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Athletic Club of BC was created in 2009.
Boys' Athletic Booster Club = Athletic Booster Club for boys (plural)
Boy's Athletic Booster Club = Athletic Booster Club for a boy (singular)
There's no team called Athletic Club but some clubs are called something athletic club, like
HAC: Le Havre Athletic Club,
ACBB: Athletic Club de Boulogne-Billancourt,...
Woman's Athletic Club was created in 1928.
Cardiff Athletic Club was created in 1922.