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This group can be defined in the following equivalent ways:
- It is the metaplectic group , i.e., it has degree two over the field of real numbers.
- It is the unique double cover of special linear group:SL(2,R).
|order of a group||cardinality of the continuum||Same infinite cardinality as SL(2,R).|
|exponent of a group||infinite||Same as SL(2,R).|
|composition length||3||We can construct a composition series that begins with a subgroup of order two for which the quotient is SL(2,R), then proceeds to the inverse image of the center of in , and then to the whole group. The successive quotients are cyclic group:Z2, cyclic group:Z2, and PSL(2,R). See also projective special linear group is simple.|
|chief length||3||The unique chief series is the same as the composition series above.|
|dimension of a real Lie group||3|| As |
As double cover of : Same as dimension of , which is 3.
|quasisimple group||Yes||The center is isomorphic to cyclic group:Z4 and the inner automorphism group is isomorphic to PSL(2,R).|
|simple group||No||The center is proper and nontrivial|