Standard generators of the Conway group Co1 are a and b where a is in class 2B, b is in class 3C, ab has order 40 and ababb has order 6. (Source: ATLAS)
Standard generators of the double cover 2.Co1 = Co0 are preimages A and B where B has order 3 and ABABABABBABABBABB has order 33. (Source: ATLAS)