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Derek Holt
Derek Holt
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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)$$2^8:{\rm Sp}(8,2)$ and $2^8\cdot ({\rm Sp}(8,2)$$2^8\cdot {\rm Sp}(8,2)$.

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 $2^8\cdot ({\rm Sp}(8,2)$.

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 $2^8\cdot {\rm Sp}(8,2)$.

Source Link
Derek Holt
Derek Holt
  • 37.4k
  • 4
  • 96
  • 150

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 $2^8\cdot ({\rm Sp}(8,2)$.