Regular 3-polytopes of order 2^𝑛𝑝

Abstract In a paper from 2006, Schulte and Weiss initiated the problem of characterizing the regular polytopes of orders <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msup> <m:mn>2</m:mn> <m:mi>n</m:mi> </m:msup> <m:mo>⁢</m:mo> <m:mi>p</m:mi> </m:mrow> </m:math> 2^{n}p for 𝑛 a positive integer and 𝑝 an odd prime. In this paper, we first prove that if a 3-polytope of order <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msup> <m:mn>2</m:mn> <m:mi>n</m:mi> </m:msup> <m:mo>⁢</m:mo> <m:mi>p</m:mi> </m:mrow> </m:math> 2^{n}p has Schläfli type <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo stretchy="false">{</m:mo> <m:msub> <m:mi>k</m:mi> <m:mn>1</m:mn> </m:msub> <m:mo>,</m:mo> <m:msub> <m:mi>k</m:mi> <m:mn>2</m:mn> </m:msub> <m:mo stretchy="false">}</m:mo> </m:mrow> </m:math> \{k_{1},k_{2}\} , then <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>p</m:mi> <m:mo>∣</m:mo> <m:msub> <m:mi>k</m:mi> <m:mn>1</m:mn> </m:msub> </m:mrow> </m:math> p\mid k_{1} or <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>p</m:mi> <m:mo>∣</m:mo> <m:msub> <m:mi>k</m:mi> <m:mn>2</m:mn> </m:msub> </m:mrow> </m:math> p\mid k_{2} . This leads to two classes, up to duality, for the Schläfli type, namely Type (1) where <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi>k</m:mi> <m:mn>1</m:mn> </m:msub> <m:mo>=</m:mo> <m:mrow> <m:msup> <m:mn>2</m:mn> <m:mi>s</m:mi> </m:msup> <m:mo>⁢</m:mo> <m:mi>p</m:mi> </m:mrow> </m:mrow> </m:math> k_{1}=2^{s}p and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi>k</m:mi> <m:mn>2</m:mn> </m:msub> <m:mo>=</m:mo> <m:msup> <m:mn>2</m:mn> <m:mi>t</m:mi> </m:msup> </m:mrow> </m:math> k_{2}=2^{t} and Type (2) where <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi>k</m:mi> <m:mn>1</m:mn> </m:msub> <m:mo>=</m:mo> <m:mrow> <m:msup> <m:mn>2</m:mn> <m:mi>s</m:mi> </m:msup> <m:mo>⁢</m:mo> <m:mi>p</m:mi> </m:mrow> </m:mrow> </m:math> k_{1}=2^{s}p and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi>k</m:mi> <m:mn>2</m:mn> </m:msub> <m:mo>=</m:mo> <m:mrow> <m:msup> <m:mn>2</m:mn> <m:mi>t</m:mi> </m:msup> <m:mo>⁢</m:mo> <m:mi>p</m:mi> </m:mrow> </m:mrow> </m:math> k_{2}=2^{t}p . We then show that there exists a regular 3-polytope of order <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msup> <m:mn>2</m:mn> <m:mi>n</m:mi> </m:msup> <m:mo>⁢</m:mo> <m:mi>p</m:mi> </m:mrow> </m:math> 2^{n}p with Type (1) when <m:math xmlns:m="http://www.w3.org/1998