Unitary group U3(3)
Derived Chevalley group G2(2)'
Order = 6 048 = 25.33.7.
Mult = 1.
Out = 2.
Robert Wilson's ATLAS page for U3(3) is available here.
U3(3): Length 61, 2-generator, 5-relator.
< x, y | x2 = y6 = (xy)7 = (xy2)3(xy-2)3 = (xyxy-2)3xy(xy-1)2 = 1 >
Remark: CMY-presentation 6.1. We have that [x, y] has order 3. x and y are R.A.Wilson's standard generators for U3(3).
U3(3): Length 80, 2-generator, 6-relator.
< x, y | x2 = y7 = (xy2)8 = [x, y]2 = xy[x, y2]2 = [x, y3]3 = 1 >
Remark: xy and xy3 have orders 12 and 8 respectively.
U3(3):2: Length 93, 5-generator, 16-relator.
< a, b, c, d, e | a2 = b2 = c2 = d2 = e2 = 1 = (ab)3 = [a, c] = [a, d] = [a, e] = (bc)3 = [b, d] = [b, e] = a(cd)4 = [c, e] = (de)3 = (bcde)8 >
Remark: LHS-presentation. We have that H = < a, b, c, d > = L3(2):2 = PGL2(7) has index 36 in G = U3(3):2 = G2(2).
U3(3) x 7: Length 56, 2-generator, 5-relator.
< x, y | x2 = y7 = [x, y]2 = xy[x, y2]2 = [x, y3]3 = 1 >
(U3(3) x 3):2: Length 61, 5-generator, 15-relator.
< a, b, c, d, e | a2 = b2 = c2 = d2 = e2 = 1 = (ab)3 = [a, c] = [a, d] = [a, e] = (bc)3 = [b, d] = [b, e] = a(cd)4 = [c, e] = (de)3 >
Remark: LHS-presentation. We have that H = < a, b, c, d > = L3(2):2 = PGL2(7) has index 108 in G = (U3(3) x 3):2.
Last updated 16th July, 1997
John N. Bray