== Q    
k=3 |G_k|=2^5  d(G_k)=3  |G_k^ab|=8  G_k^ab elementary abelian: True  |[G_k,G_k]|=2^2
k=4 |G_k|=2^8  d(G_k)=3  |G_k^ab|=8  G_k^ab elementary abelian: True  |[G_k,G_k]|=2^5
k=5 |G_k|=2^11  d(G_k)=3  |G_k^ab|=8  G_k^ab elementary abelian: True  |[G_k,G_k]|=2^8
k=6 |G_k|=2^14  d(G_k)=3  |G_k^ab|=8  G_k^ab elementary abelian: True  |[G_k,G_k]|=2^11
== P_PSL
k=3 |G_k|=2^5  d(G_k)=2  |G_k^ab|=8  G_k^ab elementary abelian: False  |[G_k,G_k]|=2^2
k=4 |G_k|=2^8  d(G_k)=2  |G_k^ab|=16  G_k^ab elementary abelian: False  |[G_k,G_k]|=2^4
k=5 |G_k|=2^11  d(G_k)=2  |G_k^ab|=16  G_k^ab elementary abelian: False  |[G_k,G_k]|=2^7
k=6 |G_k|=2^14  d(G_k)=2  |G_k^ab|=16  G_k^ab elementary abelian: False  |[G_k,G_k]|=2^10
== P_PGL
k=3 |G_k|=2^7  d(G_k)=4  |G_k^ab|=16  G_k^ab elementary abelian: True  |[G_k,G_k]|=2^3
k=4 |G_k|=2^10  d(G_k)=4  |G_k^ab|=16  G_k^ab elementary abelian: True  |[G_k,G_k]|=2^6
k=5 |G_k|=2^13  d(G_k)=4  |G_k^ab|=16  G_k^ab elementary abelian: True  |[G_k,G_k]|=2^9
k=6 |G_k|=2^16  d(G_k)=4  |G_k^ab|=16  G_k^ab elementary abelian: True  |[G_k,G_k]|=2^12
== brute-force cross-check (all commutators / all squares of all elements)
Q     k=4 |G_k^ab| = 8  exponent of G_k^ab = 2  d(G_k) = 3
P_PSL k=4 |G_k^ab| = 16  exponent of G_k^ab = 8  d(G_k) = 2
P_PGL k=3 |G_k^ab| = 16  exponent of G_k^ab = 2  d(G_k) = 4
