Actually, there really is no order-specific generalization you can make regarding primate social-sexual systems. The social-sexual system you described would most closely resemble gorilla behaviour. Male gorillas compete for harems of females. One or two gorillas usually outcompete all other males for the complete control of several females for as long as they can. However, the variation among all primates is truly overwhelming. For example, chimpanzees organize themselves in multi-male, multi-female systems. Although alpha males or top ranking males will generally have more mating opportunity and generally sire more offspring, females have sex with all the males to increase paternity confusion (decreasing the chance of infanticide). Bonobos on the other hand are famously matriarchal. Females control sex and use it to dominate males. As a result every female will usually have sex with most other males and even many other females.

Two more interesting examples:

Gibbons are lesser apes and they are almost completely monogamous. They find one partner and usually mate for life. There is some evidence of external copulations - 'cheating' - but they are rare and the other partner is generally unaware of these external copulations.

Ring-tailed lemurs are also female dominant. They have also have very strict breeding seasons. During a very brief 2 day window all the males will compete for access to females - but it is the female who will make the final choice of who to mate with and for how long.

An interesting evolutionary rule to make sense of all this sexual primate mayhem is the general principle that social-sexual system can be dictated by sexual dimorphism. Sexual dismorphism is the size difference between the sexes. In species with extreme dimorphism (like gorillas) a few males usually dominate and control the reproduction of all females (this is true for other mammalian orders as well). However, when there is no sexual dimorphism (like gibbons) males and females tend to be monogamous. When there is sexual dimorphism - but it is not pronounced (like chimpanzees and bonobos) there is some other interesting variant of the two extremes. However, it is still unknown how ring-tailed lemurs and bonobo evolved female dominant social-sexual systems.