Two Classes Of Regular Polytopes Of Order 2^n

In[Problems on polytopes,their groups,and realizations,Periodica Math.Hungarica 2006,53:231-255.]Schulte and Weiss proposed the following open problem:Characterize the groups of order 2n,with n a positive integer,which are automorphism groups of regular polytopes.Hou,Feng and Dimitri have done some work on the above problem.They classified the regular 3-polytopes of order 2n with type {4,2n-3},{4,2n-4} and {4,2n-5}in[Existence of regular 3-polytopes of order 2n,Journal of Group Theory 2019,22:5 79616.].In this paper,we continue the work of Hou,Feng and Dimitri,constructing two classes of regular polytopes of order 2n with type ｛4,2n-6}.This thesis is organized as follows.In section 1,we introduce the background of regular polytopes and our results in this thesis.In section 2,we give some basic definitions and results of regular polytope and string C-groups.Moreover,we restate some wellknown results about groups.Proof of the main results is given in section 3.