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We initiate the study of the groups(  l  ,  m  ∣  n  ,  k  ∣  p  ,  q  )(l,m\mid n,k\mid p,q)    defined by the presentation⟨   a  ,  b   ∣    a  l   ,   b  m   ,    (   a  ⁢  b   )   n   ,    (    a  p   ⁢   b  q    )   k    ⟩\langle a,b\mid a^{l},b^{m},(ab)^{n},(a^{p}b^{q})^{k}\rangle   . Whenp  =  1p=1    andq  =   m  -  1q=m-1   , we obtain the group(  l  ,   m  ∣   n  ,  k    )(l,m\mid n,k)   , first systematically studied by Coxeter in 1939. In this paper, we restrict ourselves to the casel  =  2l=2    and1  n   +   1  k    ≤   1  2\frac{1}{n}+\frac{1}{k}\leq\frac{1}{2}    and give a complete determination as to which of the resulting groups are finite. We also, under certain broadly defined conditions, calculate generating sets for the second homotopy groupπ  2   ⁢   (  Z  )\pi_{2}(Z)   , where  is the space formed by attaching 2-cells corresponding to(   a  ⁢  b   )   n(ab)^{n}    and(   a  ⁢   b  q    )   k(ab^{q})^{k}    to the wedge sum of the Eilenberg–MacLane spaces  and , whereπ  1   ⁢   (  X  )    ≅   C  2\pi_{1}(X)\cong C_{2}    andπ  1   ⁢   (  Y  )    ≅   C  m\pi_{1}(Y)\cong C_{m}   ; in particular,π  1    (  Z  )   ≅   (  2  ,  m  ∣  n  ,  k  ∣  1  ,  q  )\pi_{1}(Z)\cong(2,m\mid n,k\mid 1,q)   .

The groups (2, 𝑚 | 𝑛, 𝑘 | 1, 𝑞): Finiteness and homotopy