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# It is the [[metaplectic group]] <math>Mp(2,\R)</math>, 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)]].

==Arithmetic functions==

{| class="sortable" border="1"

! Function !! Value !! Similar groups !! Explanation

|-

| [[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)]].

|-

| {{arithmetic function value|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 <math>SL(2,\R)</math> in <math>Mp(2,\R)</math>, 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]].

|-

| {{arithmetic function value|chief length|3}} || || The unique chief series is the same as the composition series above.

|-

| {{arithmetic function value|dimension of a real Lie group|3}} || || As <math>Mp(n,\R), n = 2: n(n+1)/2 = 2(2 + 1)/2 = 3</math><br>As double cover of <math>SL(2,\R)</math>: Same as dimension of <math>SL(2,\R)</math>, which is 3.

|}

==Group properties==

{| class="sortable" border="1"

! Property !! Satisfied? !! Explanation

|-

| [[dissatisfies property::abelian group]] || No ||

|-

| [[dissatisfies property::nilpotent group]] || No ||

|-

| [[dissatisfies property::solvable group]] || No ||

|-

| [[satisfies property::quasisimple group]] || Yes || The center is isomorphic to [[cyclic group:Z4]] and the [[inner automorphism group]] is isomorphic to [[PSL(2,R)]].

|-

| [[dissatisfies property::simple group]] || No || The center is proper and nontrivial

|}

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