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GroupTheory

 ProjectiveSymplecticGroup
 construct a permutation group isomorphic to a projective symplectic group

 Calling Sequence ProjectiveSymplecticGroup(n, q) PSp(n, q)

Parameters

 n - an even positive integer q - power of a prime number

Description

 • The projective symplectic group $PSp\left(n,q\right)$ is the quotient of the symplectic group $Sp\left(n,q\right)$ by its center.
 • The groups $PSp\left(n,q\right)$ are simple except for the group $PSp\left(2,2\right)$ , which is isomorphic to ${\mathbf{S}}_{3}$ , the group $PSp\left(2,3\right)$ , isomorphic to ${\mathbf{A}}_{4}$ , and the group $PSp\left(4,2\right)$ which is isomorphic to ${\mathbf{S}}_{6}$ .
 • Note that for $n=2$ the groups $PSp\left(n,q\right)$ and $PSL\left(n,q\right)$ are isomorphic.
 • The integer $n$ must be even.
 • The ProjectiveSymplecticGroup( n, q ) command returns a permutation group isomorphic to the projective symplectic group $PSp\left(n,q\right)$ .
 • The PSp( n, q ) command is provided as an abbreviation.
 • In the Standard Worksheet interface, you can insert this group into a document or worksheet by using the Group Constructors palette.

Examples

 > $\mathrm{with}\left(\mathrm{GroupTheory}\right):$
 > $G≔\mathrm{ProjectiveSymplecticGroup}\left(2,64\right)$
 ${\mathrm{PSL}}{}\left({2}{,}{64}\right)$ (1)
 > $\mathrm{Degree}\left(G\right)$
 ${65}$ (2)
 > $\mathrm{GroupOrder}\left(G\right)$
 ${262080}$ (3)
 > $\mathrm{IsTransitive}\left(G\right)$
 ${\mathrm{true}}$ (4)
 > $\mathrm{AreIsomorphic}\left(\mathrm{PSp}\left(2,2\right),\mathrm{Symm}\left(3\right)\right)$
 ${\mathrm{true}}$ (5)
 > $\mathrm{AreIsomorphic}\left(\mathrm{PSp}\left(2,3\right),\mathrm{Alt}\left(4\right)\right)$
 ${\mathrm{true}}$ (6)
 > $\mathrm{GroupOrder}\left(\mathrm{PSp}\left(4,3\right)\right)$
 ${25920}$ (7)
 > $\mathrm{IsSimple}\left(\mathrm{PSp}\left(4,3\right)\right)$
 ${\mathrm{true}}$ (8)
 > $\mathrm{Display}\left(\mathrm{CharacterTable}\left(\mathrm{PSp}\left(4,3\right)\right)\right)$

 C 1a 2a 2b 3a 3b 3c 3d 4a 4b 5a 6a 6b 6c 6d 6e 6f 9a 9b 12a 12b |C| 1 45 270 40 40 240 480 540 3240 5184 360 360 720 720 1440 2160 2880 2880 2160 2160 $\mathrm{χ__1}$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $\mathrm{χ__2}$ $5$ $-3$ $1$ $\frac{1}{2}-\frac{3I\sqrt{3}}{2}$ $\frac{1}{2}+\frac{3I\sqrt{3}}{2}$ $-1$ $2$ $1$ $-1$ $0$ $-\frac{3}{2}-\frac{I\sqrt{3}}{2}$ $-\frac{3}{2}+\frac{I\sqrt{3}}{2}$ $-I\sqrt{3}$ $I\sqrt{3}$ $0$ $1$ $\frac{1}{2}+\frac{\sqrt{-3}}{2}$ $\frac{1}{2}-\frac{\sqrt{-3}}{2}$ $-\frac{1}{2}+\frac{I\sqrt{3}}{2}$ $-\frac{1}{2}-\frac{I\sqrt{3}}{2}$ $\mathrm{χ__3}$ $5$ $-3$ $1$ $\frac{1}{2}+\frac{3I\sqrt{3}}{2}$ $\frac{1}{2}-\frac{3I\sqrt{3}}{2}$ $-1$ $2$ $1$ $-1$ $0$ $-\frac{3}{2}+\frac{I\sqrt{3}}{2}$ $-\frac{3}{2}-\frac{I\sqrt{3}}{2}$ $I\sqrt{3}$ $-I\sqrt{3}$ $0$ $1$ $\frac{1}{2}-\frac{\sqrt{-3}}{2}$ $\frac{1}{2}+\frac{\sqrt{-3}}{2}$ $-\frac{1}{2}-\frac{I\sqrt{3}}{2}$ $-\frac{1}{2}+\frac{I\sqrt{3}}{2}$ $\mathrm{χ__4}$ $6$ $-2$ $2$ $-3$ $-3$ $3$ $0$ $2$ $0$ $1$ $1$ $1$ $1$ $1$ $-2$ $-1$ $0$ $0$ $-1$ $-1$ $\mathrm{χ__5}$ $10$ $2$ $-2$ $-\frac{7}{2}-\frac{3I\sqrt{3}}{2}$ $-\frac{7}{2}+\frac{3I\sqrt{3}}{2}$ $1$ $1$ $2$ $0$ $0$ $\frac{1}{2}+\frac{3I\sqrt{3}}{2}$ $\frac{1}{2}-\frac{3I\sqrt{3}}{2}$ $-1$ $-1$ $-1$ $1$ $-\frac{1}{2}+\frac{\sqrt{-3}}{2}$ $-\frac{1}{2}-\frac{\sqrt{-3}}{2}$ $\frac{1}{2}-\frac{I\sqrt{3}}{2}$ $\frac{1}{2}+\frac{I\sqrt{3}}{2}$ $\mathrm{χ__6}$ $10$ $2$ $-2$ $-\frac{7}{2}+\frac{3I\sqrt{3}}{2}$ $-\frac{7}{2}-\frac{3I\sqrt{3}}{2}$ $1$ $1$ $2$ $0$ $0$ $\frac{1}{2}-\frac{3I\sqrt{3}}{2}$ $\frac{1}{2}+\frac{3I\sqrt{3}}{2}$ $-1$ $-1$ $-1$ $1$ $-\frac{1}{2}-\frac{\sqrt{-3}}{2}$ $-\frac{1}{2}+\frac{\sqrt{-3}}{2}$ $\frac{1}{2}+\frac{I\sqrt{3}}{2}$ $\frac{1}{2}-\frac{I\sqrt{3}}{2}$ $\mathrm{χ__7}$ $15$ $-1$ $-1$ $6$ $6$ $3$ $0$ $3$ $-1$ $0$ $2$ $2$ $-1$ $-1$ $2$ $-1$ $0$ $0$ $0$ $0$ $\mathrm{χ__8}$ $15$ $7$ $3$ $-3$ $-3$ $0$ $3$ $-1$ $1$ $0$ $1$ $1$ $-2$ $-2$ $1$ $0$ $0$ $0$ $-1$ $-1$ $\mathrm{χ__9}$ $20$ $4$ $4$ $2$ $2$ $5$ $-1$ $0$ $0$ $0$ $-2$ $-2$ $1$ $1$ $1$ $1$ $-1$ $-1$ $0$ $0$ $\mathrm{χ__10}$ $24$ $8$ $0$ $6$ $6$ $0$ $3$ $0$ $0$ $-1$ $2$ $2$ $2$ $2$ $-1$ $0$ $0$ $0$ $0$ $0$ $\mathrm{χ__11}$ $30$ $6$ $2$ $-\frac{3}{2}-\frac{9I\sqrt{3}}{2}$ $-\frac{3}{2}+\frac{9I\sqrt{3}}{2}$ $-3$ $0$ $2$ $0$ $0$ $-\frac{3}{2}+\frac{I\sqrt{3}}{2}$ $-\frac{3}{2}-\frac{I\sqrt{3}}{2}$ $I\sqrt{3}$ $-I\sqrt{3}$ $0$ $-1$ $0$ $0$ $\frac{1}{2}+\frac{I\sqrt{3}}{2}$ $\frac{1}{2}-\frac{I\sqrt{3}}{2}$ $\mathrm{χ__12}$ $30$ $6$ $2$ $-\frac{3}{2}+\frac{9I\sqrt{3}}{2}$ $-\frac{3}{2}-\frac{9I\sqrt{3}}{2}$ $-3$ $0$ $2$ $0$ $0$ $-\frac{3}{2}-\frac{I\sqrt{3}}{2}$ $-\frac{3}{2}+\frac{I\sqrt{3}}{2}$ $-I\sqrt{3}$ $I\sqrt{3}$ $0$ $-1$ $0$ $0$ $\frac{1}{2}-\frac{I\sqrt{3}}{2}$ $\frac{1}{2}+\frac{I\sqrt{3}}{2}$ $\mathrm{χ__13}$ $30$ $-10$ $2$ $3$ $3$ $3$ $3$ $-2$ $0$ $0$ $-1$ $-1$ $-1$ $-1$ $-1$ $-1$ $0$ $0$ $1$ $1$ $\mathrm{χ__14}$ $40$ $-8$ $0$ $-5-3I\sqrt{3}$ $-5+3I\sqrt{3}$ $-2$ $1$ $0$ $0$ $0$ $1+I\sqrt{3}$ $1-I\sqrt{3}$ $1-I\sqrt{3}$ $1+I\sqrt{3}$ $1$ $0$ $-\frac{1}{2}-\frac{\sqrt{-3}}{2}$ $-\frac{1}{2}+\frac{\sqrt{-3}}{2}$ $0$ $0$ $\mathrm{χ__15}$ $40$ $-8$ $0$ $-5+3I\sqrt{3}$ $-5-3I\sqrt{3}$ $-2$ $1$ $0$ $0$ $0$ $1-I\sqrt{3}$ $1+I\sqrt{3}$ $1+I\sqrt{3}$ $1-I\sqrt{3}$ $1$ $0$ $-\frac{1}{2}+\frac{\sqrt{-3}}{2}$ $-\frac{1}{2}-\frac{\sqrt{-3}}{2}$ $0$ $0$ $\mathrm{χ__16}$ $45$ $-3$ $-3$ $\frac{9}{2}-\frac{9I\sqrt{3}}{2}$ $\frac{9}{2}+\frac{9I\sqrt{3}}{2}$ $0$ $0$ $1$ $1$ $0$ $-\frac{3}{2}-\frac{3I\sqrt{3}}{2}$ $-\frac{3}{2}+\frac{3I\sqrt{3}}{2}$ $0$ $0$ $0$ $0$ $0$ $0$ $-\frac{1}{2}-\frac{I\sqrt{3}}{2}$ $-\frac{1}{2}+\frac{I\sqrt{3}}{2}$ $\mathrm{χ__17}$ $45$ $-3$ $-3$ $\frac{9}{2}+\frac{9I\sqrt{3}}{2}$ $\frac{9}{2}-\frac{9I\sqrt{3}}{2}$ $0$ $0$ $1$ $1$ $0$ $-\frac{3}{2}+\frac{3I\sqrt{3}}{2}$ $-\frac{3}{2}-\frac{3I\sqrt{3}}{2}$ $0$ $0$ $0$ $0$ $0$ $0$ $-\frac{1}{2}+\frac{I\sqrt{3}}{2}$ $-\frac{1}{2}-\frac{I\sqrt{3}}{2}$ $\mathrm{χ__18}$ $60$ $-4$ $4$ $6$ $6$ $-3$ $-3$ $0$ $0$ $0$ $2$ $2$ $-1$ $-1$ $-1$ $1$ $0$ $0$ $0$ $0$ $\mathrm{χ__19}$ $64$ $0$ $0$ $-8$ $-8$ $4$ $-2$ $0$ $0$ $-1$ $0$ $0$ $0$ $0$ $0$ $0$ $1$ $1$ $0$ $0$ $\mathrm{χ__20}$ $81$ $9$ $-3$ $0$ $0$ $0$ $0$ $-3$ $-1$ $1$ $0$ $0$ $0$ $0$ $0$ $0$ $0$ $0$ $0$ $0$

 > $\mathrm{IsSimple}\left(\mathrm{PSp}\left(4,2\right)\right)$
 ${\mathrm{false}}$ (9)
 > $\mathrm{AreIsomorphic}\left(\mathrm{PSp}\left(4,2\right),\mathrm{Symm}\left(6\right)\right)$
 ${\mathrm{true}}$ (10)

The smallest simple group whose order is a perfect square.

 > $G≔\mathrm{PSp}\left(4,7\right)$
 ${\mathrm{GroupTheory}}{:-}{\mathrm{PSp}}{}\left({4}{,}{7}\right)$ (11)
 > $\mathrm{IsSimple}\left(G\right)$
 ${\mathrm{true}}$ (12)
 > $\mathrm{GroupOrder}\left(G\right)={11760}^{2}$
 ${138297600}{=}{138297600}$ (13)
 > $\mathrm{ClassifyFiniteSimpleGroup}\left(\mathrm{PSp}\left(2,4\right)\right)$
 ${\mathrm{AlternatingCFSG}}{}\left({5}\right)$ (14)
 > $\mathrm{GroupOrder}\left(\mathrm{PSp}\left(4,q\right)\right)$
 $\frac{{{q}}^{{4}}{}\left({{q}}^{{2}}{-}{1}\right){}\left({{q}}^{{4}}{-}{1}\right)}{{\mathrm{igcd}}{}\left({2}{,}{q}{-}{1}\right)}$ (15)
 > $\mathrm{IsSimple}\left(\mathrm{PSp}\left(n,q\right)\right)$
 $\left\{\begin{array}{cc}\left\{\begin{array}{cc}{\mathrm{false}}& {q}{=}{2}\\ {\mathrm{false}}& {q}{=}{3}\\ {\mathrm{true}}& {\mathrm{otherwise}}\end{array}\right\& {n}{=}{2}\\ \left\{\begin{array}{cc}{\mathrm{false}}& {q}{=}{2}\\ {\mathrm{true}}& {\mathrm{otherwise}}\end{array}\right\& {n}{=}{4}\\ {\mathrm{true}}& {\mathrm{otherwise}}\end{array}\right\$ (16)

Compatibility

 • The GroupTheory[ProjectiveSymplecticGroup] command was introduced in Maple 17.
 • For more information on Maple 17 changes, see Updates in Maple 17.
 • The GroupTheory[ProjectiveSymplecticGroup] command was updated in Maple 2020.