challenge: According to the Gorenstein–Walter theorem, if a finite group G has a dihedral Sylow 2-subgroup, what are the possible isomorphism types of G/O(G)?
  
  A. A 2-group, or a subgroup of PSL2(q) for q an even prime power, or the alternating group A8
  B. A 2-group, or the alternating group A6, or a subgroup of PSL2(q) for q an odd prime power
  C. A 2-group, or the alternating group A7, or a subgroup of PGL2(q) containing PSL2(q) for q an odd prime power
  D. A 2-group, or the symmetric group S7, or a subgroup of PSL2(q) for q an even prime power
  Answer: 