How to create symbol "cross product with dot under it"

Using \ooalign (one of my favorite toys) together with \mathpalette for properly scaling the symbol if in subscripts or superscripts. Note that the dot is properly centered, contrary to what the image shows.

\documentclass{article}
\usepackage{amsmath}

\makeatletter
\newcommand{\timesdot}{\mathbin{\mathpalette\timesdot@\relax}}
\newcommand{\timesdot@}[2]{%
  \ooalign{$\m@th#1\times$\cr\hidewidth$\m@th#1.$\hidewidth\cr}%
}
\makeatother

\begin{document}

$\mathcal{I}\timesdot\mathcal{J}$

\end{document}

With stackengine:

\documentclass{article}
\usepackage{amsfonts}
\usepackage{stackengine}
\newcommand\timesdot{\stackMath\mathbin{\stackinset{c}{0ex}{c}{-0.49ex}{\scriptscriptstyle.}{\times}}}

\begin{document}

$\mathcal I \timesdot \mathcal J$

\end{document}