By talking to her in the Contest Hall.
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She is the 5th, gym leader, you have to beat pastoria gym first the vielstone! then go back to harthome city and you can find her in her gym
after that you go to hearthome city and fantina (the next gym leader) will be there. P.S. i found this info on serebii.net. you should check it out.
You go to hearthome city and go to the contest hall. you should see fantina. then she will go away and go to pastareia city and fight the gym leader. then at the beach you might see a grunt. talk to him, fallow him, and find citheia. then do what she asked.
Fantina is in Hearthome City. If she isn't there. There is something you have to do before. I think you have to go to pastoria GYM first. Then she'll be available, but have been into the Pokemon contest hall before, she appears in there with your mum.
fantina ia a ghost type gym leader Pokemon : Drifblim Lvl.32, Gengar Lvl.34, and Mismagius Lvl.36