This is an Omegaverse story

Omegaverse is an alternate reality where men can have children.  

Typical omegaverse archetypes are as follows:

Alpha-aggressive, top of the ladder. Weak to omega hormones. Marks an omega by biting them on the back of the neck.

Beta-Normal typical human. Immune to pheromones. Usually has low chance of impregnating an omega.

Omega-Submissive, gives off pheromones, can give birth. Has a heat cycle where their pheromones drive all alphas crazy. Use suppressants to calm it down.  Unlike werewolf stories, not necessarily physically or mentally weak.

This story is slightly different in that:

There isn't a heat cycle per se. It's more like ovulation with heightened body temperature and sexual sensitivity during the full and new moons (like werewolves.)

Alpha-Females cannot carry children but can make them. Males and females can impregnate a gamma or omega.

Beta-still laid back and low key. Women cannot carry children BUT males and females can make them with a gamma or omega.

Gamma-Is a gender fluid character. They possess male and female parts and their gender can change based on the preferences of their mate. CANNOT have children with other gammas.

Omega-Are genetically engineered clones created for the sole purpose of procreation. Can procreate with alpha, beta or gamma but not another omega.