ5ζ1312-ζ5ζ138-ζ138 | ζ54ζ138-ζ54ζ135+ζ53ζ1312-ζ53ζ135-ζ52ζ135+ζ52ζ13-ζ135 | orthogonal faithful |
| ρ14 | 4 | 0 | 0 | 0 | -1 | ζ139+ζ137+ζ136+ζ134 | ζ1311+ζ1310+ζ133+ζ132 | ζ1312+ζ138+ζ135+ζ13 | -ζ54ζ137+ζ54ζ134-ζ53ζ137+ζ53ζ136+ζ5ζ139-ζ5ζ137-ζ137 | -ζ54ζ138+ζ54ζ135-ζ53ζ138+ζ53ζ13+ζ52ζ1312-ζ52ζ138-ζ138 | ζ53ζ139-ζ53ζ136-ζ52ζ136+ζ52ζ134+ζ5ζ137-ζ5ζ136-ζ136 | ζ54ζ137-ζ54ζ134+ζ52ζ139-ζ52ζ134+ζ5ζ136-ζ5ζ134-ζ134 | ζ54ζ138-ζ54ζ135+ζ53ζ1312-ζ53ζ135-ζ52ζ135+ζ52ζ13-ζ135 | ζ54ζ139-ζ54ζ136+ζ53ζ137-ζ53ζ136-ζ5ζ136+ζ5ζ134-ζ136 | ζ54ζ1310-ζ54ζ132+ζ52ζ1311-ζ52ζ132+ζ5ζ133-ζ5ζ132-ζ132 | ζ54ζ1311-ζ54ζ133+ζ53ζ1310-ζ53ζ133-ζ5ζ133+ζ5ζ132-ζ133 | -ζ53ζ1312+ζ53ζ135-ζ52ζ1312+ζ52ζ138-ζ5ζ1312+ζ5ζ13-ζ1312 | -ζ54ζ138+ζ54ζ13-ζ52ζ138+ζ52ζ135+ζ5ζ1312-ζ5ζ138-ζ138 | ζ53ζ133-ζ53ζ132+ζ52ζ1310-ζ52ζ132+ζ5ζ1311-ζ5ζ132-ζ132 | -ζ54ζ1311+ζ54ζ133-ζ52ζ1311+ζ52ζ132-ζ5ζ1311+ζ5ζ1310-ζ1311 | orthogonal faithful |
| ρ15 | 4 | 0 | 0 | 0 | -1 | ζ1312+ζ138+ζ135+ζ13 | ζ139+ζ137+ζ136+ζ134 | ζ1311+ζ1310+ζ133+ζ132 | -ζ53ζ1312+ζ53ζ135-ζ52ζ1312+ζ52ζ138-ζ5ζ1312+ζ5ζ13-ζ1312 | ζ53ζ133-ζ53ζ132+ζ52ζ1310-ζ52ζ132+ζ5ζ1311-ζ5ζ132-ζ132 | -ζ54ζ138+ζ54ζ135-ζ53ζ138+ζ53ζ13+ζ52ζ1312-ζ52ζ138-ζ138 | ζ54ζ138-ζ54ζ135+ζ53ζ1312-ζ53ζ135-ζ52ζ135+ζ52ζ13-ζ135 | ζ54ζ1311-ζ54ζ133+ζ53ζ1310-ζ53ζ133-ζ5ζ133+ζ5ζ132-ζ133 | -ζ54ζ138+ζ54ζ13-ζ52ζ138+ζ52ζ135+ζ5ζ1312-ζ5ζ138-ζ138 | -ζ54ζ137+ζ54ζ134-ζ53ζ137+ζ53ζ136+ζ5ζ139-ζ5ζ137-ζ137 | ζ54ζ137-ζ54ζ134+ζ52ζ139-ζ52ζ134+ζ5ζ136-ζ5ζ134-ζ134 | ζ54ζ1310-ζ54ζ132+ζ52ζ1311-ζ52ζ132+ζ5ζ133-ζ5ζ132-ζ132 | -ζ54ζ1311+ζ54ζ133-ζ52ζ1311+ζ52ζ132-ζ5ζ1311+ζ5ζ1310-ζ1311 | ζ53ζ139-ζ53ζ136-ζ52ζ136+ζ52ζ134+ζ5ζ137-ζ5ζ136-ζ136 | ζ54ζ139-ζ54ζ136+ζ53ζ137-ζ53ζ136-ζ5ζ136+ζ5ζ134-ζ136 | orthogonal faithful |
| ρ16 | 4 | 0 | 0 | 0 | -1 | ζ1312+ζ138+ζ135+ζ13 | ζ139+ζ137+ζ136+ζ134 | ζ1311+ζ1310+ζ133+ζ132 | ζ54ζ138-ζ54ζ135+ζ53ζ1312-ζ53ζ135-ζ52ζ135+ζ52ζ13-ζ135 | ζ54ζ1310-ζ54ζ132+ζ52ζ1311-ζ52ζ132+ζ5ζ133-ζ5ζ132-ζ132 | -ζ53ζ1312+ζ53ζ135-ζ52ζ1312+ζ52ζ138-ζ5ζ1312+ζ5ζ13-ζ1312 | -ζ54ζ138+ζ54ζ13-ζ52ζ138+ζ52ζ135+ζ5ζ1312-ζ5ζ138-ζ138 | -ζ54ζ1311+ζ54ζ133-ζ52ζ1311+ζ52ζ132-ζ5ζ1311+ζ5ζ1310-ζ1311 | -ζ54ζ138+ζ54ζ135-ζ53ζ138+ζ53ζ13+ζ52ζ1312-ζ52ζ138-ζ138 | ζ54ζ137-ζ54ζ134+ζ52ζ139-ζ52ζ134+ζ5ζ136-ζ5ζ134-ζ134 | ζ54ζ139-ζ54ζ136+ζ53ζ137-ζ53ζ136-ζ5ζ136+ζ5ζ134-ζ136 | ζ54ζ1311-ζ54ζ133+ζ53ζ1310-ζ53ζ133-ζ5ζ133+ζ5ζ132-ζ133 | ζ53ζ133-ζ53ζ132+ζ52ζ1310-ζ52ζ132+ζ5ζ1311-ζ5ζ132-ζ132 | -ζ54ζ137+ζ54ζ134-ζ53ζ137+ζ53ζ136+ζ5ζ139-ζ5ζ137-ζ137 | ζ53ζ139-ζ53ζ136-ζ52ζ136+ζ52ζ134+ζ5ζ137-ζ5ζ136-ζ136 | orthogonal faithful |
| ρ17 | 4 | 0 | 0 | 0 | -1 | ζ139+ζ137+ζ136+ζ134 | ζ1311+ζ1310+ζ133+ζ132 | ζ1312+ζ138+ζ135+ζ13 | ζ54ζ137-ζ54ζ134+ζ52ζ139-ζ52ζ134+ζ5ζ136-ζ5ζ134-ζ134 | -ζ53ζ1312+ζ53ζ135-ζ52ζ1312+ζ52ζ138-ζ5ζ1312+ζ5ζ13-ζ1312 | -ζ54ζ137+ζ54ζ134-ζ53ζ137+ζ53ζ136+ζ5ζ139-ζ5ζ137-ζ137 | ζ54ζ139-ζ54ζ136+ζ53ζ137-ζ53ζ136-ζ5ζ136+ζ5ζ134-ζ136 | -ζ54ζ138+ζ54ζ13-ζ52ζ138+ζ52ζ135+ζ5ζ1312-ζ5ζ138-ζ138 | ζ53ζ139-ζ53ζ136-ζ52ζ136+ζ52ζ134+ζ5ζ137-ζ5ζ136-ζ136 | ζ54ζ1311-ζ54ζ133+ζ53ζ1310-ζ53ζ133-ζ5ζ133+ζ5ζ132-ζ133 | -ζ54ζ1311+ζ54ζ133-ζ52ζ1311+ζ52ζ132-ζ5ζ1311+ζ5ζ1310-ζ1311 | ζ54ζ138-ζ54ζ135+ζ53ζ1312-ζ53ζ135-ζ52ζ135+ζ52ζ13-ζ135 | -ζ54ζ138+ζ54ζ135-ζ53ζ138+ζ53ζ13+ζ52ζ1312-ζ52ζ138-ζ138 | ζ54ζ1310-ζ54ζ132+ζ52ζ1311-ζ52ζ132+ζ5ζ133-ζ5ζ132-ζ132 | ζ53ζ133-ζ53ζ132+ζ52ζ1310-ζ52ζ132+ζ5ζ1311-ζ5ζ132-ζ132 | orthogonal faithful |
| ρ18 | 4 | 0 | 0 | 0 | -1 | ζ139+ζ137+ζ136+ζ134 | ζ1311+ζ1310+ζ133+ζ132 | ζ1312+ζ138+ζ135+ζ13 | ζ53ζ139-ζ53ζ136-ζ52ζ136+ζ52ζ134+ζ5ζ137-ζ5ζ136-ζ136 | -ζ54ζ138+ζ54ζ13-ζ52ζ138+ζ52ζ135+ζ5ζ1312-ζ5ζ138-ζ138 | ζ54ζ139-ζ54ζ136+ζ53ζ137-ζ53ζ136-ζ5ζ136+ζ5ζ134-ζ136 | -ζ54ζ137+ζ54ζ134-ζ53ζ137+ζ53ζ136+ζ5ζ139-ζ5ζ137-ζ137 | -ζ53ζ1312+ζ53ζ135-ζ52ζ1312+ζ52ζ138-ζ5ζ1312+ζ5ζ13-ζ1312 | ζ54ζ137-ζ54ζ134+ζ52ζ139-ζ52ζ134+ζ5ζ136-ζ5ζ134-ζ134 | ζ53ζ133-ζ53ζ132+ζ52ζ1310-ζ52ζ132+ζ5ζ1311-ζ5ζ132-ζ132 | ζ54ζ1310-ζ54ζ132+ζ52ζ1311-ζ52ζ132+ζ5ζ133-ζ5ζ132-ζ132 | -ζ54ζ138+ζ54ζ135-ζ53ζ138+ζ53ζ13+ζ52ζ1312-ζ52ζ138-ζ138 | ζ54ζ138-ζ54ζ135+ζ53ζ1312-ζ53ζ135-ζ52ζ135+ζ52ζ13-ζ135 | -ζ54ζ1311+ζ54ζ133-ζ52ζ1311+ζ52ζ132-ζ5ζ1311+ζ5ζ1310-ζ1311 | ζ54ζ1311-ζ54ζ133+ζ53ζ1310-ζ53ζ133-ζ5ζ133+ζ5ζ132-ζ133 | orthogonal faithful |
| ρ19 | 4 | 0 | 0 | 0 | -1 | ζ1311+ζ1310+ζ133+ζ132 | ζ1312+ζ138+ζ135+ζ13 | ζ139+ζ137+ζ136+ζ134 | -ζ54ζ1311+ζ54ζ133-ζ52ζ1311+ζ52ζ132-ζ5ζ1311+ζ5ζ1310-ζ1311 | ζ54ζ137-ζ54ζ134+ζ52ζ139-ζ52ζ134+ζ5ζ136-ζ5ζ134-ζ134 | ζ54ζ1311-ζ54ζ133+ζ53ζ1310-ζ53ζ133-ζ5ζ133+ζ5ζ132-ζ133 | ζ53ζ133-ζ53ζ132+ζ52ζ1310-ζ52ζ132+ζ5ζ1311-ζ5ζ132-ζ132 | ζ53ζ139-ζ53ζ136-ζ52ζ136+ζ52ζ134+ζ5ζ137-ζ5ζ136-ζ136 | ζ54ζ1310-ζ54ζ132+ζ52ζ1311-ζ52ζ132+ζ5ζ133-ζ5ζ132-ζ132 | -ζ54ζ138+ζ54ζ13-ζ52ζ138+ζ52ζ135+ζ5ζ1312-ζ5ζ138-ζ138 | -ζ54ζ138+ζ54ζ135-ζ53ζ138+ζ53ζ13+ζ52ζ1312-ζ52ζ138-ζ138 | ζ54ζ139-ζ54ζ136+ζ53ζ137-ζ53ζ136-ζ5ζ136+ζ5ζ134-ζ136 | -ζ54ζ137+ζ54ζ134-ζ53ζ137+ζ53ζ136+ζ5ζ139-ζ5ζ137-ζ137 | ζ54ζ138-ζ54ζ135+ζ53ζ1312-ζ53ζ135-ζ52ζ135+ζ52ζ13-ζ135 | -ζ53ζ1312+ζ53ζ135-ζ52ζ1312+ζ52ζ138-ζ5ζ1312+ζ5ζ13-ζ1312 | orthogonal faithful |
| ρ20 | 4 | 0 | 0 | 0 | -1 | ζ1311+ζ1310+ζ133+ζ132 | ζ1312+ζ138+ζ135+ζ13 | ζ139+ζ137+ζ136+ζ134 | ζ54ζ1311-ζ54ζ133+ζ53ζ1310-ζ53ζ133-ζ5ζ133+ζ5ζ132-ζ133 | -ζ54ζ137+ζ54ζ134-ζ53ζ137+ζ53ζ136+ζ5ζ139-ζ5ζ137-ζ137 | ζ54ζ1310-ζ54ζ132+ζ52ζ1311-ζ52ζ132+ζ5ζ133-ζ5ζ132-ζ132 | -ζ54ζ1311+ζ54ζ133-ζ52ζ1311+ζ52ζ132-ζ5ζ1311+ζ5ζ1310-ζ1311 | ζ54ζ139-ζ54ζ136+ζ53ζ137-ζ53ζ136-ζ5ζ136+ζ5ζ134-ζ136 | ζ53ζ133-ζ53ζ132+ζ52ζ1310-ζ52ζ132+ζ5ζ1311-ζ5ζ132-ζ132 | ζ54ζ138-ζ54ζ135+ζ53ζ1312-ζ53ζ135-ζ52ζ135+ζ52ζ13-ζ135 | -ζ54ζ138+ζ54ζ13-ζ52ζ138+ζ52ζ135+ζ5ζ1312-ζ5ζ138-ζ138 | ζ54ζ137-ζ54ζ134+ζ52ζ139-ζ52ζ134+ζ5ζ136-ζ5ζ134-ζ134 | ζ53ζ139-ζ53ζ136-ζ52ζ136+ζ52ζ134+ζ5ζ137-ζ5ζ136-ζ136 | -ζ53ζ1312+ζ53ζ135-ζ52ζ1312+ζ52ζ138-ζ5ζ1312+ζ5ζ13-ζ1312 | -ζ54ζ138+ζ54ζ135-ζ53ζ138+ζ53ζ13+ζ52ζ1312-ζ52ζ138-ζ138 | orthogonal 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)
(1 15 37 52 62)(2 16 38 40 63)(3 17 39 41 64)(4 18 27 42 65)(5 19 28 43 53)(6 20 29 44 54)(7 21 30 45 55)(8 22 31 46 56)(9 23 32 47 57)(10 24 33 48 58)(11 25 34 49 59)(12 26 35 50 60)(13 14 36 51 61)
(2 9 13 6)(3 4 12 11)(5 7 10 8)(14 29 63 47)(15 37 62 52)(16 32 61 44)(17 27 60 49)(18 35 59 41)(19 30 58 46)(20 38 57 51)(21 33 56 43)(22 28 55 48)(23 36 54 40)(24 31 53 45)(25 39 65 50)(26 34 64 42)
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), (1,15,37,52,62)(2,16,38,40,63)(3,17,39,41,64)(4,18,27,42,65)(5,19,28,43,53)(6,20,29,44,54)(7,21,30,45,55)(8,22,31,46,56)(9,23,32,47,57)(10,24,33,48,58)(11,25,34,49,59)(12,26,35,50,60)(13,14,36,51,61), (2,9,13,6)(3,4,12,11)(5,7,10,8)(14,29,63,47)(15,37,62,52)(16,32,61,44)(17,27,60,49)(18,35,59,41)(19,30,58,46)(20,38,57,51)(21,33,56,43)(22,28,55,48)(23,36,54,40)(24,31,53,45)(25,39,65,50)(26,34,64,42)>;
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), (1,15,37,52,62)(2,16,38,40,63)(3,17,39,41,64)(4,18,27,42,65)(5,19,28,43,53)(6,20,29,44,54)(7,21,30,45,55)(8,22,31,46,56)(9,23,32,47,57)(10,24,33,48,58)(11,25,34,49,59)(12,26,35,50,60)(13,14,36,51,61), (2,9,13,6)(3,4,12,11)(5,7,10,8)(14,29,63,47)(15,37,62,52)(16,32,61,44)(17,27,60,49)(18,35,59,41)(19,30,58,46)(20,38,57,51)(21,33,56,43)(22,28,55,48)(23,36,54,40)(24,31,53,45)(25,39,65,50)(26,34,64,42) );
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)], [(1,15,37,52,62),(2,16,38,40,63),(3,17,39,41,64),(4,18,27,42,65),(5,19,28,43,53),(6,20,29,44,54),(7,21,30,45,55),(8,22,31,46,56),(9,23,32,47,57),(10,24,33,48,58),(11,25,34,49,59),(12,26,35,50,60),(13,14,36,51,61)], [(2,9,13,6),(3,4,12,11),(5,7,10,8),(14,29,63,47),(15,37,62,52),(16,32,61,44),(17,27,60,49),(18,35,59,41),(19,30,58,46),(20,38,57,51),(21,33,56,43),(22,28,55,48),(23,36,54,40),(24,31,53,45),(25,39,65,50),(26,34,64,42)]])
Matrix representation of C13⋊F5 ►in GL4(𝔽521) generated by
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 520 | 126 | 150 | 126 |
| 261 | 177 | 455 | 320 |
| 201 | 464 | 245 | 137 |
| 384 | 270 | 174 | 314 |
| 207 | 352 | 480 | 142 |
| 1 | 0 | 0 | 0 |
| 395 | 245 | 270 | 396 |
| 150 | 251 | 275 | 125 |
| 0 | 0 | 1 | 0 |
G:=sub<GL(4,GF(521))| [0,0,0,520,1,0,0,126,0,1,0,150,0,0,1,126],[261,201,384,207,177,464,270,352,455,245,174,480,320,137,314,142],[1,395,150,0,0,245,251,0,0,270,275,1,0,396,125,0] >;
C13⋊F5 in GAP, Magma, Sage, TeX
C_{13}\rtimes F_5 % in TeX
G:=Group("C13:F5"); // GroupNames label
G:=SmallGroup(260,9);
// by ID
G=gap.SmallGroup(260,9);
# by ID
G:=PCGroup([4,-2,-2,-5,-13,8,98,102,2563,1927]);
// Polycyclic
G:=Group<a,b,c|a^13=b^5=c^4=1,a*b=b*a,c*a*c^-1=a^5,c*b*c^-1=b^3>;
// generators/relations
Export
Subgroup lattice of C13⋊F5 in TeX
Character table of C13⋊F5 in TeX