metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: D52, C4⋊D13, C13⋊1D4, C52⋊1C2, D26⋊1C2, C2.4D26, C26.3C22, sometimes denoted D104 or Dih52 or Dih104, SmallGroup(104,6)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D52
G = < a,b | a52=b2=1, bab=a-1 >
Character table of D52
| class | 1 | 2A | 2B | 2C | 4 | 13A | 13B | 13C | 13D | 13E | 13F | 26A | 26B | 26C | 26D | 26E | 26F | 52A | 52B | 52C | 52D | 52E | 52F | 52G | 52H | 52I | 52J | 52K | 52L | |
| size | 1 | 1 | 26 | 26 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | |
| 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 | -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 |
| ρ4 | 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 |
| ρ5 | 2 | -2 | 0 | 0 | 0 | 2 | 2 | 2 | 2 | 2 | 2 | -2 | -2 | -2 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ6 | 2 | 2 | 0 | 0 | 2 | ζ1311+ζ132 | ζ1310+ζ133 | ζ137+ζ136 | ζ139+ζ134 | ζ1312+ζ13 | ζ138+ζ135 | ζ1311+ζ132 | ζ139+ζ134 | ζ1310+ζ133 | ζ137+ζ136 | ζ1312+ζ13 | ζ138+ζ135 | ζ138+ζ135 | ζ1311+ζ132 | ζ139+ζ134 | ζ1310+ζ133 | ζ1310+ζ133 | ζ139+ζ134 | ζ1311+ζ132 | ζ138+ζ135 | ζ1312+ζ13 | ζ137+ζ136 | ζ137+ζ136 | ζ1312+ζ13 | orthogonal lifted from D13 |
| ρ7 | 2 | 2 | 0 | 0 | -2 | ζ1312+ζ13 | ζ138+ζ135 | ζ1310+ζ133 | ζ1311+ζ132 | ζ137+ζ136 | ζ139+ζ134 | ζ1312+ζ13 | ζ1311+ζ132 | ζ138+ζ135 | ζ1310+ζ133 | ζ137+ζ136 | ζ139+ζ134 | -ζ139-ζ134 | -ζ1312-ζ13 | -ζ1311-ζ132 | -ζ138-ζ135 | -ζ138-ζ135 | -ζ1311-ζ132 | -ζ1312-ζ13 | -ζ139-ζ134 | -ζ137-ζ136 | -ζ1310-ζ133 | -ζ1310-ζ133 | -ζ137-ζ136 | orthogonal lifted from D26 |
| ρ8 | 2 | 2 | 0 | 0 | 2 | ζ138+ζ135 | ζ1312+ζ13 | ζ1311+ζ132 | ζ1310+ζ133 | ζ139+ζ134 | ζ137+ζ136 | ζ138+ζ135 | ζ1310+ζ133 | ζ1312+ζ13 | ζ1311+ζ132 | ζ139+ζ134 | ζ137+ζ136 | ζ137+ζ136 | ζ138+ζ135 | ζ1310+ζ133 | ζ1312+ζ13 | ζ1312+ζ13 | ζ1310+ζ133 | ζ138+ζ135 | ζ137+ζ136 | ζ139+ζ134 | ζ1311+ζ132 | ζ1311+ζ132 | ζ139+ζ134 | orthogonal lifted from D13 |
| ρ9 | 2 | 2 | 0 | 0 | -2 | ζ1311+ζ132 | ζ1310+ζ133 | ζ137+ζ136 | ζ139+ζ134 | ζ1312+ζ13 | ζ138+ζ135 | ζ1311+ζ132 | ζ139+ζ134 | ζ1310+ζ133 | ζ137+ζ136 | ζ1312+ζ13 | ζ138+ζ135 | -ζ138-ζ135 | -ζ1311-ζ132 | -ζ139-ζ134 | -ζ1310-ζ133 | -ζ1310-ζ133 | -ζ139-ζ134 | -ζ1311-ζ132 | -ζ138-ζ135 | -ζ1312-ζ13 | -ζ137-ζ136 | -ζ137-ζ136 | -ζ1312-ζ13 | orthogonal lifted from D26 |
| ρ10 | 2 | 2 | 0 | 0 | -2 | ζ139+ζ134 | ζ137+ζ136 | ζ1312+ζ13 | ζ138+ζ135 | ζ1311+ζ132 | ζ1310+ζ133 | ζ139+ζ134 | ζ138+ζ135 | ζ137+ζ136 | ζ1312+ζ13 | ζ1311+ζ132 | ζ1310+ζ133 | -ζ1310-ζ133 | -ζ139-ζ134 | -ζ138-ζ135 | -ζ137-ζ136 | -ζ137-ζ136 | -ζ138-ζ135 | -ζ139-ζ134 | -ζ1310-ζ133 | -ζ1311-ζ132 | -ζ1312-ζ13 | -ζ1312-ζ13 | -ζ1311-ζ132 | orthogonal lifted from D26 |
| ρ11 | 2 | 2 | 0 | 0 | 2 | ζ1312+ζ13 | ζ138+ζ135 | ζ1310+ζ133 | ζ1311+ζ132 | ζ137+ζ136 | ζ139+ζ134 | ζ1312+ζ13 | ζ1311+ζ132 | ζ138+ζ135 | ζ1310+ζ133 | ζ137+ζ136 | ζ139+ζ134 | ζ139+ζ134 | ζ1312+ζ13 | ζ1311+ζ132 | ζ138+ζ135 | ζ138+ζ135 | ζ1311+ζ132 | ζ1312+ζ13 | ζ139+ζ134 | ζ137+ζ136 | ζ1310+ζ133 | ζ1310+ζ133 | ζ137+ζ136 | orthogonal lifted from D13 |
| ρ12 | 2 | 2 | 0 | 0 | 2 | ζ1310+ζ133 | ζ1311+ζ132 | ζ139+ζ134 | ζ137+ζ136 | ζ138+ζ135 | ζ1312+ζ13 | ζ1310+ζ133 | ζ137+ζ136 | ζ1311+ζ132 | ζ139+ζ134 | ζ138+ζ135 | ζ1312+ζ13 | ζ1312+ζ13 | ζ1310+ζ133 | ζ137+ζ136 | ζ1311+ζ132 | ζ1311+ζ132 | ζ137+ζ136 | ζ1310+ζ133 | ζ1312+ζ13 | ζ138+ζ135 | ζ139+ζ134 | ζ139+ζ134 | ζ138+ζ135 | orthogonal lifted from D13 |
| ρ13 | 2 | 2 | 0 | 0 | -2 | ζ138+ζ135 | ζ1312+ζ13 | ζ1311+ζ132 | ζ1310+ζ133 | ζ139+ζ134 | ζ137+ζ136 | ζ138+ζ135 | ζ1310+ζ133 | ζ1312+ζ13 | ζ1311+ζ132 | ζ139+ζ134 | ζ137+ζ136 | -ζ137-ζ136 | -ζ138-ζ135 | -ζ1310-ζ133 | -ζ1312-ζ13 | -ζ1312-ζ13 | -ζ1310-ζ133 | -ζ138-ζ135 | -ζ137-ζ136 | -ζ139-ζ134 | -ζ1311-ζ132 | -ζ1311-ζ132 | -ζ139-ζ134 | orthogonal lifted from D26 |
| ρ14 | 2 | 2 | 0 | 0 | 2 | ζ137+ζ136 | ζ139+ζ134 | ζ138+ζ135 | ζ1312+ζ13 | ζ1310+ζ133 | ζ1311+ζ132 | ζ137+ζ136 | ζ1312+ζ13 | ζ139+ζ134 | ζ138+ζ135 | ζ1310+ζ133 | ζ1311+ζ132 | ζ1311+ζ132 | ζ137+ζ136 | ζ1312+ζ13 | ζ139+ζ134 | ζ139+ζ134 | ζ1312+ζ13 | ζ137+ζ136 | ζ1311+ζ132 | ζ1310+ζ133 | ζ138+ζ135 | ζ138+ζ135 | ζ1310+ζ133 | orthogonal lifted from D13 |
| ρ15 | 2 | 2 | 0 | 0 | -2 | ζ1310+ζ133 | ζ1311+ζ132 | ζ139+ζ134 | ζ137+ζ136 | ζ138+ζ135 | ζ1312+ζ13 | ζ1310+ζ133 | ζ137+ζ136 | ζ1311+ζ132 | ζ139+ζ134 | ζ138+ζ135 | ζ1312+ζ13 | -ζ1312-ζ13 | -ζ1310-ζ133 | -ζ137-ζ136 | -ζ1311-ζ132 | -ζ1311-ζ132 | -ζ137-ζ136 | -ζ1310-ζ133 | -ζ1312-ζ13 | -ζ138-ζ135 | -ζ139-ζ134 | -ζ139-ζ134 | -ζ138-ζ135 | orthogonal lifted from D26 |
| ρ16 | 2 | 2 | 0 | 0 | -2 | ζ137+ζ136 | ζ139+ζ134 | ζ138+ζ135 | ζ1312+ζ13 | ζ1310+ζ133 | ζ1311+ζ132 | ζ137+ζ136 | ζ1312+ζ13 | ζ139+ζ134 | ζ138+ζ135 | ζ1310+ζ133 | ζ1311+ζ132 | -ζ1311-ζ132 | -ζ137-ζ136 | -ζ1312-ζ13 | -ζ139-ζ134 | -ζ139-ζ134 | -ζ1312-ζ13 | -ζ137-ζ136 | -ζ1311-ζ132 | -ζ1310-ζ133 | -ζ138-ζ135 | -ζ138-ζ135 | -ζ1310-ζ133 | orthogonal lifted from D26 |
| ρ17 | 2 | 2 | 0 | 0 | 2 | ζ139+ζ134 | ζ137+ζ136 | ζ1312+ζ13 | ζ138+ζ135 | ζ1311+ζ132 | ζ1310+ζ133 | ζ139+ζ134 | ζ138+ζ135 | ζ137+ζ136 | ζ1312+ζ13 | ζ1311+ζ132 | ζ1310+ζ133 | ζ1310+ζ133 | ζ139+ζ134 | ζ138+ζ135 | ζ137+ζ136 | ζ137+ζ136 | ζ138+ζ135 | ζ139+ζ134 | ζ1310+ζ133 | ζ1311+ζ132 | ζ1312+ζ13 | ζ1312+ζ13 | ζ1311+ζ132 | orthogonal lifted from D13 |
| ρ18 | 2 | -2 | 0 | 0 | 0 | ζ138+ζ135 | ζ1312+ζ13 | ζ1311+ζ132 | ζ1310+ζ133 | ζ139+ζ134 | ζ137+ζ136 | -ζ138-ζ135 | -ζ1310-ζ133 | -ζ1312-ζ13 | -ζ1311-ζ132 | -ζ139-ζ134 | -ζ137-ζ136 | -ζ4ζ137+ζ4ζ136 | -ζ43ζ138+ζ43ζ135 | -ζ43ζ1310+ζ43ζ133 | -ζ43ζ1312+ζ43ζ13 | ζ43ζ1312-ζ43ζ13 | ζ43ζ1310-ζ43ζ133 | ζ43ζ138-ζ43ζ135 | ζ4ζ137-ζ4ζ136 | ζ4ζ139-ζ4ζ134 | ζ4ζ1311-ζ4ζ132 | -ζ4ζ1311+ζ4ζ132 | -ζ4ζ139+ζ4ζ134 | orthogonal faithful |
| ρ19 | 2 | -2 | 0 | 0 | 0 | ζ138+ζ135 | ζ1312+ζ13 | ζ1311+ζ132 | ζ1310+ζ133 | ζ139+ζ134 | ζ137+ζ136 | -ζ138-ζ135 | -ζ1310-ζ133 | -ζ1312-ζ13 | -ζ1311-ζ132 | -ζ139-ζ134 | -ζ137-ζ136 | ζ4ζ137-ζ4ζ136 | ζ43ζ138-ζ43ζ135 | ζ43ζ1310-ζ43ζ133 | ζ43ζ1312-ζ43ζ13 | -ζ43ζ1312+ζ43ζ13 | -ζ43ζ1310+ζ43ζ133 | -ζ43ζ138+ζ43ζ135 | -ζ4ζ137+ζ4ζ136 | -ζ4ζ139+ζ4ζ134 | -ζ4ζ1311+ζ4ζ132 | ζ4ζ1311-ζ4ζ132 | ζ4ζ139-ζ4ζ134 | orthogonal faithful |
| ρ20 | 2 | -2 | 0 | 0 | 0 | ζ1310+ζ133 | ζ1311+ζ132 | ζ139+ζ134 | ζ137+ζ136 | ζ138+ζ135 | ζ1312+ζ13 | -ζ1310-ζ133 | -ζ137-ζ136 | -ζ1311-ζ132 | -ζ139-ζ134 | -ζ138-ζ135 | -ζ1312-ζ13 | ζ43ζ1312-ζ43ζ13 | -ζ43ζ1310+ζ43ζ133 | -ζ4ζ137+ζ4ζ136 | -ζ4ζ1311+ζ4ζ132 | ζ4ζ1311-ζ4ζ132 | ζ4ζ137-ζ4ζ136 | ζ43ζ1310-ζ43ζ133 | -ζ43ζ1312+ζ43ζ13 | -ζ43ζ138+ζ43ζ135 | -ζ4ζ139+ζ4ζ134 | ζ4ζ139-ζ4ζ134 | ζ43ζ138-ζ43ζ135 | orthogonal faithful |
| ρ21 | 2 | -2 | 0 | 0 | 0 | ζ1310+ζ133 | ζ1311+ζ132 | ζ139+ζ134 | ζ137+ζ136 | ζ138+ζ135 | ζ1312+ζ13 | -ζ1310-ζ133 | -ζ137-ζ136 | -ζ1311-ζ132 | -ζ139-ζ134 | -ζ138-ζ135 | -ζ1312-ζ13 | -ζ43ζ1312+ζ43ζ13 | ζ43ζ1310-ζ43ζ133 | ζ4ζ137-ζ4ζ136 | ζ4ζ1311-ζ4ζ132 | -ζ4ζ1311+ζ4ζ132 | -ζ4ζ137+ζ4ζ136 | -ζ43ζ1310+ζ43ζ133 | ζ43ζ1312-ζ43ζ13 | ζ43ζ138-ζ43ζ135 | ζ4ζ139-ζ4ζ134 | -ζ4ζ139+ζ4ζ134 | -ζ43ζ138+ζ43ζ135 | orthogonal faithful |
| ρ22 | 2 | -2 | 0 | 0 | 0 | ζ139+ζ134 | ζ137+ζ136 | ζ1312+ζ13 | ζ138+ζ135 | ζ1311+ζ132 | ζ1310+ζ133 | -ζ139-ζ134 | -ζ138-ζ135 | -ζ137-ζ136 | -ζ1312-ζ13 | -ζ1311-ζ132 | -ζ1310-ζ133 | ζ43ζ1310-ζ43ζ133 | -ζ4ζ139+ζ4ζ134 | ζ43ζ138-ζ43ζ135 | -ζ4ζ137+ζ4ζ136 | ζ4ζ137-ζ4ζ136 | -ζ43ζ138+ζ43ζ135 | ζ4ζ139-ζ4ζ134 | -ζ43ζ1310+ζ43ζ133 | ζ4ζ1311-ζ4ζ132 | -ζ43ζ1312+ζ43ζ13 | ζ43ζ1312-ζ43ζ13 | -ζ4ζ1311+ζ4ζ132 | orthogonal faithful |
| ρ23 | 2 | -2 | 0 | 0 | 0 | ζ1312+ζ13 | ζ138+ζ135 | ζ1310+ζ133 | ζ1311+ζ132 | ζ137+ζ136 | ζ139+ζ134 | -ζ1312-ζ13 | -ζ1311-ζ132 | -ζ138-ζ135 | -ζ1310-ζ133 | -ζ137-ζ136 | -ζ139-ζ134 | -ζ4ζ139+ζ4ζ134 | ζ43ζ1312-ζ43ζ13 | ζ4ζ1311-ζ4ζ132 | -ζ43ζ138+ζ43ζ135 | ζ43ζ138-ζ43ζ135 | -ζ4ζ1311+ζ4ζ132 | -ζ43ζ1312+ζ43ζ13 | ζ4ζ139-ζ4ζ134 | -ζ4ζ137+ζ4ζ136 | ζ43ζ1310-ζ43ζ133 | -ζ43ζ1310+ζ43ζ133 | ζ4ζ137-ζ4ζ136 | orthogonal faithful |
| ρ24 | 2 | -2 | 0 | 0 | 0 | ζ1311+ζ132 | ζ1310+ζ133 | ζ137+ζ136 | ζ139+ζ134 | ζ1312+ζ13 | ζ138+ζ135 | -ζ1311-ζ132 | -ζ139-ζ134 | -ζ1310-ζ133 | -ζ137-ζ136 | -ζ1312-ζ13 | -ζ138-ζ135 | ζ43ζ138-ζ43ζ135 | ζ4ζ1311-ζ4ζ132 | -ζ4ζ139+ζ4ζ134 | -ζ43ζ1310+ζ43ζ133 | ζ43ζ1310-ζ43ζ133 | ζ4ζ139-ζ4ζ134 | -ζ4ζ1311+ζ4ζ132 | -ζ43ζ138+ζ43ζ135 | ζ43ζ1312-ζ43ζ13 | ζ4ζ137-ζ4ζ136 | -ζ4ζ137+ζ4ζ136 | -ζ43ζ1312+ζ43ζ13 | orthogonal faithful |
| ρ25 | 2 | -2 | 0 | 0 | 0 | ζ1311+ζ132 | ζ1310+ζ133 | ζ137+ζ136 | ζ139+ζ134 | ζ1312+ζ13 | ζ138+ζ135 | -ζ1311-ζ132 | -ζ139-ζ134 | -ζ1310-ζ133 | -ζ137-ζ136 | -ζ1312-ζ13 | -ζ138-ζ135 | -ζ43ζ138+ζ43ζ135 | -ζ4ζ1311+ζ4ζ132 | ζ4ζ139-ζ4ζ134 | ζ43ζ1310-ζ43ζ133 | -ζ43ζ1310+ζ43ζ133 | -ζ4ζ139+ζ4ζ134 | ζ4ζ1311-ζ4ζ132 | ζ43ζ138-ζ43ζ135 | -ζ43ζ1312+ζ43ζ13 | -ζ4ζ137+ζ4ζ136 | ζ4ζ137-ζ4ζ136 | ζ43ζ1312-ζ43ζ13 | orthogonal faithful |
| ρ26 | 2 | -2 | 0 | 0 | 0 | ζ1312+ζ13 | ζ138+ζ135 | ζ1310+ζ133 | ζ1311+ζ132 | ζ137+ζ136 | ζ139+ζ134 | -ζ1312-ζ13 | -ζ1311-ζ132 | -ζ138-ζ135 | -ζ1310-ζ133 | -ζ137-ζ136 | -ζ139-ζ134 | ζ4ζ139-ζ4ζ134 | -ζ43ζ1312+ζ43ζ13 | -ζ4ζ1311+ζ4ζ132 | ζ43ζ138-ζ43ζ135 | -ζ43ζ138+ζ43ζ135 | ζ4ζ1311-ζ4ζ132 | ζ43ζ1312-ζ43ζ13 | -ζ4ζ139+ζ4ζ134 | ζ4ζ137-ζ4ζ136 | -ζ43ζ1310+ζ43ζ133 | ζ43ζ1310-ζ43ζ133 | -ζ4ζ137+ζ4ζ136 | orthogonal faithful |
| ρ27 | 2 | -2 | 0 | 0 | 0 | ζ137+ζ136 | ζ139+ζ134 | ζ138+ζ135 | ζ1312+ζ13 | ζ1310+ζ133 | ζ1311+ζ132 | -ζ137-ζ136 | -ζ1312-ζ13 | -ζ139-ζ134 | -ζ138-ζ135 | -ζ1310-ζ133 | -ζ1311-ζ132 | -ζ4ζ1311+ζ4ζ132 | ζ4ζ137-ζ4ζ136 | -ζ43ζ1312+ζ43ζ13 | -ζ4ζ139+ζ4ζ134 | ζ4ζ139-ζ4ζ134 | ζ43ζ1312-ζ43ζ13 | -ζ4ζ137+ζ4ζ136 | ζ4ζ1311-ζ4ζ132 | ζ43ζ1310-ζ43ζ133 | -ζ43ζ138+ζ43ζ135 | ζ43ζ138-ζ43ζ135 | -ζ43ζ1310+ζ43ζ133 | orthogonal faithful |
| ρ28 | 2 | -2 | 0 | 0 | 0 | ζ137+ζ136 | ζ139+ζ134 | ζ138+ζ135 | ζ1312+ζ13 | ζ1310+ζ133 | ζ1311+ζ132 | -ζ137-ζ136 | -ζ1312-ζ13 | -ζ139-ζ134 | -ζ138-ζ135 | -ζ1310-ζ133 | -ζ1311-ζ132 | ζ4ζ1311-ζ4ζ132 | -ζ4ζ137+ζ4ζ136 | ζ43ζ1312-ζ43ζ13 | ζ4ζ139-ζ4ζ134 | -ζ4ζ139+ζ4ζ134 | -ζ43ζ1312+ζ43ζ13 | ζ4ζ137-ζ4ζ136 | -ζ4ζ1311+ζ4ζ132 | -ζ43ζ1310+ζ43ζ133 | ζ43ζ138-ζ43ζ135 | -ζ43ζ138+ζ43ζ135 | ζ43ζ1310-ζ43ζ133 | orthogonal faithful |
| ρ29 | 2 | -2 | 0 | 0 | 0 | ζ139+ζ134 | ζ137+ζ136 | ζ1312+ζ13 | ζ138+ζ135 | ζ1311+ζ132 | ζ1310+ζ133 | -ζ139-ζ134 | -ζ138-ζ135 | -ζ137-ζ136 | -ζ1312-ζ13 | -ζ1311-ζ132 | -ζ1310-ζ133 | -ζ43ζ1310+ζ43ζ133 | ζ4ζ139-ζ4ζ134 | -ζ43ζ138+ζ43ζ135 | ζ4ζ137-ζ4ζ136 | -ζ4ζ137+ζ4ζ136 | ζ43ζ138-ζ43ζ135 | -ζ4ζ139+ζ4ζ134 | ζ43ζ1310-ζ43ζ133 | -ζ4ζ1311+ζ4ζ132 | ζ43ζ1312-ζ43ζ13 | -ζ43ζ1312+ζ43ζ13 | ζ4ζ1311-ζ4ζ132 | orthogonal faithful |
►On 52 points
Generators in S52
(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)
(1 52)(2 51)(3 50)(4 49)(5 48)(6 47)(7 46)(8 45)(9 44)(10 43)(11 42)(12 41)(13 40)(14 39)(15 38)(16 37)(17 36)(18 35)(19 34)(20 33)(21 32)(22 31)(23 30)(24 29)(25 28)(26 27)
G:=sub<Sym(52)| (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), (1,52)(2,51)(3,50)(4,49)(5,48)(6,47)(7,46)(8,45)(9,44)(10,43)(11,42)(12,41)(13,40)(14,39)(15,38)(16,37)(17,36)(18,35)(19,34)(20,33)(21,32)(22,31)(23,30)(24,29)(25,28)(26,27)>;
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), (1,52)(2,51)(3,50)(4,49)(5,48)(6,47)(7,46)(8,45)(9,44)(10,43)(11,42)(12,41)(13,40)(14,39)(15,38)(16,37)(17,36)(18,35)(19,34)(20,33)(21,32)(22,31)(23,30)(24,29)(25,28)(26,27) );
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)], [(1,52),(2,51),(3,50),(4,49),(5,48),(6,47),(7,46),(8,45),(9,44),(10,43),(11,42),(12,41),(13,40),(14,39),(15,38),(16,37),(17,36),(18,35),(19,34),(20,33),(21,32),(22,31),(23,30),(24,29),(25,28),(26,27)]])
D52 is a maximal subgroup of
C104⋊C2 D104 D4⋊D13 Q8⋊D13 D52⋊5C2 D4×D13 D52⋊C2 D52⋊C3 C3⋊D52 D156
D52 is a maximal quotient of
C104⋊C2 D104 Dic52 C52⋊3C4 D26⋊C4 C3⋊D52 D156
Matrix representation of D52 ►in GL2(𝔽53) generated by
| 21 | 38 |
| 15 | 7 |
| 21 | 38 |
| 47 | 32 |
G:=sub<GL(2,GF(53))| [21,15,38,7],[21,47,38,32] >;
D52 in GAP, Magma, Sage, TeX
D_{52} % in TeX
G:=Group("D52"); // GroupNames label
G:=SmallGroup(104,6);
// by ID
G=gap.SmallGroup(104,6);
# by ID
G:=PCGroup([4,-2,-2,-2,-13,49,21,1539]);
// Polycyclic
G:=Group<a,b|a^52=b^2=1,b*a*b=a^-1>;
// generators/relations
Export
Subgroup lattice of D52 in TeX
Character table of D52 in TeX