I am preparing lecture notes for an abstract algebra course I am teaching. I would like to draw the following diagram, which is from Zassenhaus's Butterfly lemma.
$K$ in the diagram is the group $(G'\cap H)(G\cap H')$. I did not write it there because I thought it would clutter the diagram for people who will look at it and help me on this site.
I am not very well-versed in LaTeX. I have the following structure to my code so far.
\documentclass[12pt, a4paper]{book}
\usepackage[utf8]{inputenc}
\usepackage[margin=0.6in]{geometry}
\usepackage{graphicx}
\usepackage[none]{hyphenat}
\usepackage{amssymb, tikz}
\usepackage{amsmath}
\usepackage{amsthm}
\usepackage{physics}
\usepackage{mathrsfs}
\usepackage{fourier-orns}
\usepackage{tikz-cd}
\usepackage{blindtext}
\usepackage{mathtools}
\usepackage{pdfrender,xcolor}
\usepackage[T1]{fontenc}
\usepackage{crimson}
\setlength{\parskip}{0.3em}
\setlength{\parindent}{2em}
\newcommand\bulletpoint[1][.5]{\mathbin{\vcenter{\hbox{\scalebox{#1}{$\bullet$}}}}}
\newcommand\proofend[1][.75]{\mathbin{\vcenter{\hbox{\scalebox{#1}{$\qed$}}}}}
\newcommand\Thicc{\fontsize{35}{50}\selectfont}
\newcommand\textbox[1]{%
\parbox{.404\textwidth}{#1}%
}
\newcommand\textbonk[1]{%
\parbox{.333\textwidth}{#1}%
}
\newcommand{\mysetminusD}{\hbox{\tikz{\draw[line width=0.6pt,line cap=round] (3pt,0) -- (0,6pt);}}}
\newcommand{\mysetminusT}{\mysetminusD}
\newcommand{\mysetminusS}{\hbox{\tikz{\draw[line width=0.45pt,line cap=round] (2pt,0) -- (0,4pt);}}}
\newcommand{\mysetminusSS}{\hbox{\tikz{\draw[line width=0.4pt,line cap=round] (1.5pt,0) -- (0,3pt);}}}
\newcommand{\mysetminus}{\mathbin{\mathchoice{\mysetminusD}{\mysetminusT}{\mysetminusS}{\mysetminusSS}}}
\DeclareSymbolFont{AMSa}{U}{msa}{m}{n}
\DeclareMathSymbol{\normal}{\mathrel}{AMSa}{"45}
\let\l\langle
\let\r\rangle
\let\phi\varphi
\let\varphi\phi
\makeatletter
\renewcommand*\env@matrix[1][*\c@MaxMatrixCols c]{%
\hskip -\arraycolsep
\let\@ifnextchar\new@ifnextchar
\array{#1}}
\makeatother
\begin{document}
\emergencystretch 1em
\pdfrender{StrokeColor=black,TextRenderingMode=2,LineWidth=0.4pt}
\[\begin{tikzcd}[column sep=30pt]G\arrow[d,"\,"]&\,&H\arrow[d,"\,"]\\G'(G\cap H)\arrow[dd,"\,"]\arrow[dr,"\,"]&\,&(G\cap H)H'\arrow[dd,"\,"]\arrow[dl,"\,"]\\\,&G\cap H\arrow[dd,"\,"]&\,\\G'(G\cap H')\arrow[dr,"\,"]&\,&(G'\cap H)H'\arrow[dl,"\,"]\\\,&K&\,\end{tikzcd}\]
\end{document}
Please help me draw it. Any help is really appreciated.
Edit: I am sorry, I did not know that the community here values looking at the original poster's attempts before asking for help. I have included what little I could do with my limited knowledge of LaTeX. Of course, I do not want them to be arrows, and I want the intersections to be points rather than the notations for the groups themselves.
