Count the number of automorphisms in the graph

[scorpius1782]

Homework Statement

I had a different problem before about this and I figured it out. I'd like to know if I'm doing this one correctly as well.
Count the number of automorphisms in the graph.
The graph is attached, now.

The Attempt at a Solution

I know I can rearrange the (a,m,b) 3! ways. I also know that I can only arrange (g,h) and (e,d) 4 ways. So, for each arrangement of (a,m,b) I have 4 arrangements of (g,h) and (e,d). Then 6*4=24 total.

Have I got this sorted out correctly?

 

[phinds]

Just FYI, the graph is NOT attached.