metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C39⋊1C8, C78.1C4, C26.Dic3, Dic13.2S3, C13⋊(C3⋊C8), C3⋊(C13⋊C8), C6.(C13⋊C4), C2.(C39⋊C4), (C3×Dic13).3C2, SmallGroup(312,14)
Series: Derived ►Chief ►Lower central ►Upper central
| C39 — C39⋊C8 |
Generators and relations for C39⋊C8
G = < a,b | a39=b8=1, bab-1=a8 >
Character table of C39⋊C8
| class | 1 | 2 | 3 | 4A | 4B | 6 | 8A | 8B | 8C | 8D | 12A | 12B | 13A | 13B | 13C | 26A | 26B | 26C | 39A | 39B | 39C | 39D | 39E | 39F | 78A | 78B | 78C | 78D | 78E | 78F | |
| size | 1 | 1 | 2 | 13 | 13 | 2 | 39 | 39 | 39 | 39 | 26 | 26 | 4 | 4 | 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 | 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 | 1 | linear of order 2 |
| ρ3 | 1 | 1 | 1 | -1 | -1 | 1 | i | -i | -i | i | -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 | 1 | -1 | -1 | 1 | -i | i | i | -i | -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 | 1 | -1 | 1 | -i | i | -1 | ζ83 | ζ85 | ζ8 | ζ87 | i | -i | 1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | linear of order 8 |
| ρ6 | 1 | -1 | 1 | i | -i | -1 | ζ85 | ζ83 | ζ87 | ζ8 | -i | i | 1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | linear of order 8 |
| ρ7 | 1 | -1 | 1 | i | -i | -1 | ζ8 | ζ87 | ζ83 | ζ85 | -i | i | 1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | linear of order 8 |
| ρ8 | 1 | -1 | 1 | -i | i | -1 | ζ87 | ζ8 | ζ85 | ζ83 | i | -i | 1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | linear of order 8 |
| ρ9 | 2 | 2 | -1 | 2 | 2 | -1 | 0 | 0 | 0 | 0 | -1 | -1 | 2 | 2 | 2 | 2 | 2 | 2 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ10 | 2 | 2 | -1 | -2 | -2 | -1 | 0 | 0 | 0 | 0 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | symplectic lifted from Dic3, Schur index 2 |
| ρ11 | 2 | -2 | -1 | -2i | 2i | 1 | 0 | 0 | 0 | 0 | -i | i | 2 | 2 | 2 | -2 | -2 | -2 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | complex lifted from C3⋊C8 |
| ρ12 | 2 | -2 | -1 | 2i | -2i | 1 | 0 | 0 | 0 | 0 | i | -i | 2 | 2 | 2 | -2 | -2 | -2 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | complex lifted from C3⋊C8 |
| ρ13 | 4 | 4 | 4 | 0 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | ζ139+ζ137+ζ136+ζ134 | ζ1312+ζ138+ζ135+ζ13 | ζ1311+ζ1310+ζ133+ζ132 | ζ1311+ζ1310+ζ133+ζ132 | ζ139+ζ137+ζ136+ζ134 | ζ1312+ζ138+ζ135+ζ13 | ζ1312+ζ138+ζ135+ζ13 | ζ1311+ζ1310+ζ133+ζ132 | ζ1312+ζ138+ζ135+ζ13 | ζ1311+ζ1310+ζ133+ζ132 | ζ139+ζ137+ζ136+ζ134 | ζ139+ζ137+ζ136+ζ134 | 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 |
| ρ14 | 4 | 4 | 4 | 0 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | ζ1311+ζ1310+ζ133+ζ132 | ζ139+ζ137+ζ136+ζ134 | ζ1312+ζ138+ζ135+ζ13 | ζ1312+ζ138+ζ135+ζ13 | ζ1311+ζ1310+ζ133+ζ132 | ζ139+ζ137+ζ136+ζ134 | ζ139+ζ137+ζ136+ζ134 | ζ1312+ζ138+ζ135+ζ13 | ζ139+ζ137+ζ136+ζ134 | ζ1312+ζ138+ζ135+ζ13 | ζ1311+ζ1310+ζ133+ζ132 | ζ1311+ζ1310+ζ133+ζ132 | 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 |
| ρ15 | 4 | 4 | 4 | 0 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | ζ1312+ζ138+ζ135+ζ13 | ζ1311+ζ1310+ζ133+ζ132 | ζ139+ζ137+ζ136+ζ134 | ζ139+ζ137+ζ136+ζ134 | ζ1312+ζ138+ζ135+ζ13 | ζ1311+ζ1310+ζ133+ζ132 | ζ1311+ζ1310+ζ133+ζ132 | ζ139+ζ137+ζ136+ζ134 | ζ1311+ζ1310+ζ133+ζ132 | ζ139+ζ137+ζ136+ζ134 | ζ1312+ζ138+ζ135+ζ13 | ζ1312+ζ138+ζ135+ζ13 | 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 |
| ρ16 | 4 | -4 | 4 | 0 | 0 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | ζ1311+ζ1310+ζ133+ζ132 | ζ139+ζ137+ζ136+ζ134 | ζ1312+ζ138+ζ135+ζ13 | -ζ1312-ζ138-ζ135-ζ13 | -ζ1311-ζ1310-ζ133-ζ132 | -ζ139-ζ137-ζ136-ζ134 | ζ139+ζ137+ζ136+ζ134 | ζ1312+ζ138+ζ135+ζ13 | ζ139+ζ137+ζ136+ζ134 | ζ1312+ζ138+ζ135+ζ13 | ζ1311+ζ1310+ζ133+ζ132 | ζ1311+ζ1310+ζ133+ζ132 | -ζ139-ζ137-ζ136-ζ134 | -ζ1312-ζ138-ζ135-ζ13 | -ζ1311-ζ1310-ζ133-ζ132 | -ζ1312-ζ138-ζ135-ζ13 | -ζ1311-ζ1310-ζ133-ζ132 | -ζ139-ζ137-ζ136-ζ134 | symplectic lifted from C13⋊C8, Schur index 2 |
| ρ17 | 4 | -4 | 4 | 0 | 0 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | ζ139+ζ137+ζ136+ζ134 | ζ1312+ζ138+ζ135+ζ13 | ζ1311+ζ1310+ζ133+ζ132 | -ζ1311-ζ1310-ζ133-ζ132 | -ζ139-ζ137-ζ136-ζ134 | -ζ1312-ζ138-ζ135-ζ13 | ζ1312+ζ138+ζ135+ζ13 | ζ1311+ζ1310+ζ133+ζ132 | ζ1312+ζ138+ζ135+ζ13 | ζ1311+ζ1310+ζ133+ζ132 | ζ139+ζ137+ζ136+ζ134 | ζ139+ζ137+ζ136+ζ134 | -ζ1312-ζ138-ζ135-ζ13 | -ζ1311-ζ1310-ζ133-ζ132 | -ζ139-ζ137-ζ136-ζ134 | -ζ1311-ζ1310-ζ133-ζ132 | -ζ139-ζ137-ζ136-ζ134 | -ζ1312-ζ138-ζ135-ζ13 | symplectic lifted from C13⋊C8, Schur index 2 |
| ρ18 | 4 | -4 | 4 | 0 | 0 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | ζ1312+ζ138+ζ135+ζ13 | ζ1311+ζ1310+ζ133+ζ132 | ζ139+ζ137+ζ136+ζ134 | -ζ139-ζ137-ζ136-ζ134 | -ζ1312-ζ138-ζ135-ζ13 | -ζ1311-ζ1310-ζ133-ζ132 | ζ1311+ζ1310+ζ133+ζ132 | ζ139+ζ137+ζ136+ζ134 | ζ1311+ζ1310+ζ133+ζ132 | ζ139+ζ137+ζ136+ζ134 | ζ1312+ζ138+ζ135+ζ13 | ζ1312+ζ138+ζ135+ζ13 | -ζ1311-ζ1310-ζ133-ζ132 | -ζ139-ζ137-ζ136-ζ134 | -ζ1312-ζ138-ζ135-ζ13 | -ζ139-ζ137-ζ136-ζ134 | -ζ1312-ζ138-ζ135-ζ13 | -ζ1311-ζ1310-ζ133-ζ132 | symplectic lifted from C13⋊C8, Schur index 2 |
| ρ19 | 4 | 4 | -2 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | ζ1311+ζ1310+ζ133+ζ132 | ζ139+ζ137+ζ136+ζ134 | ζ1312+ζ138+ζ135+ζ13 | ζ1312+ζ138+ζ135+ζ13 | ζ1311+ζ1310+ζ133+ζ132 | ζ139+ζ137+ζ136+ζ134 | ζ3ζ139-ζ3ζ137-ζ3ζ136+ζ3ζ134-ζ137-ζ136 | -ζ32ζ1312+ζ32ζ138+ζ32ζ135-ζ32ζ13-ζ1312-ζ13 | ζ32ζ139-ζ32ζ137-ζ32ζ136+ζ32ζ134-ζ137-ζ136 | ζ32ζ1312-ζ32ζ138-ζ32ζ135+ζ32ζ13-ζ138-ζ135 | -ζ3ζ1311+ζ3ζ1310+ζ3ζ133-ζ3ζ132-ζ1311-ζ132 | ζ3ζ1311-ζ3ζ1310-ζ3ζ133+ζ3ζ132-ζ1310-ζ133 | ζ32ζ139-ζ32ζ137-ζ32ζ136+ζ32ζ134-ζ137-ζ136 | ζ32ζ1312-ζ32ζ138-ζ32ζ135+ζ32ζ13-ζ138-ζ135 | -ζ3ζ1311+ζ3ζ1310+ζ3ζ133-ζ3ζ132-ζ1311-ζ132 | -ζ32ζ1312+ζ32ζ138+ζ32ζ135-ζ32ζ13-ζ1312-ζ13 | ζ3ζ1311-ζ3ζ1310-ζ3ζ133+ζ3ζ132-ζ1310-ζ133 | ζ3ζ139-ζ3ζ137-ζ3ζ136+ζ3ζ134-ζ137-ζ136 | complex lifted from C39⋊C4 |
| ρ20 | 4 | 4 | -2 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | ζ1312+ζ138+ζ135+ζ13 | ζ1311+ζ1310+ζ133+ζ132 | ζ139+ζ137+ζ136+ζ134 | ζ139+ζ137+ζ136+ζ134 | ζ1312+ζ138+ζ135+ζ13 | ζ1311+ζ1310+ζ133+ζ132 | ζ3ζ1311-ζ3ζ1310-ζ3ζ133+ζ3ζ132-ζ1310-ζ133 | ζ32ζ139-ζ32ζ137-ζ32ζ136+ζ32ζ134-ζ137-ζ136 | -ζ3ζ1311+ζ3ζ1310+ζ3ζ133-ζ3ζ132-ζ1311-ζ132 | ζ3ζ139-ζ3ζ137-ζ3ζ136+ζ3ζ134-ζ137-ζ136 | ζ32ζ1312-ζ32ζ138-ζ32ζ135+ζ32ζ13-ζ138-ζ135 | -ζ32ζ1312+ζ32ζ138+ζ32ζ135-ζ32ζ13-ζ1312-ζ13 | -ζ3ζ1311+ζ3ζ1310+ζ3ζ133-ζ3ζ132-ζ1311-ζ132 | ζ3ζ139-ζ3ζ137-ζ3ζ136+ζ3ζ134-ζ137-ζ136 | ζ32ζ1312-ζ32ζ138-ζ32ζ135+ζ32ζ13-ζ138-ζ135 | ζ32ζ139-ζ32ζ137-ζ32ζ136+ζ32ζ134-ζ137-ζ136 | -ζ32ζ1312+ζ32ζ138+ζ32ζ135-ζ32ζ13-ζ1312-ζ13 | ζ3ζ1311-ζ3ζ1310-ζ3ζ133+ζ3ζ132-ζ1310-ζ133 | complex lifted from C39⋊C4 |
| ρ21 | 4 | -4 | -2 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | ζ1312+ζ138+ζ135+ζ13 | ζ1311+ζ1310+ζ133+ζ132 | ζ139+ζ137+ζ136+ζ134 | -ζ139-ζ137-ζ136-ζ134 | -ζ1312-ζ138-ζ135-ζ13 | -ζ1311-ζ1310-ζ133-ζ132 | -ζ3ζ1311+ζ3ζ1310+ζ3ζ133-ζ3ζ132-ζ1311-ζ132 | ζ3ζ139-ζ3ζ137-ζ3ζ136+ζ3ζ134-ζ137-ζ136 | ζ3ζ1311-ζ3ζ1310-ζ3ζ133+ζ3ζ132-ζ1310-ζ133 | ζ32ζ139-ζ32ζ137-ζ32ζ136+ζ32ζ134-ζ137-ζ136 | -ζ32ζ1312+ζ32ζ138+ζ32ζ135-ζ32ζ13-ζ1312-ζ13 | ζ32ζ1312-ζ32ζ138-ζ32ζ135+ζ32ζ13-ζ138-ζ135 | -ζ3ζ1311+ζ3ζ1310+ζ3ζ133-ζ3ζ132+ζ1310+ζ133 | -ζ32ζ139+ζ32ζ137+ζ32ζ136-ζ32ζ134+ζ137+ζ136 | -ζ3ζ1312+ζ3ζ138+ζ3ζ135-ζ3ζ13+ζ138+ζ135 | ζ32ζ139-ζ32ζ137-ζ32ζ136+ζ32ζ134+ζ139+ζ134 | ζ3ζ1312-ζ3ζ138-ζ3ζ135+ζ3ζ13+ζ1312+ζ13 | -ζ32ζ1311+ζ32ζ1310+ζ32ζ133-ζ32ζ132+ζ1310+ζ133 | complex faithful |
| ρ22 | 4 | -4 | -2 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | ζ1312+ζ138+ζ135+ζ13 | ζ1311+ζ1310+ζ133+ζ132 | ζ139+ζ137+ζ136+ζ134 | -ζ139-ζ137-ζ136-ζ134 | -ζ1312-ζ138-ζ135-ζ13 | -ζ1311-ζ1310-ζ133-ζ132 | ζ3ζ1311-ζ3ζ1310-ζ3ζ133+ζ3ζ132-ζ1310-ζ133 | ζ32ζ139-ζ32ζ137-ζ32ζ136+ζ32ζ134-ζ137-ζ136 | -ζ3ζ1311+ζ3ζ1310+ζ3ζ133-ζ3ζ132-ζ1311-ζ132 | ζ3ζ139-ζ3ζ137-ζ3ζ136+ζ3ζ134-ζ137-ζ136 | ζ32ζ1312-ζ32ζ138-ζ32ζ135+ζ32ζ13-ζ138-ζ135 | -ζ32ζ1312+ζ32ζ138+ζ32ζ135-ζ32ζ13-ζ1312-ζ13 | -ζ32ζ1311+ζ32ζ1310+ζ32ζ133-ζ32ζ132+ζ1310+ζ133 | ζ32ζ139-ζ32ζ137-ζ32ζ136+ζ32ζ134+ζ139+ζ134 | ζ3ζ1312-ζ3ζ138-ζ3ζ135+ζ3ζ13+ζ1312+ζ13 | -ζ32ζ139+ζ32ζ137+ζ32ζ136-ζ32ζ134+ζ137+ζ136 | -ζ3ζ1312+ζ3ζ138+ζ3ζ135-ζ3ζ13+ζ138+ζ135 | -ζ3ζ1311+ζ3ζ1310+ζ3ζ133-ζ3ζ132+ζ1310+ζ133 | complex faithful |
| ρ23 | 4 | 4 | -2 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | ζ139+ζ137+ζ136+ζ134 | ζ1312+ζ138+ζ135+ζ13 | ζ1311+ζ1310+ζ133+ζ132 | ζ1311+ζ1310+ζ133+ζ132 | ζ139+ζ137+ζ136+ζ134 | ζ1312+ζ138+ζ135+ζ13 | -ζ32ζ1312+ζ32ζ138+ζ32ζ135-ζ32ζ13-ζ1312-ζ13 | -ζ3ζ1311+ζ3ζ1310+ζ3ζ133-ζ3ζ132-ζ1311-ζ132 | ζ32ζ1312-ζ32ζ138-ζ32ζ135+ζ32ζ13-ζ138-ζ135 | ζ3ζ1311-ζ3ζ1310-ζ3ζ133+ζ3ζ132-ζ1310-ζ133 | ζ3ζ139-ζ3ζ137-ζ3ζ136+ζ3ζ134-ζ137-ζ136 | ζ32ζ139-ζ32ζ137-ζ32ζ136+ζ32ζ134-ζ137-ζ136 | ζ32ζ1312-ζ32ζ138-ζ32ζ135+ζ32ζ13-ζ138-ζ135 | ζ3ζ1311-ζ3ζ1310-ζ3ζ133+ζ3ζ132-ζ1310-ζ133 | ζ3ζ139-ζ3ζ137-ζ3ζ136+ζ3ζ134-ζ137-ζ136 | -ζ3ζ1311+ζ3ζ1310+ζ3ζ133-ζ3ζ132-ζ1311-ζ132 | ζ32ζ139-ζ32ζ137-ζ32ζ136+ζ32ζ134-ζ137-ζ136 | -ζ32ζ1312+ζ32ζ138+ζ32ζ135-ζ32ζ13-ζ1312-ζ13 | complex lifted from C39⋊C4 |
| ρ24 | 4 | -4 | -2 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | ζ1311+ζ1310+ζ133+ζ132 | ζ139+ζ137+ζ136+ζ134 | ζ1312+ζ138+ζ135+ζ13 | -ζ1312-ζ138-ζ135-ζ13 | -ζ1311-ζ1310-ζ133-ζ132 | -ζ139-ζ137-ζ136-ζ134 | ζ3ζ139-ζ3ζ137-ζ3ζ136+ζ3ζ134-ζ137-ζ136 | -ζ32ζ1312+ζ32ζ138+ζ32ζ135-ζ32ζ13-ζ1312-ζ13 | ζ32ζ139-ζ32ζ137-ζ32ζ136+ζ32ζ134-ζ137-ζ136 | ζ32ζ1312-ζ32ζ138-ζ32ζ135+ζ32ζ13-ζ138-ζ135 | -ζ3ζ1311+ζ3ζ1310+ζ3ζ133-ζ3ζ132-ζ1311-ζ132 | ζ3ζ1311-ζ3ζ1310-ζ3ζ133+ζ3ζ132-ζ1310-ζ133 | -ζ32ζ139+ζ32ζ137+ζ32ζ136-ζ32ζ134+ζ137+ζ136 | ζ3ζ1312-ζ3ζ138-ζ3ζ135+ζ3ζ13+ζ1312+ζ13 | -ζ32ζ1311+ζ32ζ1310+ζ32ζ133-ζ32ζ132+ζ1310+ζ133 | -ζ3ζ1312+ζ3ζ138+ζ3ζ135-ζ3ζ13+ζ138+ζ135 | -ζ3ζ1311+ζ3ζ1310+ζ3ζ133-ζ3ζ132+ζ1310+ζ133 | ζ32ζ139-ζ32ζ137-ζ32ζ136+ζ32ζ134+ζ139+ζ134 | complex faithful |
| ρ25 | 4 | 4 | -2 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | ζ1311+ζ1310+ζ133+ζ132 | ζ139+ζ137+ζ136+ζ134 | ζ1312+ζ138+ζ135+ζ13 | ζ1312+ζ138+ζ135+ζ13 | ζ1311+ζ1310+ζ133+ζ132 | ζ139+ζ137+ζ136+ζ134 | ζ32ζ139-ζ32ζ137-ζ32ζ136+ζ32ζ134-ζ137-ζ136 | ζ32ζ1312-ζ32ζ138-ζ32ζ135+ζ32ζ13-ζ138-ζ135 | ζ3ζ139-ζ3ζ137-ζ3ζ136+ζ3ζ134-ζ137-ζ136 | -ζ32ζ1312+ζ32ζ138+ζ32ζ135-ζ32ζ13-ζ1312-ζ13 | ζ3ζ1311-ζ3ζ1310-ζ3ζ133+ζ3ζ132-ζ1310-ζ133 | -ζ3ζ1311+ζ3ζ1310+ζ3ζ133-ζ3ζ132-ζ1311-ζ132 | ζ3ζ139-ζ3ζ137-ζ3ζ136+ζ3ζ134-ζ137-ζ136 | -ζ32ζ1312+ζ32ζ138+ζ32ζ135-ζ32ζ13-ζ1312-ζ13 | ζ3ζ1311-ζ3ζ1310-ζ3ζ133+ζ3ζ132-ζ1310-ζ133 | ζ32ζ1312-ζ32ζ138-ζ32ζ135+ζ32ζ13-ζ138-ζ135 | -ζ3ζ1311+ζ3ζ1310+ζ3ζ133-ζ3ζ132-ζ1311-ζ132 | ζ32ζ139-ζ32ζ137-ζ32ζ136+ζ32ζ134-ζ137-ζ136 | complex lifted from C39⋊C4 |
| ρ26 | 4 | 4 | -2 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | ζ139+ζ137+ζ136+ζ134 | ζ1312+ζ138+ζ135+ζ13 | ζ1311+ζ1310+ζ133+ζ132 | ζ1311+ζ1310+ζ133+ζ132 | ζ139+ζ137+ζ136+ζ134 | ζ1312+ζ138+ζ135+ζ13 | ζ32ζ1312-ζ32ζ138-ζ32ζ135+ζ32ζ13-ζ138-ζ135 | ζ3ζ1311-ζ3ζ1310-ζ3ζ133+ζ3ζ132-ζ1310-ζ133 | -ζ32ζ1312+ζ32ζ138+ζ32ζ135-ζ32ζ13-ζ1312-ζ13 | -ζ3ζ1311+ζ3ζ1310+ζ3ζ133-ζ3ζ132-ζ1311-ζ132 | ζ32ζ139-ζ32ζ137-ζ32ζ136+ζ32ζ134-ζ137-ζ136 | ζ3ζ139-ζ3ζ137-ζ3ζ136+ζ3ζ134-ζ137-ζ136 | -ζ32ζ1312+ζ32ζ138+ζ32ζ135-ζ32ζ13-ζ1312-ζ13 | -ζ3ζ1311+ζ3ζ1310+ζ3ζ133-ζ3ζ132-ζ1311-ζ132 | ζ32ζ139-ζ32ζ137-ζ32ζ136+ζ32ζ134-ζ137-ζ136 | ζ3ζ1311-ζ3ζ1310-ζ3ζ133+ζ3ζ132-ζ1310-ζ133 | ζ3ζ139-ζ3ζ137-ζ3ζ136+ζ3ζ134-ζ137-ζ136 | ζ32ζ1312-ζ32ζ138-ζ32ζ135+ζ32ζ13-ζ138-ζ135 | complex lifted from C39⋊C4 |
| ρ27 | 4 | 4 | -2 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | ζ1312+ζ138+ζ135+ζ13 | ζ1311+ζ1310+ζ133+ζ132 | ζ139+ζ137+ζ136+ζ134 | ζ139+ζ137+ζ136+ζ134 | ζ1312+ζ138+ζ135+ζ13 | ζ1311+ζ1310+ζ133+ζ132 | -ζ3ζ1311+ζ3ζ1310+ζ3ζ133-ζ3ζ132-ζ1311-ζ132 | ζ3ζ139-ζ3ζ137-ζ3ζ136+ζ3ζ134-ζ137-ζ136 | ζ3ζ1311-ζ3ζ1310-ζ3ζ133+ζ3ζ132-ζ1310-ζ133 | ζ32ζ139-ζ32ζ137-ζ32ζ136+ζ32ζ134-ζ137-ζ136 | -ζ32ζ1312+ζ32ζ138+ζ32ζ135-ζ32ζ13-ζ1312-ζ13 | ζ32ζ1312-ζ32ζ138-ζ32ζ135+ζ32ζ13-ζ138-ζ135 | ζ3ζ1311-ζ3ζ1310-ζ3ζ133+ζ3ζ132-ζ1310-ζ133 | ζ32ζ139-ζ32ζ137-ζ32ζ136+ζ32ζ134-ζ137-ζ136 | -ζ32ζ1312+ζ32ζ138+ζ32ζ135-ζ32ζ13-ζ1312-ζ13 | ζ3ζ139-ζ3ζ137-ζ3ζ136+ζ3ζ134-ζ137-ζ136 | ζ32ζ1312-ζ32ζ138-ζ32ζ135+ζ32ζ13-ζ138-ζ135 | -ζ3ζ1311+ζ3ζ1310+ζ3ζ133-ζ3ζ132-ζ1311-ζ132 | complex lifted from C39⋊C4 |
| ρ28 | 4 | -4 | -2 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | ζ1311+ζ1310+ζ133+ζ132 | ζ139+ζ137+ζ136+ζ134 | ζ1312+ζ138+ζ135+ζ13 | -ζ1312-ζ138-ζ135-ζ13 | -ζ1311-ζ1310-ζ133-ζ132 | -ζ139-ζ137-ζ136-ζ134 | ζ32ζ139-ζ32ζ137-ζ32ζ136+ζ32ζ134-ζ137-ζ136 | ζ32ζ1312-ζ32ζ138-ζ32ζ135+ζ32ζ13-ζ138-ζ135 | ζ3ζ139-ζ3ζ137-ζ3ζ136+ζ3ζ134-ζ137-ζ136 | -ζ32ζ1312+ζ32ζ138+ζ32ζ135-ζ32ζ13-ζ1312-ζ13 | ζ3ζ1311-ζ3ζ1310-ζ3ζ133+ζ3ζ132-ζ1310-ζ133 | -ζ3ζ1311+ζ3ζ1310+ζ3ζ133-ζ3ζ132-ζ1311-ζ132 | ζ32ζ139-ζ32ζ137-ζ32ζ136+ζ32ζ134+ζ139+ζ134 | -ζ3ζ1312+ζ3ζ138+ζ3ζ135-ζ3ζ13+ζ138+ζ135 | -ζ3ζ1311+ζ3ζ1310+ζ3ζ133-ζ3ζ132+ζ1310+ζ133 | ζ3ζ1312-ζ3ζ138-ζ3ζ135+ζ3ζ13+ζ1312+ζ13 | -ζ32ζ1311+ζ32ζ1310+ζ32ζ133-ζ32ζ132+ζ1310+ζ133 | -ζ32ζ139+ζ32ζ137+ζ32ζ136-ζ32ζ134+ζ137+ζ136 | complex faithful |
| ρ29 | 4 | -4 | -2 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | ζ139+ζ137+ζ136+ζ134 | ζ1312+ζ138+ζ135+ζ13 | ζ1311+ζ1310+ζ133+ζ132 | -ζ1311-ζ1310-ζ133-ζ132 | -ζ139-ζ137-ζ136-ζ134 | -ζ1312-ζ138-ζ135-ζ13 | ζ32ζ1312-ζ32ζ138-ζ32ζ135+ζ32ζ13-ζ138-ζ135 | ζ3ζ1311-ζ3ζ1310-ζ3ζ133+ζ3ζ132-ζ1310-ζ133 | -ζ32ζ1312+ζ32ζ138+ζ32ζ135-ζ32ζ13-ζ1312-ζ13 | -ζ3ζ1311+ζ3ζ1310+ζ3ζ133-ζ3ζ132-ζ1311-ζ132 | ζ32ζ139-ζ32ζ137-ζ32ζ136+ζ32ζ134-ζ137-ζ136 | ζ3ζ139-ζ3ζ137-ζ3ζ136+ζ3ζ134-ζ137-ζ136 | -ζ3ζ1312+ζ3ζ138+ζ3ζ135-ζ3ζ13+ζ138+ζ135 | -ζ32ζ1311+ζ32ζ1310+ζ32ζ133-ζ32ζ132+ζ1310+ζ133 | -ζ32ζ139+ζ32ζ137+ζ32ζ136-ζ32ζ134+ζ137+ζ136 | -ζ3ζ1311+ζ3ζ1310+ζ3ζ133-ζ3ζ132+ζ1310+ζ133 | ζ32ζ139-ζ32ζ137-ζ32ζ136+ζ32ζ134+ζ139+ζ134 | ζ3ζ1312-ζ3ζ138-ζ3ζ135+ζ3ζ13+ζ1312+ζ13 | complex faithful |
| ρ30 | 4 | -4 | -2 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | ζ139+ζ137+ζ136+ζ134 | ζ1312+ζ138+ζ135+ζ13 | ζ1311+ζ1310+ζ133+ζ132 | -ζ1311-ζ1310-ζ133-ζ132 | -ζ139-ζ137-ζ136-ζ134 | -ζ1312-ζ138-ζ135-ζ13 | -ζ32ζ1312+ζ32ζ138+ζ32ζ135-ζ32ζ13-ζ1312-ζ13 | -ζ3ζ1311+ζ3ζ1310+ζ3ζ133-ζ3ζ132-ζ1311-ζ132 | ζ32ζ1312-ζ32ζ138-ζ32ζ135+ζ32ζ13-ζ138-ζ135 | ζ3ζ1311-ζ3ζ1310-ζ3ζ133+ζ3ζ132-ζ1310-ζ133 | ζ3ζ139-ζ3ζ137-ζ3ζ136+ζ3ζ134-ζ137-ζ136 | ζ32ζ139-ζ32ζ137-ζ32ζ136+ζ32ζ134-ζ137-ζ136 | ζ3ζ1312-ζ3ζ138-ζ3ζ135+ζ3ζ13+ζ1312+ζ13 | -ζ3ζ1311+ζ3ζ1310+ζ3ζ133-ζ3ζ132+ζ1310+ζ133 | ζ32ζ139-ζ32ζ137-ζ32ζ136+ζ32ζ134+ζ139+ζ134 | -ζ32ζ1311+ζ32ζ1310+ζ32ζ133-ζ32ζ132+ζ1310+ζ133 | -ζ32ζ139+ζ32ζ137+ζ32ζ136-ζ32ζ134+ζ137+ζ136 | -ζ3ζ1312+ζ3ζ138+ζ3ζ135-ζ3ζ13+ζ138+ζ135 | complex faithful |
►Regular action on 312 points
Generators in S312
(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 66 67 68 69 70 71 72 73 74 75 76 77 78)(79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117)(118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156)(157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195)(196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234)(235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273)(274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312)
(1 274 133 226 52 253 115 190)(2 279 119 234 53 258 101 159)(3 284 144 203 54 263 87 167)(4 289 130 211 55 268 112 175)(5 294 155 219 56 273 98 183)(6 299 141 227 57 239 84 191)(7 304 127 196 58 244 109 160)(8 309 152 204 59 249 95 168)(9 275 138 212 60 254 81 176)(10 280 124 220 61 259 106 184)(11 285 149 228 62 264 92 192)(12 290 135 197 63 269 117 161)(13 295 121 205 64 235 103 169)(14 300 146 213 65 240 89 177)(15 305 132 221 66 245 114 185)(16 310 118 229 67 250 100 193)(17 276 143 198 68 255 86 162)(18 281 129 206 69 260 111 170)(19 286 154 214 70 265 97 178)(20 291 140 222 71 270 83 186)(21 296 126 230 72 236 108 194)(22 301 151 199 73 241 94 163)(23 306 137 207 74 246 80 171)(24 311 123 215 75 251 105 179)(25 277 148 223 76 256 91 187)(26 282 134 231 77 261 116 195)(27 287 120 200 78 266 102 164)(28 292 145 208 40 271 88 172)(29 297 131 216 41 237 113 180)(30 302 156 224 42 242 99 188)(31 307 142 232 43 247 85 157)(32 312 128 201 44 252 110 165)(33 278 153 209 45 257 96 173)(34 283 139 217 46 262 82 181)(35 288 125 225 47 267 107 189)(36 293 150 233 48 272 93 158)(37 298 136 202 49 238 79 166)(38 303 122 210 50 243 104 174)(39 308 147 218 51 248 90 182)
G:=sub<Sym(312)| (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)(40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78)(79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117)(118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156)(157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195)(196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234)(235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273)(274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312), (1,274,133,226,52,253,115,190)(2,279,119,234,53,258,101,159)(3,284,144,203,54,263,87,167)(4,289,130,211,55,268,112,175)(5,294,155,219,56,273,98,183)(6,299,141,227,57,239,84,191)(7,304,127,196,58,244,109,160)(8,309,152,204,59,249,95,168)(9,275,138,212,60,254,81,176)(10,280,124,220,61,259,106,184)(11,285,149,228,62,264,92,192)(12,290,135,197,63,269,117,161)(13,295,121,205,64,235,103,169)(14,300,146,213,65,240,89,177)(15,305,132,221,66,245,114,185)(16,310,118,229,67,250,100,193)(17,276,143,198,68,255,86,162)(18,281,129,206,69,260,111,170)(19,286,154,214,70,265,97,178)(20,291,140,222,71,270,83,186)(21,296,126,230,72,236,108,194)(22,301,151,199,73,241,94,163)(23,306,137,207,74,246,80,171)(24,311,123,215,75,251,105,179)(25,277,148,223,76,256,91,187)(26,282,134,231,77,261,116,195)(27,287,120,200,78,266,102,164)(28,292,145,208,40,271,88,172)(29,297,131,216,41,237,113,180)(30,302,156,224,42,242,99,188)(31,307,142,232,43,247,85,157)(32,312,128,201,44,252,110,165)(33,278,153,209,45,257,96,173)(34,283,139,217,46,262,82,181)(35,288,125,225,47,267,107,189)(36,293,150,233,48,272,93,158)(37,298,136,202,49,238,79,166)(38,303,122,210,50,243,104,174)(39,308,147,218,51,248,90,182)>;
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)(40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78)(79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117)(118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156)(157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195)(196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234)(235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273)(274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312), (1,274,133,226,52,253,115,190)(2,279,119,234,53,258,101,159)(3,284,144,203,54,263,87,167)(4,289,130,211,55,268,112,175)(5,294,155,219,56,273,98,183)(6,299,141,227,57,239,84,191)(7,304,127,196,58,244,109,160)(8,309,152,204,59,249,95,168)(9,275,138,212,60,254,81,176)(10,280,124,220,61,259,106,184)(11,285,149,228,62,264,92,192)(12,290,135,197,63,269,117,161)(13,295,121,205,64,235,103,169)(14,300,146,213,65,240,89,177)(15,305,132,221,66,245,114,185)(16,310,118,229,67,250,100,193)(17,276,143,198,68,255,86,162)(18,281,129,206,69,260,111,170)(19,286,154,214,70,265,97,178)(20,291,140,222,71,270,83,186)(21,296,126,230,72,236,108,194)(22,301,151,199,73,241,94,163)(23,306,137,207,74,246,80,171)(24,311,123,215,75,251,105,179)(25,277,148,223,76,256,91,187)(26,282,134,231,77,261,116,195)(27,287,120,200,78,266,102,164)(28,292,145,208,40,271,88,172)(29,297,131,216,41,237,113,180)(30,302,156,224,42,242,99,188)(31,307,142,232,43,247,85,157)(32,312,128,201,44,252,110,165)(33,278,153,209,45,257,96,173)(34,283,139,217,46,262,82,181)(35,288,125,225,47,267,107,189)(36,293,150,233,48,272,93,158)(37,298,136,202,49,238,79,166)(38,303,122,210,50,243,104,174)(39,308,147,218,51,248,90,182) );
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),(40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78),(79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117),(118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156),(157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195),(196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234),(235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273),(274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312)], [(1,274,133,226,52,253,115,190),(2,279,119,234,53,258,101,159),(3,284,144,203,54,263,87,167),(4,289,130,211,55,268,112,175),(5,294,155,219,56,273,98,183),(6,299,141,227,57,239,84,191),(7,304,127,196,58,244,109,160),(8,309,152,204,59,249,95,168),(9,275,138,212,60,254,81,176),(10,280,124,220,61,259,106,184),(11,285,149,228,62,264,92,192),(12,290,135,197,63,269,117,161),(13,295,121,205,64,235,103,169),(14,300,146,213,65,240,89,177),(15,305,132,221,66,245,114,185),(16,310,118,229,67,250,100,193),(17,276,143,198,68,255,86,162),(18,281,129,206,69,260,111,170),(19,286,154,214,70,265,97,178),(20,291,140,222,71,270,83,186),(21,296,126,230,72,236,108,194),(22,301,151,199,73,241,94,163),(23,306,137,207,74,246,80,171),(24,311,123,215,75,251,105,179),(25,277,148,223,76,256,91,187),(26,282,134,231,77,261,116,195),(27,287,120,200,78,266,102,164),(28,292,145,208,40,271,88,172),(29,297,131,216,41,237,113,180),(30,302,156,224,42,242,99,188),(31,307,142,232,43,247,85,157),(32,312,128,201,44,252,110,165),(33,278,153,209,45,257,96,173),(34,283,139,217,46,262,82,181),(35,288,125,225,47,267,107,189),(36,293,150,233,48,272,93,158),(37,298,136,202,49,238,79,166),(38,303,122,210,50,243,104,174),(39,308,147,218,51,248,90,182)]])
Matrix representation of C39⋊C8 ►in GL4(𝔽5) generated by
| 4 | 0 | 0 | 4 |
| 4 | 1 | 3 | 2 |
| 2 | 1 | 1 | 3 |
| 0 | 4 | 4 | 2 |
| 1 | 4 | 0 | 0 |
| 1 | 0 | 0 | 4 |
| 4 | 0 | 4 | 0 |
| 1 | 0 | 2 | 0 |
G:=sub<GL(4,GF(5))| [4,4,2,0,0,1,1,4,0,3,1,4,4,2,3,2],[1,1,4,1,4,0,0,0,0,0,4,2,0,4,0,0] >;
C39⋊C8 in GAP, Magma, Sage, TeX
C_{39}\rtimes C_8 % in TeX
G:=Group("C39:C8"); // GroupNames label
G:=SmallGroup(312,14);
// by ID
G=gap.SmallGroup(312,14);
# by ID
G:=PCGroup([5,-2,-2,-2,-3,-13,10,26,323,3004,3609]);
// Polycyclic
G:=Group<a,b|a^39=b^8=1,b*a*b^-1=a^8>;
// generators/relations
Export
Subgroup lattice of C39⋊C8 in TeX
Character table of C39⋊C8 in TeX