If we define $ G $ to be $ \langle a,b,c,d \ \ | \ \ a^2=b^2=c^2=d^2=1, (ab)^3=(bc)^4=(cd)^3=1, ac=ca,ad=da,bd=db \rangle $, then what's the order of $ G $?
When $ G $ has only two generators, the problem of this type is ok for me. I just need to 'guess' all distinct elements of $ G $ and verify the conjecture. However, in this problem, I have no idea about elements of $ G $ :(
I also try to find the index of $ H :=\langle a,b,c \ \ | \ \ a^2=b^2=c^2=1, (ab)^3=(bc)^4, ac=ca \rangle $ because I know the order of $ H $. But I still failed. Any hints?