# Transitive subgroups of 𝔖6

𝔖6 (order 720): This is the group of all possible permutations of the six objects. It allows each object to be moved to any position, independently of the others (subject only to the constraint that two distinct objects cannot occupy the same position).

𝔄6 (order 360): This is the group of even permutations of the six objects, i.e., permutations which have an even number of inversions, or equivalently which can be obtained by exchanging an even number of pairs of objects, or yet equivalently such that the sum over all cycles of the cycle length minus one is even. It allows four objects to be moved to any position, after what the position of the remaining two are fixed.

𝔖5 (order 120): The symmetric group on five objects can act (up to conjugacy) in two different ways on six objects: the obvious way consists of simply fixing one of the objects and acting on the five remaining ones, but this is then obviously not transitive so not in this list. Here we have the other action (which is obtained from the first by the Sylvester automorphism of 𝔖6, see e.g. here): instead of fixing a point if fixes a synthematic pentad, meaning one of the (six possible) partitions of the set of fifteen edges between the six objects in five classes (or syntheme) in such a way that no two edges in the same class have a vertex in common (here the pentad chosen to be fixed is the only one which puts the vertices of the hexagon in only two classes). This group allows three objects to be moved to any position, after what the position of the remaining three are fixed (it is sharply 3-transitive).

(𝔖3×𝔖3):C2 (order 72): This is the group of permutations which must leave alternating objects at alternating places, i.e., red, green and blue must never be moved to adjacent places but must always form a triangle.

𝔄5 (order 60): This is the intersection of 𝔄6 and 𝔖5, i.e., even permutations which preserve a synthematic pentad. This group allows two objects to be moved to any position (it is 2-transitive).

𝔖4×C2 (order 48): This is the group of permutations which must leave opposite objects at opposite places, i.e., red and cyan must always be placed opposite, as must green and magenta, and blue and yellow. Equivalently, it is the group of permutations which preserve a partition of the six synthematic pentads (see under 𝔖5) into two plus four (where here the two pentads are the only two which place the edges connecting opposite objects in the same class/syntheme).

(C3×C3):C4 (order 36): This is the intersection of 𝔄6 and (𝔖3×𝔖3):C2, i.e., even permutations which must leave alternating objects at alternating places.

𝔖3×𝔖3 (order 36): This is the subgroup of (𝔖3×𝔖3):C2 which not only must leave alternating objects at alternating places, i.e., red, green and blue form a triangle, and yellow, cyan and magenta as well, but also must keep these triangles in identical orientations.

𝔖4 (order 24): This is the group of permutations preserving two synthematic pentads (see 𝔖5). Here we have chosen these pentads so as to make this a subgroup of 𝔖4×C2, i.e., leaving opposite objects at opposite places (the two pentads are the only two which place the edges connecting opposite objects in the same class/syntheme). This choice makes 𝔖4 a subgroup of 𝔖5 only up to conjugacy.

𝔖4+ (order 24): This is the intersection of 𝔄6 and 𝔖4×C2, i.e., even permutations leaving opposite objects at opposite places.

𝔄4×C2 (order 24): This is the subgroup of 𝔖4×C2 consisting of permutations which, seen as acting on the six synthematic pentads (see under 𝔖5), not only preserve a partition into two plus four (where here the two pentads are the only two which place the edges connecting opposite objects in the same class/syntheme) but also act on the four by an even permutation.

𝔖3×C3 (order 18): This is the subgroup of (𝔖3×𝔖3):C2 which not only must leave alternating objects at alternating places, i.e., red, green and blue form a triangle, and yellow, cyan and magenta as well, but also must keep each of these triangles in identical orientations.

D6 (order 12): This is the group of symmetries of the hexagon, i.e., permutations which leave adjacent objects in adjacent positions (either by permuting them cyclically or by reflecting them about some axis).

𝔄4 (order 12): This is the intersection of any two among 𝔖4, 𝔖4+ and 𝔖4×C2. It consists of permutations which, seen as acting on the six synthematic pentads (see under 𝔖5), fix two of them (where here the two pentads are the only two which place the edges connecting opposite objects in the same class/syntheme) and act in an even way on the remaining four. This choice makes 𝔄4 a subgroup of 𝔖5 or 𝔄5 only up to conjugacy.

C6 (order 6): This is the group of rotations (=orientation-preserving symmetries) of the hexagon, i.e., cyclic permutations. It is also the intersection of 𝔖3×C3 and D6.

𝔖3 (order 6): This is the subgroup of D6 consisting of symmetries of the hexagon which move any given side to a side with the same parity. It is also the intersection of 𝔖4 and D6. This choice makes 𝔖3 a subgroup of 𝔖3×C3 only up to conjugacy.