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For the $A$-series the Weyl group is the symmetric group $S_n$. For the $B$ and $C$ series the Weyl group is the hyperoctahedral group $\mathbb Z_2 \wr S_n$.

A) Does the $D$-series Weyl group $S_n \mathbb{Z}_2^{n-1}$ have a name?

B) For the exceptional groups, the Weyl group of $G_2$ is the dihedral group $D_6$. What about the other exceptional, do their Weyl groups have names?

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    $\begingroup$ Every Weyl group has a character that sends all reflections in long/short roots to $-1$, and reflections in short/long roots to $1$. (Which reflections are which depends on whether you call your root system $\mathsf C_n$ or $\mathsf B_n$.) If you call all roots of $\mathsf A_n$ long/short, then this is the sign homomorphism. From this point of view, I'd argue that $W(\mathsf D_n)$ is to the hyperoctahedral group as the alternating group is to $W(\mathsf A_n)$. I'd dare to suggest "alternating hyperoctahedral group", but (1) it's pretty terrible and (2) I've never seen it used. $\endgroup$
    – LSpice
    Commented Aug 4 at 16:03
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    $\begingroup$ @LSpice: on the other hand "group of even signed permutations" is totally clear and sounds fine, if a bit wordy. $\endgroup$
    – Sam Hopkins
    Commented Aug 4 at 16:21
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    $\begingroup$ @LSpice Indeed I now see how that could be confusing. For example, in this paper doi.org/10.37236/1836 the terms "even-signed permutations" and "signed even permutations" are both used, which seems unnecessarily confusing to me. $\endgroup$
    – Sam Hopkins
    Commented Aug 5 at 0:14
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    $\begingroup$ Maybe "special hyperoctahedral group"? $\endgroup$
    – მამუკა ჯიბლაძე
    Commented Aug 5 at 10:31
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    $\begingroup$ (I also take the opportunity to mention that my comment should have said that $W(\mathsf D_n)$ contains odd-as-unsigned permutations, not that it doesn't contain even-as-unsigned permutations. That is, my objection should have been that $W(\mathsf D_n) \cap W(\mathsf A_{n - 1})$ isn't the alternating subgroup of $W(\mathsf A_{n - 1})$, not that $W(\mathsf D_n)$ doesn't contain the alternating subgroup of $W(\mathsf A_{n - 1})$; of course the latter is false. @SamHopkins, thank you for reading my comment charitably as it was meant!) $\endgroup$
    – LSpice
    Commented Aug 5 at 13:48

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I don't know where I first saw it, but I have seen the term "demihyperoctahedral group" used before, so if you're doing searches, that can sometimes pick up relevant papers.

As for the exceptional groups, the type E Weyl groups are all closely related to groups of Lie type over finite fields, so from that isomorphism they have names sort of. Look at the last 3 rows of https://en.wikipedia.org/wiki/Complex_reflection_group#List_of_irreducible_complex_reflection_groups Type $E_6$ has many different realizations so not sure if any of those names would be most canonical.

As far as I know, the Weyl group of type $F_4$ is most cleanly described as the automorphisms of the "24-cell", a certain 4-dimensional polytope.

Some more detail can be found in Section 2.12 of Humphreys' book Reflection Groups and Coxeter Groups.

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