To The Given Order Of The Finite Group With The Structure Classification

We know that the basic problem of researching finite groups is deciding finite single group and finding the theory of extension.The theory of extension is how to find other groups when we know two groups.So it is one important method of researching finite groups.This paper use group action and the theory of groups extension,in the base of the special subgroup and matrix rational normal form.we obtain:when p=5,F(G) only is Z₂×Z₂×Z₂×Z₂.when p=7,F(G)only is①F(G)(?)Z₂×Z₂×Z₂②F(G)(?)Z₂×Z₂×Z₂×Z₂.When p=3,F(G) only is:①F(G)(?)Z₂×Z₂×Z₂,②F(G)(?)Q₈,③F(G)(?)Z₂×Z₂×Z₂×Z₂,④F(G)(?)＜x,y,z|x⁴=y²=z²=1,y^-1,xy=x,z^-1xz=x,[y,z]=x²＞,⑤F(G)(?)＜x,y,z|x⁴=y²=1,z²=x².y^-1xy=x,z^-1xz=x^-1[y,z]=1＞.and then by theorey in extension of groups,1.we obtain the structure,of groups of order 2⁴p.That is,Groups of order 2⁴3 has52 nonisomorphic types.Groups of order 2⁴5 has 52 nonisomorphic types.Groups of order 2⁴7 has 45 nonisomorphic types.2.By using the special structures of some groups which are dihedral groups,or semidihedral groups,or quatemion groups,and some theorems in extension of groups,we obtain the classification of groups of order 2ⁿp².