the problem is, you've got the action wrong for how G acts on {e,a,a^{2}}.

G acts "cyclically" on these elements, for example a.e = a, a.a = a^{2}, and a.a^{2}= e. so the 3x3 matrix corresponding to a should be:

[0 1 0]

[0 0 1]

[1 0 0]

it's easier to understand this if you view G as the 3-element subgroup of S3 given by:

{e, (1 2 3), (1 3 2)}, via Cayley's theorem (here "a" corresponds to the 3-cycle 1-->2-->3-->1).

for some reason, you took a.a to be e, but a is of order 3, not 2.

let's look at a "bigger example", the group D_{3}of order 6. i'll write the basis elements of the group ring Q[D_{3}] as {1,r,r^{2},s,rs,r^{2}s}.

clearly 1 maps to the 6x6 identity matrix, that's rather boring. so let's consider (left)-multiplication by r:

1-->r

r-->r^{2}

r^{2}-->1

s-->rs

rs-->r^{2}s

r^{2}s-->s, which gives the 6x6 matrix:

[0 1 0 0 0 0]

[0 0 1 0 0 0]

[1 0 0 0 0 0]

[0 0 0 0 1 0]

[0 0 0 0 0 1]

[0 0 0 1 0 0].

note that we can write this in block form as:

[A 0]

[0 A], where A is the 3x3 matrix we found for a in C_{3}above. this is no coincidence-we get this nice decomposition because <r> = C_{3}is a normal subgroup of D_{3}.

similarly, for r^{2}, we get the matrix:

[0 0 1 0 0 0]

[1 0 0 0 0 0]

[0 1 0 0 0 0]

[0 0 0 0 0 1]

[0 0 0 1 0 0]

[0 0 0 0 1 0].

look at how the reflections "mess this up", for s we get the matrix:

[0 0 0 1 0 0]

[0 0 0 0 0 1]

[0 0 0 0 1 0]

[1 0 0 0 0 0]

[0 0 1 0 0 0]

[0 1 0 0 0 0], we still have a block decomposition, but the "blocks" are in the wrong place.

i leave it to you to guess what the other 2 matrices look like, and to show this set of 6 matrices is indeed closed under matrix multiplication (hint: use their "block form").

the fact that we can reduce this group representation to 3x3 blocks might lead you to suspect that using a 6-dimensional space may not be the lowest possible dimension, and you'd be right.

EDIT: it appears that actually D&F would take the TRANSPOSE of these matrices, while J&L would use these (because of the way the different authors define the mappings).