metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C65⋊4C4, C5⋊Dic13, C13⋊3F5, D5.D13, (D5×C13).1C2, SmallGroup(260,8)
Series: Derived ►Chief ►Lower central ►Upper central
| C65 — C13⋊3F5 |
Generators and relations for C13⋊3F5
G = < a,b,c | a13=b5=c4=1, ab=ba, cac-1=a-1, cbc-1=b3 >
Character table of C13⋊3F5
| class | 1 | 2 | 4A | 4B | 5 | 13A | 13B | 13C | 13D | 13E | 13F | 26A | 26B | 26C | 26D | 26E | 26F | 65A | 65B | 65C | 65D | 65E | 65F | 65G | 65H | 65I | 65J | 65K | 65L | |
| size | 1 | 5 | 65 | 65 | 4 | 2 | 2 | 2 | 2 | 2 | 2 | 10 | 10 | 10 | 10 | 10 | 10 | 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 | 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 | 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 | 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 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 4 |
| ρ5 | 2 | 2 | 0 | 0 | 2 | ζ1312+ζ13 | ζ138+ζ135 | ζ1311+ζ132 | ζ1310+ζ133 | ζ137+ζ136 | ζ139+ζ134 | ζ138+ζ135 | ζ1310+ζ133 | ζ137+ζ136 | ζ139+ζ134 | ζ1312+ζ13 | ζ1311+ζ132 | ζ1312+ζ13 | ζ137+ζ136 | ζ1310+ζ133 | ζ1310+ζ133 | ζ1311+ζ132 | ζ138+ζ135 | ζ138+ζ135 | ζ1311+ζ132 | ζ1312+ζ13 | ζ139+ζ134 | ζ137+ζ136 | ζ139+ζ134 | orthogonal lifted from D13 |
| ρ6 | 2 | 2 | 0 | 0 | 2 | ζ139+ζ134 | ζ137+ζ136 | ζ138+ζ135 | ζ1312+ζ13 | ζ1311+ζ132 | ζ1310+ζ133 | ζ137+ζ136 | ζ1312+ζ13 | ζ1311+ζ132 | ζ1310+ζ133 | ζ139+ζ134 | ζ138+ζ135 | ζ139+ζ134 | ζ1311+ζ132 | ζ1312+ζ13 | ζ1312+ζ13 | ζ138+ζ135 | ζ137+ζ136 | ζ137+ζ136 | ζ138+ζ135 | ζ139+ζ134 | ζ1310+ζ133 | ζ1311+ζ132 | ζ1310+ζ133 | orthogonal lifted from D13 |
| ρ7 | 2 | 2 | 0 | 0 | 2 | ζ1311+ζ132 | ζ1310+ζ133 | ζ139+ζ134 | ζ137+ζ136 | ζ1312+ζ13 | ζ138+ζ135 | ζ1310+ζ133 | ζ137+ζ136 | ζ1312+ζ13 | ζ138+ζ135 | ζ1311+ζ132 | ζ139+ζ134 | ζ1311+ζ132 | ζ1312+ζ13 | ζ137+ζ136 | ζ137+ζ136 | ζ139+ζ134 | ζ1310+ζ133 | ζ1310+ζ133 | ζ139+ζ134 | ζ1311+ζ132 | ζ138+ζ135 | ζ1312+ζ13 | ζ138+ζ135 | orthogonal lifted from D13 |
| ρ8 | 2 | 2 | 0 | 0 | 2 | ζ137+ζ136 | ζ139+ζ134 | ζ1312+ζ13 | ζ138+ζ135 | ζ1310+ζ133 | ζ1311+ζ132 | ζ139+ζ134 | ζ138+ζ135 | ζ1310+ζ133 | ζ1311+ζ132 | ζ137+ζ136 | ζ1312+ζ13 | ζ137+ζ136 | ζ1310+ζ133 | ζ138+ζ135 | ζ138+ζ135 | ζ1312+ζ13 | ζ139+ζ134 | ζ139+ζ134 | ζ1312+ζ13 | ζ137+ζ136 | ζ1311+ζ132 | ζ1310+ζ133 | ζ1311+ζ132 | orthogonal lifted from D13 |
| ρ9 | 2 | 2 | 0 | 0 | 2 | ζ1310+ζ133 | ζ1311+ζ132 | ζ137+ζ136 | ζ139+ζ134 | ζ138+ζ135 | ζ1312+ζ13 | ζ1311+ζ132 | ζ139+ζ134 | ζ138+ζ135 | ζ1312+ζ13 | ζ1310+ζ133 | ζ137+ζ136 | ζ1310+ζ133 | ζ138+ζ135 | ζ139+ζ134 | ζ139+ζ134 | ζ137+ζ136 | ζ1311+ζ132 | ζ1311+ζ132 | ζ137+ζ136 | ζ1310+ζ133 | ζ1312+ζ13 | ζ138+ζ135 | ζ1312+ζ13 | orthogonal lifted from D13 |
| ρ10 | 2 | 2 | 0 | 0 | 2 | ζ138+ζ135 | ζ1312+ζ13 | ζ1310+ζ133 | ζ1311+ζ132 | ζ139+ζ134 | ζ137+ζ136 | ζ1312+ζ13 | ζ1311+ζ132 | ζ139+ζ134 | ζ137+ζ136 | ζ138+ζ135 | ζ1310+ζ133 | ζ138+ζ135 | ζ139+ζ134 | ζ1311+ζ132 | ζ1311+ζ132 | ζ1310+ζ133 | ζ1312+ζ13 | ζ1312+ζ13 | ζ1310+ζ133 | ζ138+ζ135 | ζ137+ζ136 | ζ139+ζ134 | ζ137+ζ136 | orthogonal lifted from D13 |
| ρ11 | 2 | -2 | 0 | 0 | 2 | ζ139+ζ134 | ζ137+ζ136 | ζ138+ζ135 | ζ1312+ζ13 | ζ1311+ζ132 | ζ1310+ζ133 | -ζ137-ζ136 | -ζ1312-ζ13 | -ζ1311-ζ132 | -ζ1310-ζ133 | -ζ139-ζ134 | -ζ138-ζ135 | ζ139+ζ134 | ζ1311+ζ132 | ζ1312+ζ13 | ζ1312+ζ13 | ζ138+ζ135 | ζ137+ζ136 | ζ137+ζ136 | ζ138+ζ135 | ζ139+ζ134 | ζ1310+ζ133 | ζ1311+ζ132 | ζ1310+ζ133 | symplectic lifted from Dic13, Schur index 2 |
| ρ12 | 2 | -2 | 0 | 0 | 2 | ζ137+ζ136 | ζ139+ζ134 | ζ1312+ζ13 | ζ138+ζ135 | ζ1310+ζ133 | ζ1311+ζ132 | -ζ139-ζ134 | -ζ138-ζ135 | -ζ1310-ζ133 | -ζ1311-ζ132 | -ζ137-ζ136 | -ζ1312-ζ13 | ζ137+ζ136 | ζ1310+ζ133 | ζ138+ζ135 | ζ138+ζ135 | ζ1312+ζ13 | ζ139+ζ134 | ζ139+ζ134 | ζ1312+ζ13 | ζ137+ζ136 | ζ1311+ζ132 | ζ1310+ζ133 | ζ1311+ζ132 | symplectic lifted from Dic13, Schur index 2 |
| ρ13 | 2 | -2 | 0 | 0 | 2 | ζ138+ζ135 | ζ1312+ζ13 | ζ1310+ζ133 | ζ1311+ζ132 | ζ139+ζ134 | ζ137+ζ136 | -ζ1312-ζ13 | -ζ1311-ζ132 | -ζ139-ζ134 | -ζ137-ζ136 | -ζ138-ζ135 | -ζ1310-ζ133 | ζ138+ζ135 | ζ139+ζ134 | ζ1311+ζ132 | ζ1311+ζ132 | ζ1310+ζ133 | ζ1312+ζ13 | ζ1312+ζ13 | ζ1310+ζ133 | ζ138+ζ135 | ζ137+ζ136 | ζ139+ζ134 | ζ137+ζ136 | symplectic lifted from Dic13, Schur index 2 |
| ρ14 | 2 | -2 | 0 | 0 | 2 | ζ1310+ζ133 | ζ1311+ζ132 | ζ137+ζ136 | ζ139+ζ134 | ζ138+ζ135 | ζ1312+ζ13 | -ζ1311-ζ132 | -ζ139-ζ134 | -ζ138-ζ135 | -ζ1312-ζ13 | -ζ1310-ζ133 | -ζ137-ζ136 | ζ1310+ζ133 | ζ138+ζ135 | ζ139+ζ134 | ζ139+ζ134 | ζ137+ζ136 | ζ1311+ζ132 | ζ1311+ζ132 | ζ137+ζ136 | ζ1310+ζ133 | ζ1312+ζ13 | ζ138+ζ135 | ζ1312+ζ13 | symplectic lifted from Dic13, Schur index 2 |
| ρ15 | 2 | -2 | 0 | 0 | 2 | ζ1311+ζ132 | ζ1310+ζ133 | ζ139+ζ134 | ζ137+ζ136 | ζ1312+ζ13 | ζ138+ζ135 | -ζ1310-ζ133 | -ζ137-ζ136 | -ζ1312-ζ13 | -ζ138-ζ135 | -ζ1311-ζ132 | -ζ139-ζ134 | ζ1311+ζ132 | ζ1312+ζ13 | ζ137+ζ136 | ζ137+ζ136 | ζ139+ζ134 | ζ1310+ζ133 | ζ1310+ζ133 | ζ139+ζ134 | ζ1311+ζ132 | ζ138+ζ135 | ζ1312+ζ13 | ζ138+ζ135 | symplectic lifted from Dic13, Schur index 2 |
| ρ16 | 2 | -2 | 0 | 0 | 2 | ζ1312+ζ13 | ζ138+ζ135 | ζ1311+ζ132 | ζ1310+ζ133 | ζ137+ζ136 | ζ139+ζ134 | -ζ138-ζ135 | -ζ1310-ζ133 | -ζ137-ζ136 | -ζ139-ζ134 | -ζ1312-ζ13 | -ζ1311-ζ132 | ζ1312+ζ13 | ζ137+ζ136 | ζ1310+ζ133 | ζ1310+ζ133 | ζ1311+ζ132 | ζ138+ζ135 | ζ138+ζ135 | ζ1311+ζ132 | ζ1312+ζ13 | ζ139+ζ134 | ζ137+ζ136 | ζ139+ζ134 | symplectic lifted from Dic13, Schur index 2 |
| ρ17 | 4 | 0 | 0 | 0 | -1 | 4 | 4 | 4 | 4 | 4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from F5 |
| ρ18 | 4 | 0 | 0 | 0 | -1 | 2ζ1312+2ζ13 | 2ζ138+2ζ135 | 2ζ1311+2ζ132 | 2ζ1310+2ζ133 | 2ζ137+2ζ136 | 2ζ139+2ζ134 | 0 | 0 | 0 | 0 | 0 | 0 | -ζ54ζ1312+ζ54ζ13-ζ5ζ1312+ζ5ζ13-ζ1312 | ζ53ζ137-ζ53ζ136+ζ52ζ137-ζ52ζ136-ζ136 | -ζ54ζ1310+ζ54ζ133-ζ5ζ1310+ζ5ζ133-ζ1310 | ζ54ζ1310-ζ54ζ133+ζ5ζ1310-ζ5ζ133-ζ133 | ζ54ζ1311-ζ54ζ132+ζ5ζ1311-ζ5ζ132-ζ132 | -ζ53ζ138+ζ53ζ135-ζ52ζ138+ζ52ζ135-ζ138 | ζ53ζ138-ζ53ζ135+ζ52ζ138-ζ52ζ135-ζ135 | ζ53ζ1311-ζ53ζ132+ζ52ζ1311-ζ52ζ132-ζ132 | -ζ53ζ1312+ζ53ζ13-ζ52ζ1312+ζ52ζ13-ζ1312 | ζ54ζ139-ζ54ζ134+ζ5ζ139-ζ5ζ134-ζ134 | ζ54ζ137-ζ54ζ136+ζ5ζ137-ζ5ζ136-ζ136 | -ζ54ζ139+ζ54ζ134-ζ5ζ139+ζ5ζ134-ζ139 | complex faithful |
| ρ19 | 4 | 0 | 0 | 0 | -1 | 2ζ137+2ζ136 | 2ζ139+2ζ134 | 2ζ1312+2ζ13 | 2ζ138+2ζ135 | 2ζ1310+2ζ133 | 2ζ1311+2ζ132 | 0 | 0 | 0 | 0 | 0 | 0 | ζ53ζ137-ζ53ζ136+ζ52ζ137-ζ52ζ136-ζ136 | ζ54ζ1310-ζ54ζ133+ζ5ζ1310-ζ5ζ133-ζ133 | ζ53ζ138-ζ53ζ135+ζ52ζ138-ζ52ζ135-ζ135 | -ζ53ζ138+ζ53ζ135-ζ52ζ138+ζ52ζ135-ζ138 | -ζ54ζ1312+ζ54ζ13-ζ5ζ1312+ζ5ζ13-ζ1312 | ζ54ζ139-ζ54ζ134+ζ5ζ139-ζ5ζ134-ζ134 | -ζ54ζ139+ζ54ζ134-ζ5ζ139+ζ5ζ134-ζ139 | -ζ53ζ1312+ζ53ζ13-ζ52ζ1312+ζ52ζ13-ζ1312 | ζ54ζ137-ζ54ζ136+ζ5ζ137-ζ5ζ136-ζ136 | ζ53ζ1311-ζ53ζ132+ζ52ζ1311-ζ52ζ132-ζ132 | -ζ54ζ1310+ζ54ζ133-ζ5ζ1310+ζ5ζ133-ζ1310 | ζ54ζ1311-ζ54ζ132+ζ5ζ1311-ζ5ζ132-ζ132 | complex faithful |
| ρ20 | 4 | 0 | 0 | 0 | -1 | 2ζ139+2ζ134 | 2ζ137+2ζ136 | 2ζ138+2ζ135 | 2ζ1312+2ζ13 | 2ζ1311+2ζ132 | 2ζ1310+2ζ133 | 0 | 0 | 0 | 0 | 0 | 0 | ζ54ζ139-ζ54ζ134+ζ5ζ139-ζ5ζ134-ζ134 | ζ53ζ1311-ζ53ζ132+ζ52ζ1311-ζ52ζ132-ζ132 | -ζ54ζ1312+ζ54ζ13-ζ5ζ1312+ζ5ζ13-ζ1312 | -ζ53ζ1312+ζ53ζ13-ζ52ζ1312+ζ52ζ13-ζ1312 | -ζ53ζ138+ζ53ζ135-ζ52ζ138+ζ52ζ135-ζ138 | ζ54ζ137-ζ54ζ136+ζ5ζ137-ζ5ζ136-ζ136 | ζ53ζ137-ζ53ζ136+ζ52ζ137-ζ52ζ136-ζ136 | ζ53ζ138-ζ53ζ135+ζ52ζ138-ζ52ζ135-ζ135 | -ζ54ζ139+ζ54ζ134-ζ5ζ139+ζ5ζ134-ζ139 | -ζ54ζ1310+ζ54ζ133-ζ5ζ1310+ζ5ζ133-ζ1310 | ζ54ζ1311-ζ54ζ132+ζ5ζ1311-ζ5ζ132-ζ132 | ζ54ζ1310-ζ54ζ133+ζ5ζ1310-ζ5ζ133-ζ133 | complex faithful |
| ρ21 | 4 | 0 | 0 | 0 | -1 | 2ζ137+2ζ136 | 2ζ139+2ζ134 | 2ζ1312+2ζ13 | 2ζ138+2ζ135 | 2ζ1310+2ζ133 | 2ζ1311+2ζ132 | 0 | 0 | 0 | 0 | 0 | 0 | ζ54ζ137-ζ54ζ136+ζ5ζ137-ζ5ζ136-ζ136 | -ζ54ζ1310+ζ54ζ133-ζ5ζ1310+ζ5ζ133-ζ1310 | -ζ53ζ138+ζ53ζ135-ζ52ζ138+ζ52ζ135-ζ138 | ζ53ζ138-ζ53ζ135+ζ52ζ138-ζ52ζ135-ζ135 | -ζ53ζ1312+ζ53ζ13-ζ52ζ1312+ζ52ζ13-ζ1312 | -ζ54ζ139+ζ54ζ134-ζ5ζ139+ζ5ζ134-ζ139 | ζ54ζ139-ζ54ζ134+ζ5ζ139-ζ5ζ134-ζ134 | -ζ54ζ1312+ζ54ζ13-ζ5ζ1312+ζ5ζ13-ζ1312 | ζ53ζ137-ζ53ζ136+ζ52ζ137-ζ52ζ136-ζ136 | ζ54ζ1311-ζ54ζ132+ζ5ζ1311-ζ5ζ132-ζ132 | ζ54ζ1310-ζ54ζ133+ζ5ζ1310-ζ5ζ133-ζ133 | ζ53ζ1311-ζ53ζ132+ζ52ζ1311-ζ52ζ132-ζ132 | complex faithful |
| ρ22 | 4 | 0 | 0 | 0 | -1 | 2ζ1311+2ζ132 | 2ζ1310+2ζ133 | 2ζ139+2ζ134 | 2ζ137+2ζ136 | 2ζ1312+2ζ13 | 2ζ138+2ζ135 | 0 | 0 | 0 | 0 | 0 | 0 | ζ53ζ1311-ζ53ζ132+ζ52ζ1311-ζ52ζ132-ζ132 | -ζ53ζ1312+ζ53ζ13-ζ52ζ1312+ζ52ζ13-ζ1312 | ζ53ζ137-ζ53ζ136+ζ52ζ137-ζ52ζ136-ζ136 | ζ54ζ137-ζ54ζ136+ζ5ζ137-ζ5ζ136-ζ136 | ζ54ζ139-ζ54ζ134+ζ5ζ139-ζ5ζ134-ζ134 | -ζ54ζ1310+ζ54ζ133-ζ5ζ1310+ζ5ζ133-ζ1310 | ζ54ζ1310-ζ54ζ133+ζ5ζ1310-ζ5ζ133-ζ133 | -ζ54ζ139+ζ54ζ134-ζ5ζ139+ζ5ζ134-ζ139 | ζ54ζ1311-ζ54ζ132+ζ5ζ1311-ζ5ζ132-ζ132 | ζ53ζ138-ζ53ζ135+ζ52ζ138-ζ52ζ135-ζ135 | -ζ54ζ1312+ζ54ζ13-ζ5ζ1312+ζ5ζ13-ζ1312 | -ζ53ζ138+ζ53ζ135-ζ52ζ138+ζ52ζ135-ζ138 | complex faithful |
| ρ23 | 4 | 0 | 0 | 0 | -1 | 2ζ1310+2ζ133 | 2ζ1311+2ζ132 | 2ζ137+2ζ136 | 2ζ139+2ζ134 | 2ζ138+2ζ135 | 2ζ1312+2ζ13 | 0 | 0 | 0 | 0 | 0 | 0 | -ζ54ζ1310+ζ54ζ133-ζ5ζ1310+ζ5ζ133-ζ1310 | ζ53ζ138-ζ53ζ135+ζ52ζ138-ζ52ζ135-ζ135 | ζ54ζ139-ζ54ζ134+ζ5ζ139-ζ5ζ134-ζ134 | -ζ54ζ139+ζ54ζ134-ζ5ζ139+ζ5ζ134-ζ139 | ζ54ζ137-ζ54ζ136+ζ5ζ137-ζ5ζ136-ζ136 | ζ54ζ1311-ζ54ζ132+ζ5ζ1311-ζ5ζ132-ζ132 | ζ53ζ1311-ζ53ζ132+ζ52ζ1311-ζ52ζ132-ζ132 | ζ53ζ137-ζ53ζ136+ζ52ζ137-ζ52ζ136-ζ136 | ζ54ζ1310-ζ54ζ133+ζ5ζ1310-ζ5ζ133-ζ133 | -ζ54ζ1312+ζ54ζ13-ζ5ζ1312+ζ5ζ13-ζ1312 | -ζ53ζ138+ζ53ζ135-ζ52ζ138+ζ52ζ135-ζ138 | -ζ53ζ1312+ζ53ζ13-ζ52ζ1312+ζ52ζ13-ζ1312 | complex faithful |
| ρ24 | 4 | 0 | 0 | 0 | -1 | 2ζ1311+2ζ132 | 2ζ1310+2ζ133 | 2ζ139+2ζ134 | 2ζ137+2ζ136 | 2ζ1312+2ζ13 | 2ζ138+2ζ135 | 0 | 0 | 0 | 0 | 0 | 0 | ζ54ζ1311-ζ54ζ132+ζ5ζ1311-ζ5ζ132-ζ132 | -ζ54ζ1312+ζ54ζ13-ζ5ζ1312+ζ5ζ13-ζ1312 | ζ54ζ137-ζ54ζ136+ζ5ζ137-ζ5ζ136-ζ136 | ζ53ζ137-ζ53ζ136+ζ52ζ137-ζ52ζ136-ζ136 | -ζ54ζ139+ζ54ζ134-ζ5ζ139+ζ5ζ134-ζ139 | ζ54ζ1310-ζ54ζ133+ζ5ζ1310-ζ5ζ133-ζ133 | -ζ54ζ1310+ζ54ζ133-ζ5ζ1310+ζ5ζ133-ζ1310 | ζ54ζ139-ζ54ζ134+ζ5ζ139-ζ5ζ134-ζ134 | ζ53ζ1311-ζ53ζ132+ζ52ζ1311-ζ52ζ132-ζ132 | -ζ53ζ138+ζ53ζ135-ζ52ζ138+ζ52ζ135-ζ138 | -ζ53ζ1312+ζ53ζ13-ζ52ζ1312+ζ52ζ13-ζ1312 | ζ53ζ138-ζ53ζ135+ζ52ζ138-ζ52ζ135-ζ135 | complex faithful |
| ρ25 | 4 | 0 | 0 | 0 | -1 | 2ζ1312+2ζ13 | 2ζ138+2ζ135 | 2ζ1311+2ζ132 | 2ζ1310+2ζ133 | 2ζ137+2ζ136 | 2ζ139+2ζ134 | 0 | 0 | 0 | 0 | 0 | 0 | -ζ53ζ1312+ζ53ζ13-ζ52ζ1312+ζ52ζ13-ζ1312 | ζ54ζ137-ζ54ζ136+ζ5ζ137-ζ5ζ136-ζ136 | ζ54ζ1310-ζ54ζ133+ζ5ζ1310-ζ5ζ133-ζ133 | -ζ54ζ1310+ζ54ζ133-ζ5ζ1310+ζ5ζ133-ζ1310 | ζ53ζ1311-ζ53ζ132+ζ52ζ1311-ζ52ζ132-ζ132 | ζ53ζ138-ζ53ζ135+ζ52ζ138-ζ52ζ135-ζ135 | -ζ53ζ138+ζ53ζ135-ζ52ζ138+ζ52ζ135-ζ138 | ζ54ζ1311-ζ54ζ132+ζ5ζ1311-ζ5ζ132-ζ132 | -ζ54ζ1312+ζ54ζ13-ζ5ζ1312+ζ5ζ13-ζ1312 | -ζ54ζ139+ζ54ζ134-ζ5ζ139+ζ5ζ134-ζ139 | ζ53ζ137-ζ53ζ136+ζ52ζ137-ζ52ζ136-ζ136 | ζ54ζ139-ζ54ζ134+ζ5ζ139-ζ5ζ134-ζ134 | complex faithful |
| ρ26 | 4 | 0 | 0 | 0 | -1 | 2ζ138+2ζ135 | 2ζ1312+2ζ13 | 2ζ1310+2ζ133 | 2ζ1311+2ζ132 | 2ζ139+2ζ134 | 2ζ137+2ζ136 | 0 | 0 | 0 | 0 | 0 | 0 | -ζ53ζ138+ζ53ζ135-ζ52ζ138+ζ52ζ135-ζ138 | ζ54ζ139-ζ54ζ134+ζ5ζ139-ζ5ζ134-ζ134 | ζ54ζ1311-ζ54ζ132+ζ5ζ1311-ζ5ζ132-ζ132 | ζ53ζ1311-ζ53ζ132+ζ52ζ1311-ζ52ζ132-ζ132 | ζ54ζ1310-ζ54ζ133+ζ5ζ1310-ζ5ζ133-ζ133 | -ζ53ζ1312+ζ53ζ13-ζ52ζ1312+ζ52ζ13-ζ1312 | -ζ54ζ1312+ζ54ζ13-ζ5ζ1312+ζ5ζ13-ζ1312 | -ζ54ζ1310+ζ54ζ133-ζ5ζ1310+ζ5ζ133-ζ1310 | ζ53ζ138-ζ53ζ135+ζ52ζ138-ζ52ζ135-ζ135 | ζ54ζ137-ζ54ζ136+ζ5ζ137-ζ5ζ136-ζ136 | -ζ54ζ139+ζ54ζ134-ζ5ζ139+ζ5ζ134-ζ139 | ζ53ζ137-ζ53ζ136+ζ52ζ137-ζ52ζ136-ζ136 | complex faithful |
| ρ27 | 4 | 0 | 0 | 0 | -1 | 2ζ139+2ζ134 | 2ζ137+2ζ136 | 2ζ138+2ζ135 | 2ζ1312+2ζ13 | 2ζ1311+2ζ132 | 2ζ1310+2ζ133 | 0 | 0 | 0 | 0 | 0 | 0 | -ζ54ζ139+ζ54ζ134-ζ5ζ139+ζ5ζ134-ζ139 | ζ54ζ1311-ζ54ζ132+ζ5ζ1311-ζ5ζ132-ζ132 | -ζ53ζ1312+ζ53ζ13-ζ52ζ1312+ζ52ζ13-ζ1312 | -ζ54ζ1312+ζ54ζ13-ζ5ζ1312+ζ5ζ13-ζ1312 | ζ53ζ138-ζ53ζ135+ζ52ζ138-ζ52ζ135-ζ135 | ζ53ζ137-ζ53ζ136+ζ52ζ137-ζ52ζ136-ζ136 | ζ54ζ137-ζ54ζ136+ζ5ζ137-ζ5ζ136-ζ136 | -ζ53ζ138+ζ53ζ135-ζ52ζ138+ζ52ζ135-ζ138 | ζ54ζ139-ζ54ζ134+ζ5ζ139-ζ5ζ134-ζ134 | ζ54ζ1310-ζ54ζ133+ζ5ζ1310-ζ5ζ133-ζ133 | ζ53ζ1311-ζ53ζ132+ζ52ζ1311-ζ52ζ132-ζ132 | -ζ54ζ1310+ζ54ζ133-ζ5ζ1310+ζ5ζ133-ζ1310 | complex faithful |
| ρ28 | 4 | 0 | 0 | 0 | -1 | 2ζ1310+2ζ133 | 2ζ1311+2ζ132 | 2ζ137+2ζ136 | 2ζ139+2ζ134 | 2ζ138+2ζ135 | 2ζ1312+2ζ13 | 0 | 0 | 0 | 0 | 0 | 0 | ζ54ζ1310-ζ54ζ133+ζ5ζ1310-ζ5ζ133-ζ133 | -ζ53ζ138+ζ53ζ135-ζ52ζ138+ζ52ζ135-ζ138 | -ζ54ζ139+ζ54ζ134-ζ5ζ139+ζ5ζ134-ζ139 | ζ54ζ139-ζ54ζ134+ζ5ζ139-ζ5ζ134-ζ134 | ζ53ζ137-ζ53ζ136+ζ52ζ137-ζ52ζ136-ζ136 | ζ53ζ1311-ζ53ζ132+ζ52ζ1311-ζ52ζ132-ζ132 | ζ54ζ1311-ζ54ζ132+ζ5ζ1311-ζ5ζ132-ζ132 | ζ54ζ137-ζ54ζ136+ζ5ζ137-ζ5ζ136-ζ136 | -ζ54ζ1310+ζ54ζ133-ζ5ζ1310+ζ5ζ133-ζ1310 | -ζ53ζ1312+ζ53ζ13-ζ52ζ1312+ζ52ζ13-ζ1312 | ζ53ζ138-ζ53ζ135+ζ52ζ138-ζ52ζ135-ζ135 | -ζ54ζ1312+ζ54ζ13-ζ5ζ1312+ζ5ζ13-ζ1312 | complex faithful |
| ρ29 | 4 | 0 | 0 | 0 | -1 | 2ζ138+2ζ135 | 2ζ1312+2ζ13 | 2ζ1310+2ζ133 | 2ζ1311+2ζ132 | 2ζ139+2ζ134 | 2ζ137+2ζ136 | 0 | 0 | 0 | 0 | 0 | 0 | ζ53ζ138-ζ53ζ135+ζ52ζ138-ζ52ζ135-ζ135 | -ζ54ζ139+ζ54ζ134-ζ5ζ139+ζ5ζ134-ζ139 | ζ53ζ1311-ζ53ζ132+ζ52ζ1311-ζ52ζ132-ζ132 | ζ54ζ1311-ζ54ζ132+ζ5ζ1311-ζ5ζ132-ζ132 | -ζ54ζ1310+ζ54ζ133-ζ5ζ1310+ζ5ζ133-ζ1310 | -ζ54ζ1312+ζ54ζ13-ζ5ζ1312+ζ5ζ13-ζ1312 | -ζ53ζ1312+ζ53ζ13-ζ52ζ1312+ζ52ζ13-ζ1312 | ζ54ζ1310-ζ54ζ133+ζ5ζ1310-ζ5ζ133-ζ133 | -ζ53ζ138+ζ53ζ135-ζ52ζ138+ζ52ζ135-ζ138 | ζ53ζ137-ζ53ζ136+ζ52ζ137-ζ52ζ136-ζ136 | ζ54ζ139-ζ54ζ134+ζ5ζ139-ζ5ζ134-ζ134 | ζ54ζ137-ζ54ζ136+ζ5ζ137-ζ5ζ136-ζ136 | 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)
(1 16 38 52 57)(2 17 39 40 58)(3 18 27 41 59)(4 19 28 42 60)(5 20 29 43 61)(6 21 30 44 62)(7 22 31 45 63)(8 23 32 46 64)(9 24 33 47 65)(10 25 34 48 53)(11 26 35 49 54)(12 14 36 50 55)(13 15 37 51 56)
(2 13)(3 12)(4 11)(5 10)(6 9)(7 8)(14 27 55 41)(15 39 56 40)(16 38 57 52)(17 37 58 51)(18 36 59 50)(19 35 60 49)(20 34 61 48)(21 33 62 47)(22 32 63 46)(23 31 64 45)(24 30 65 44)(25 29 53 43)(26 28 54 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), (1,16,38,52,57)(2,17,39,40,58)(3,18,27,41,59)(4,19,28,42,60)(5,20,29,43,61)(6,21,30,44,62)(7,22,31,45,63)(8,23,32,46,64)(9,24,33,47,65)(10,25,34,48,53)(11,26,35,49,54)(12,14,36,50,55)(13,15,37,51,56), (2,13)(3,12)(4,11)(5,10)(6,9)(7,8)(14,27,55,41)(15,39,56,40)(16,38,57,52)(17,37,58,51)(18,36,59,50)(19,35,60,49)(20,34,61,48)(21,33,62,47)(22,32,63,46)(23,31,64,45)(24,30,65,44)(25,29,53,43)(26,28,54,42)>;
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), (1,16,38,52,57)(2,17,39,40,58)(3,18,27,41,59)(4,19,28,42,60)(5,20,29,43,61)(6,21,30,44,62)(7,22,31,45,63)(8,23,32,46,64)(9,24,33,47,65)(10,25,34,48,53)(11,26,35,49,54)(12,14,36,50,55)(13,15,37,51,56), (2,13)(3,12)(4,11)(5,10)(6,9)(7,8)(14,27,55,41)(15,39,56,40)(16,38,57,52)(17,37,58,51)(18,36,59,50)(19,35,60,49)(20,34,61,48)(21,33,62,47)(22,32,63,46)(23,31,64,45)(24,30,65,44)(25,29,53,43)(26,28,54,42) );
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)], [(1,16,38,52,57),(2,17,39,40,58),(3,18,27,41,59),(4,19,28,42,60),(5,20,29,43,61),(6,21,30,44,62),(7,22,31,45,63),(8,23,32,46,64),(9,24,33,47,65),(10,25,34,48,53),(11,26,35,49,54),(12,14,36,50,55),(13,15,37,51,56)], [(2,13),(3,12),(4,11),(5,10),(6,9),(7,8),(14,27,55,41),(15,39,56,40),(16,38,57,52),(17,37,58,51),(18,36,59,50),(19,35,60,49),(20,34,61,48),(21,33,62,47),(22,32,63,46),(23,31,64,45),(24,30,65,44),(25,29,53,43),(26,28,54,42)]])
Matrix representation of C13⋊3F5 ►in GL4(𝔽521) generated by
| 0 | 1 | 0 | 0 |
| 520 | 7 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 520 | 7 |
| 172 | 174 | 520 | 0 |
| 347 | 348 | 0 | 520 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 7 | 520 | 0 | 0 |
| 349 | 347 | 349 | 347 |
| 185 | 172 | 185 | 172 |
G:=sub<GL(4,GF(521))| [0,520,0,0,1,7,0,0,0,0,0,520,0,0,1,7],[172,347,1,0,174,348,0,1,520,0,0,0,0,520,0,0],[1,7,349,185,0,520,347,172,0,0,349,185,0,0,347,172] >;
C13⋊3F5 in GAP, Magma, Sage, TeX
C_{13}\rtimes_3F_5 % in TeX
G:=Group("C13:3F5"); // GroupNames label
G:=SmallGroup(260,8);
// by ID
G=gap.SmallGroup(260,8);
# by ID
G:=PCGroup([4,-2,-2,-5,-13,8,146,102,3843]);
// Polycyclic
G:=Group<a,b,c|a^13=b^5=c^4=1,a*b=b*a,c*a*c^-1=a^-1,c*b*c^-1=b^3>;
// generators/relations
Export
Subgroup lattice of C13⋊3F5 in TeX
Character table of C13⋊3F5 in TeX