The following information is available for L_{2}(7) = L_{3}(2):
Standard generators of L_{2}(7):2 = PGL_{2}(7) = L_{3}(2):2 are
c and d where
c is in class 2B, d has order 3,
cd has order 8 and cdcdd has order 4. These conditions imply that cd is in class 8A.
Standard generators of either of the double covers 2.PGL_{2}(7) are
preimages C and D where
D has order 3.
To obtain our standard generators for L_{2}(7):2 = L_{3}(2):2 we may take c to be the above automorphism and d = b^{ababb}.
This forces a = [c, d]^{2} = (cddcd)^{2} = (ddcdc)^{2} and b = (dc)^{3}(ddc)^{3}.
< a, b | a^{2} = b^{3} = (ab)^{7} = [a, b]^{4} = 1 >.
< c, d | c^{2} = d^{3} = (cd)^{8} = [c, d]^{4} = 1 >.
Last updated 11th February 1998,