The yearling, a daughter of 2001 Derby winner Galileo and Alluring Park, was also the most expensive horse of any age sold at public auction in Europe, beating a record set in 2006 by Magical Romance (4.6m guineas).
Sheikh Al Thani, son of the Emir of Qatar, was much taken by her pedigree, being a sister of 2012 Epsom Oaks winner Was.
Mandore International's Nicolas de Watrigant, the agent who bought the filly on the Sheikh's behalf said she will be trained in France by 24-time champion trainer Andre Fabre.
"She is a magnificent specimen," he said.
"We bought her sister last year and we think she could be special.
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"We didn't want to miss this one - let's hope she is the next Treve.
"We didn't expect to pay that sort of price, but Al Shaqab Racing is keen to invest.
"She goes into training with Monsieur Fabre."
Fabre receives the horse despite the Sheikh winning France's prestigious race the Prix de l'Arc de Triomphe, Europe's premier flat all-age race with Treve, an unbeaten filly trained by Fabre's French rival Criquette Head-Maarek at Longchamp on Sunday.
Treve was ridden to victory last weekend by French jockey Thiery Jarnet after British-based Italian rider Dettori fractured his ankle just days before the race.
Treve's latest victory saw her emulate Zarkava's 2008 feat of winning the French Oaks, Prix Vermeille and Arc all in the same season.