Olympia is the leader of the Anistar City Gym. She uses Psychic-type pokémon and awards the Psychic Badge.
Gym Leader Olympia
Olympia has a level 44 Sigilyph, a level 45 Slowking, and a level 48 Meowstic.
The Sigilyph is weak to Rock, Ghost, Electric, Ice, and Dark moves. It uses Psychic and Flying attacks.
The Slowking is weak to Bug, Ghost, Grass, Electric, and Dark moves. It uses Psychic and Rock attacks. In addition, it knows Yawn, so it would be helpful to bring a pokémon that has the Insomnia ability, such as Murkrow, or at least give your pokémon a Chesto Berry to hold.
The Meowstic is weak to Bug, Ghost, and Dark moves. It uses Normal, Psychic, and Ghost attacks.
When you win, you get the Psychic Badge, which causes traded pokémon up to level 90 to obey you. You also receive TM04 Calm Mind. Olympia will ask if you want to be warped to the entrance. Say yes if you don't want to take the long way.
Outside, you'll notice a Team Flare Grunt just standing there. Your neighbor runs up and wants a rematch, but you receive a threatening Holo Clip from the leader of Team Flare. Your neighbor says you should look for the Team Flare hideout and suggests going back to where you met the Team Flare leader in Lumiose City. Talk to the Team Flare Grunts next to the gym for some more clues about where to look.
Find Team Flare's Hideout
Fly back to the center of Lumiose City. Flying to the center will put you close to Team Flare's hideout. If you flew to the correct spot, you will land in front of a Pokémon Center near Magenta Plaza. To your character's left, you can see a Team Flare Grunt standing there. If you have made all of the preparations that you want to make, go into the Café near the Grunt in Magenta Plaza to enter Lysandre Café.
Talk to the waiter to battle him. He's a Team Flare Grunt. He has a level 46 Scrafty.
Go north and talk to the waitress, who is also a Team Flare Grunt. She has a level 46 Liepard.
Check on the shelves next to the waitress to open the secret door to Lysandre Labs.