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In characteristic $2$, a Sylow subgroup $P$ of $A_5$ is a Klein four-group, and its normalizer is $A_4$. Since $k{A}_4$ has one block, Brauer first main theorem gives a unique block of $k{A}_5$ with defect $P$. No order-two subgroup $D$ can be a defect group of a block: both $C_{A_5}(D)$ and $C_{A_5}(P)$ equal $P$, so the Brauer morphisms of a central block idempotent at $D$ and at $P$ are equal. A nonzero image is $1$, since a group algebra of a p-group in characteristic p is local, and consequently $D$ is not maximal among subgroups with nonzero Brauer image.

The full symmetry group of a cube acts transitively and faithfully on its twelve edges. For vertices $(\pm 1,\pm 1,\pm 1)$, its signed permutation matrices act transitively on the edges. The stabilizer subgroup of $\{(1,1,z):-1\le z\le 1\}$ consists of independently swapping $x,y$ and reversing $z$, and is a Klein four-group. The orbit-stabilizer theorem gives order $12\cdot 4=48$. Fixing every edge fixes every vertex, as each vertex is the unique intersection of its three incident edges, so the group action is faithful. The resulting subgroup of $S_{12}$ has index $12!/48$ and is not normal: central inversion has cycle type $2^6$, whose conjugacy class in $S_{12}$ has $12!/(2^{6}6!)=10395>48$ elements.

This degree-four rational map obeys $R(iz)=1/R(z)$ and $R((iz+1)/(1-iz))=e^{2\pi i/3}R(z)$. These pair the generators of the rotational symmetry group of a cube with target rotations, giving combined spatial-isospin symmetry in the Skyrme model. Its spatial half-turn kernel is a Klein four-group. The map gives a cubic four-Skyrmion approximation; preserving only its Wronskian zero set does not guarantee the same rotational symmetry.

For a depressed quartic $x^4+p{x}^2+qx+r$ with roots ${\alpha }_1,\dots ,{\alpha }_4$, the three pair-partition quantities ${\alpha }_1{\alpha }_2+{\alpha }_3{\alpha }_4$ and its two analogues are the roots ofThe Galois action on these roots is the quotient of its action on the four quartic roots by the Klein four-group.

The degree-four rational map $F(z)=z^2+z^{-2}$ has branch values $-2,2,\infty$. Away from their inverse images, the transformationsare commuting deck involutions. They generate a Klein four-group acting transitively on each fibreConsequently the monodromy group is $V_4$, acting by the identity and the three double transpositions.