Revealing mistake: During the beach scene, when Supergirl flies in front of Ethan, the wires holding her are visible.
Revealing mistake: When Kara picks up the Omegahedron, you can briefly see a ring underneath, around her fingers, to make it turn around.
Revealing mistake: When Supergirl falls on the planet ground and gets released from the mirror trap, she gets up and the hard floor stretches down, revealing it's a rubber mat to soften her fall.
Revealing mistake: When Selena uses her magical powers to accelerate the fairground ride to superfast speed, the twinkling fairground lights in the background are also flashing much faster. They obviously played the tape back at high speed to achieve this.
Answer: The Omegaahedron is the power source for the whole city. It supplies the air, the dome that protects them, the lights, and the machines they need to survive.
New this monthThe question was why would Zoltar need permission to use the Omegahedron, not what it could do.