I have a non-split extension $2^8\mathbin.(2^7\mathbin:\operatorname{Sp}(6,2))$ of $2^8$ by $2^7\mathbin:\operatorname{Sp}(6,2)$. The question is how does $2^7\mathbin:\operatorname{Sp}(6,2)$ act on $2^8$. This group sits maximally inside the unique nonsplit extension $2^8\mathbin.\operatorname{Sp}(8,2)$.
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1$\begingroup$ Assuming this is usual Atlas notation, $2^8$ does not act on $(2^7:Sp(6,2))$, since the latter is a quotient, not a normal subgroup. $\endgroup$– verretCommented Nov 8, 2019 at 18:37
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$\begingroup$ Also the title of your post does not agree with the body. $\endgroup$– Derek HoltCommented Nov 8, 2019 at 19:59
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1$\begingroup$ It is not possible to answer this question without further information. There is more than one isomorphism type of group that fits that description. $\endgroup$– Derek HoltCommented Nov 9, 2019 at 8:22
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1$\begingroup$ Where are these various kinds of notations for extensions (the colon and dot, as well as, e.g., whether $2^8$ means, say, the elementary Abelian 2-group of order $2^8$, as I assume, or a cyclic group of order $2^8$, or whatever else) defined? $\endgroup$– LSpiceCommented Nov 9, 2019 at 14:52
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4$\begingroup$ @LSpice It's known as the ATLAS notation for group structures defined in the Atlas of Finite Groups. Yes, $2^8$ denotes an elementary abelian group. The cyclic group of that order would be denoted by $256$. You can also use $[256]$ to denote a group of that order with unspecified structure. $\endgroup$– Derek HoltCommented Nov 9, 2019 at 22:53
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1 Answer
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OK, your question makes sense now. The group $2^7:{\rm Sp}(6,2)$ is a maximal subgroup of ${\rm Sp}(8,2)$, and is the stabilizer of a vector in the action of ${\rm Sp}(8,2)$ on its natural $8$-dimensional module. This immediately defines its action on that module. This action is the same in the split and nonsplit extensions $2^8:{\rm Sp}(8,2)$ and $2^8\cdot {\rm Sp}(8,2)$.
answered Nov 9, 2019 at 13:12
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$\begingroup$ So we can then consider $2^8$ as the vector space V(8,2) where upon $2^7:$Sp (6,2) acts as an eight dimensional matrix group over GF (2). I suppose this action is not irreducible. $\endgroup$– IsaacCommented Nov 9, 2019 at 16:14
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$\begingroup$ No, as I said , that subgroup is the stabilizer of a vector. It also stabilizes a $7$-dimensional subspace (which is the space orthogonal to the fixed vector under the symplectic form), and the subgroup ${\rm Sp}(6,2)$ acts in its natural action on the $6$-dimensional quotient of the $7$- by the $1$-dimensional fixed spaces. $\endgroup$ Commented Nov 9, 2019 at 22:51
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$\begingroup$ The action of $2^7:Sp (6,2) $ ( constructed as a 8×8 matrix group within Sp (8,2)) on $2^8$ forms 4 orbits of elements of $2^8$. I checked it with GAP. What do we call this action? $\endgroup$– IsaacCommented Nov 10, 2019 at 6:18
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$\begingroup$ I am still waiting on a response to the last question. $\endgroup$– IsaacCommented Nov 20, 2019 at 20:46