In how many ways can seven beads of distinct colors be put on the hexagonal grid shown, if reflections and rotations of an arrangement are considered equivalent?

[asy]
size(50);
dot((0,0));
dot((1,0));
dot((-1,0));
dot((.5,sqrt(3)/2));
dot((.5,-sqrt(3)/2));
dot((-.5,sqrt(3)/2));
dot((-.5,-sqrt(3)/2));
[/asy]
Question:
In how many ways can seven beads of distinct colors be put on the hexagonal grid shown, if reflections and rotations of an arrangement are considered equivalent?

[asy]
size(50);
dot((0,0));
dot((1,0));
dot((-1,0));
dot((.5,sqrt(3)/2));
dot((.5,-sqrt(3)/2));
dot((-.5,sqrt(3)/2));
dot((-.5,-sqrt(3)/2));
[/asy]
Answer:
Let's think step by step.

user
420
In how many ways can seven beads of distinct colors be put on the hexagonal grid shown, if reflections and rotations of an arrangement are considered equivalent?

[asy]
size(50);
dot((0,0));
dot((1,0));
dot((-1,0));
dot((.5,sqrt(3)/2));
dot((.5,-sqrt(3)/2));
dot((-.5,sqrt(3)/2));
dot((-.5,-sqrt(3)/2));
[/asy]
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