Group W(F_{4}) = GO_{4}^{+}(3) Order = 1152 = 2^{7}.3^{2}. Mult = 2^{2}. Out = D_{8} [I think].^{ } |
Group PGO_{4}^{+}(3) = (A_{4} × A_{4}):2^{2} Order = 576 = 2^{6}.3^{2}. Mult = 2^{2}. Out = 2.^{ } |
Group O_{4}^{+}(3) = A_{4} × A_{4} Order = 144 = 2^{4}.3^{2}. Mult = 2^{2} × 3. Out = D_{8}.^{ } |
The following information is available for W(F_{4}):
We shall give faithful representations of W(F_{4}) = GO_{4}^{+}(3) = 2^{1+4}:(S_{3} × S_{3}) and PGO_{4}^{+}(3) = (A_{4} × A_{4}):2^{2} = 2^{4}:(S_{3} × S_{3}). Any other image of W(F_{4}) is a quotient of S_{3} × S_{3}. (However, note that PGO_{4}^{+}(3) is not isomorphic to S_{4} × S_{4}.)
Type II standard generators of W(F_{4}) are x and y where x is in class 2C (see below), y has order 6, xy has order 6 and xyy has order 4.
We may take
x = bd and y = acd.
Conversely, we have
a = y^{3},
b = x(xy^{3})^{3} = y^{3}xy^{3}xy^{3},
c = yxy^{3}xy^{3}x
and d = (xy^{3})^{3} = (y^{3}x)^{3}.
(These maps are exact inverses of each other, not merely inverses up to
automorphisms.)
Standard generators of the image group PGO_{4}^{+}(3) are images of standard generators of W(F_{4}), and have been given the same labelling.
< a, b, c, d | a^{2} = b^{2} = c^{2} = d^{2} = (ab)^{3} = [a, c] = [a, d] = (bc)^{4} = [b, d] = (cd)^{3} = 1 >.
The centre is generated by (abcd)^{6}.
< x, y | x^{2} = y^{6} = (xy)^{6} = (xy^{2})^{4} = (xyxyxy^{-2})^{2} = 1 >.
The centre is generated by [x, y]^{3}.