Chevalley group G2(4)
Order = 251 596 800 = 212.33.52.7.13.
Mult = 2.
Out = 2.
Robert Wilson's ATLAS page for G2(4) is available here.
G2(4): Length 82, 3-generator, 9-relator.
< x, y, t | x3 = y3 = (xy)4 = (xy-1)5 = t2 = (xt)3 = (xyt)3 = [t, y-1xyx] = (xy-1t)6 = 1 >
Remark: This is the progenitor (22)*6:3.A6 factored out by the relation [(0, 1, 2, 3, 4)t0]6 = 1. We have that N = < x, y > = 3.A6 of order 1 080 and index 232 960 (in G = G2(4)) is the control subgroup and that H = < t, tx, y > = J2 has order 604 800 and index 416 in G = G2(4).
G2(4):2: Length 145, 7-generator, 29-relator.
< a, b, c, d, e, f, g | a2 = b2 = c2 = d2 = e2 = f2 = g2 = 1 = (ab)3 = [a, c] = [a, d] = [a, e] = [a, f] = [a, g] = (bc)3 = [b, d] = [b, e] = [b, f] = [b, g] = a(cd)4 = [c, e] = [c, f] = [c, g] = (de)3 = [d, f] = [d, g] = (ef)3 = [e, g] = (fg)3 = (bcde)8 >
Remark: LHS-presentation. We have that H = < a, b, c, d, e, f > = J2:2 has index 416 in G = G2(4):2.
Last updated 16th July, 1997
John N. Bray