π_{6} (orderΒ 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).

π_{6} (orderΒ 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.

π_{5} (orderΒ 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
π_{6}, 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).

(π_{3}Γπ_{3}):`C`_{2}
(orderΒ 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.

π_{5} (orderΒ 60): This is the intersection of
π_{6} and π_{5}, i.e., even permutations which
preserve a synthematic pentad. This group allows two objects to be
moved to any position (it is 2-transitive).

π_{4}Γ`C`_{2} (orderΒ 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 π_{5}) 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).

(`C`_{3}Γ`C`_{3}):`C`_{4}
(order 36): This is the intersection of π_{6} and
(π_{3}Γπ_{3}):`C`_{2}, i.e., even
permutations which must leave alternating objects at alternating
places.

π_{3}Γπ_{3} (orderΒ 36): This is the
subgroup of (π_{3}Γπ_{3}):`C`_{2}
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.

π_{4}^{β} (orderΒ 24): This is the
group of permutations preserving two synthematic pentads (see
π_{5}). Here we have chosen these pentads so as to make this
a subgroup of π_{4}Γ`C`_{2}, 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 π_{4}^{β} a
subgroup of π_{5} only up to conjugacy.

π_{4}^{+} (orderΒ 24): This is the
intersection of π_{6} and
π_{4}Γ`C`_{2}, i.e., even permutations
leaving opposite objects at opposite places.

π_{4}Γ`C`_{2} (orderΒ 24):
This is the subgroup of π_{4}Γ`C`_{2}
consisting of permutations which, seen as acting on the six
synthematic pentads (see under π_{5}), 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.

π_{3}Γ`C`_{3} (orderΒ 18):
This is the subgroup of
(π_{3}Γπ_{3}):`C`_{2} 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.

`D`_{6} (orderΒ 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).

π_{4} (orderΒ 12): This is the intersection of
any two among π_{4}^{β}, π_{4}^{+} and
π_{4}Γ`C`_{2}. It consists of permutations
which, seen as acting on the six synthematic pentads (see under
π_{5}), 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 π_{4} a subgroup of π_{5} or
π_{5} only up to conjugacy.

`C`_{6} (orderΒ 6): This is the group
of rotations (=orientation-preserving symmetries) of the hexagon,
i.e., cyclic permutations. It is also the intersection of
π_{3}Γ`C`_{3}
and `D`_{6}.

π_{3} (orderΒ 6): This is the subgroup
of `D`_{6} consisting of symmetries of the hexagon
which move any given side to a side with the same parity. It is also
the intersection of π_{4}^{β}
and `D`_{6}. This choice makes π_{3} a
subgroup of π_{3}Γ`C`_{3} only up to
conjugacy.