  We discuss three important classes of three-qubit entangled states and their
encoding into quantum gates, finite groups and Lie algebras. States of the GHZ
and W-type correspond to pure tripartite and bipartite entanglement,
respectively. We introduce another generic class B of three-qubit states, that
have balanced entanglement over two and three parties. We show how to realize
the largest cristallographic group $W(E_8)$ in terms of three-qubit gates (with
real entries) encoding states of type GHZ or W [M. Planat, {\it Clifford group
dipoles and the enactment of Weyl/Coxeter group $W(E_8)$ by entangling gates},
Preprint 0904.3691 (quant-ph)]. Then, we describe a peculiar "condensation" of
$W(E_8)$ into the four-letter alternating group $A_4$, obtained from a chain of
maximal subgroups. Group $A_4$ is realized from two B-type generators and found
to correspond to the Lie algebra $sl(3,\mathbb{C})\oplus u(1)$. Possible
applications of our findings to particle physics and the structure of genetic
code are also mentioned.
