Group order from a presentation

[Barre]

Hello.

I have been looking at some questions from old exams that I am preparing for, and I have some trouble with the kind of problems that I will now give an example of.

Homework Statement

Let $G = (a,b,c | a^4 = 1, b^2 = a^2, bab^{-1} = a^{-1}, c^3 = 1, cac^{-1} = b, cbc^{-1} = ab)$. Determine the order of this group.

Homework Equations

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The Attempt at a Solution

The relations imply that one can move c to the left past all a,b. Also, b can be moved to the left past all a, and hence we can express all elements of this group as products of a power of a, power of b and power of c (in that order). There are at most 24 elements. Now, I have not found any relations that imply orders of a and c are less than 4 and 3, so I assume the group has order 24, but how can I prove this? Easiest would be to find a group generated by 3 (or less) elements that satisfy these properties, and map G surjectively into it, but I cannot expect to memorize all low order groups.

On groups with 2 generators and simpler relations, I usually just do mind-numbing computation of the left regular representation of this group presentation. Then if the relations do not collapse the group, I can map it surjectively into the permutation group I obtained and draw conclusions about order. What can I do in this case, when there are 3 generators and relations pretty much complicated enough so that working out a regular representation on paper is out of question. Are there any popular tricks?

 

[Vargo]

Here is a hint. Identify a normal subgroup and write down G as a semidirect product. Once you do that you can look at the following tables to find the rest of what you need.

http://en.wikipedia.org/wiki/List_of_small_groups