Quasisimple group
In mathematics, a quasisimple group (also known as a covering group) is a group that is a perfect central extension E of a simple group S. In other words, there is a short exact sequence 1 → Z ( E ) → E → S → 1 {\displaystyle 1\to Z(E)\to E\to S\to 1} such that E = [ E , E ] {\displaystyle E=[E,E]} , where Z ( E ) {\displaystyle Z(E)} denotes the center of E and [ , ] denotes the commutator. Equivalently, a group is quasisimple if it is equal to its commutator subgroup and its inner automorphism group Inn(G) (its quotient by its center) is simple (and it follows Inn(G) must be non-abelian simple, as inner automorphism groups are never non-trivial cyclic).