# A question regarding groups of order $p^2qr$

When considering finite groups $$G$$ of order, $$|G|=p^2qr$$, where $$p,q,r$$ are distinct primes, let $$F$$ be a Fitting subgroup of $$G$$. Then $$F$$ and $$G/F$$ are both non-trivial and $$G/F$$ acts faithfully on $$\bar{F}:=F/ \phi(F)$$ so that no non-trivial normal subgroup of $$G/F$$ stabilizes a series through $$\bar{F}$$.

a) Case $$|F|=p^2$$ and $$F \cong C_{p}^2$$. Then $$G=C_{p}^2 \rtimes H$$ with $$H$$ of order $$qr$$ and $$H$$ embeds into $$Aut(C_{p}^2)=GL(2,p)$$.

b) Case $$|F|=pr$$. In this case $$\phi(F)=1$$ and $$Aut(F)=C_{p-1} \times C_{r-1}$$. Thus $$G/F$$ is abelian and $$G/F \cong C_{p} \times C_{q}$$.

Can some one please help clarify how to represent the above two groups using notation form? i.e. is it correct if I represent them using notations as follows?

a) $$G \cong C_{p}^2 \rtimes (C_{q} \times C_{r})$$ or $$G \cong C_{p}^2 \rtimes (C_{q} \rtimes C_{r})$$ or $$G \cong C_{p}^2 \rtimes (C_{r} \rtimes C_{q})$$

b) $$G \cong (C_{p} \times C_{r}) \rtimes (C_{p} \times C_{q})$$ or $$G \cong (C_{p} \rtimes C_{r}) \rtimes (C_{p} \times C_{q})$$ or $$G \cong (C_{r} \rtimes C_{p}) \rtimes (C_{p} \times C_{q})$$

Moreover there exists two nilpotent groups of order $$p^2qr$$. What are the representations for the two nilpotent groups of order $$p^2qr$$?

Since $$C_{pq} \cong C_p \times C_q$$ for coprime $$p$$ and $$q$$, you could use wither of those notations - they are both correct. So, there are multiple correct answers to your final question.