Let $n \geq 3$, $(n, q) \neq (3,3)$ and $q$ be an odd prime power. It is known that $SO_{2n+1}(q)=\{A\in SL_{2n+1}(q): A^tJA=J\}$, where $J$ is a matrix with anti diagonal 1, 0 otherwise and $(SO_{2n+1}(q))'= B_n(q)$. Where can I find a complete description of the structure of the double cover of $B_n(q)$? Is this double cover subgroup of $GL_{2n+1}(q)$?
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For $n\geq4$, the derived subgroup of $S_n$ is $A_n$, which is simple and has a double cover (which is not $S_n$) Is that also strange? – Mariano Suárez-Álvarez Jan 17 '16 at 09:38
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The double cover of $B_n(q)$ has twice as many elements as $B_n(q)$, so it has less than twice as many elements as $SO_{2n+1}(q)$ has, and that is much less than the number of elements $GL$ has. – Mariano Suárez-Álvarez Jan 17 '16 at 09:43
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Thank you. Actually I am satisfied with $A_n$, but since I always compar the $B_n(q)$ with $C_n(q)$, it seems strange to me. Is there any matrix representation for the double cover of $B_n(q)$ corresponding to the matrix representation of $B_n(q)$? – Sara Jan 17 '16 at 11:11
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Honestly, I have no idea what you mean by "strange". – Mariano Suárez-Álvarez Jan 17 '16 at 20:43
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I 'm going to change this word in the question. I need to know about the structure of this group. Where can I find a complete description. – Sara Jan 18 '16 at 05:38