= Deleted permutation module in characteristic two
{title2=$\{x\in\mathbb F_2^{2m+2}:\sum x_i=0\}/\langle(1,\ldots,1)\rangle$}
For even degree the augmentation hyperplane has an alternating dot product whose radical is the all-ones line. Quotienting by that line gives a nondegenerate symplectic space of dimension two less than the permutation degree. Coordinate permutations preserve the form. In degree at least six their quotient action is faithful, since two distinct weight-two coordinate vectors cannot differ by the all-ones vector. In degree six this gives $S_6\cong Sp_4(2)$; in degree four the quotient action instead has Klein-four kernel.
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