The stable behavior of the augmentation quotients of some groups of order p⁴, I

Abstract

We list the stable structure of these six groups below.
(1) G₁=Z_p׋x, y, z|x^p=y^p=z^p=[x, z]=[y, z]=1, [x, y]=z›,
n₀=3p-2, π=2
Q_3p-2(G₁)=Z_p^{(1/2)(p+1)(p2+p+1)}, Q_3p-1(G₁)=Z_p^{(1/2)(p+1)(p2+p+1)+1}
(2) G₂=‹u, x, y|u^p=x^p=y^p2=[u, y]=[x, y]=1, [u, x]=y^p›,
n₀=3p-2, π=1,
Q_3p-2(G₂)=Z_p^(2p2+p+1).
(3) G₃=Z_p׋x, y|x^p=y^p2=1, [x, y]=y^p›,
n₀=3p-2, π=1,
Q_3p-2(G₃)=Z_p^(2p2+p+1).
(4) G₄=Z_p⁽²⁾×Z_p²
n₀=3p-2, π=1,
Q_3p-2(G₄)=Z_p²×Z_p^(2p2+p-1).
(5) G₅=‹u, x, y|u^p=x^p=y^p2=[u, x]=[u, y^p]=1, [u, y]=x, [x, y]=y^p›,
n₀=4p-3, π=2,
Q_4p-3(G₅)=Z_p^{(1/2)(3p2+1)+p}, Q_4p-2(G₅)=Z_p^{(3/2)(p2+1)+p}
(6) G₆=‹u, x, y|u^p=x^p=y^p2=[u, x]=[x, y]=1, [u, y]=x›,
n₀=4p-3, π=2,
Q_4p-3(G₆)=Z_p²×Z_p^{(3/2)(p2-1)+p}, Q_4p-2(G₆)=Z_p²×Z_p^{(1/2)(3p2-1)+p}.