ProjectiveSymplecticGroup

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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 $PSp (n, q)$ is the quotient of the symplectic group $Sp (n, q)$ by its center.

•	The groups $PSp (n, q)$ are simple except for the group $PSp (2, 2)$ , which is isomorphic to $S_{3}$ , the group $PSp (2, 3)$ , isomorphic to $A_{4}$ , and the group $PSp (4, 2)$ which is isomorphic to $S_{6}$ .

•	Note that for $n = 2$ the groups $PSp (n, q)$ and $PSL (n, q)$ are isomorphic.

•	The integer $n$ must be even.

•	The ProjectiveSymplecticGroup( n, q ) command returns a permutation group isomorphic to the projective symplectic group $PSp (n, q)$ .

•	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

>	$with (GroupTheory) &colon;$

>	$G ≔ ProjectiveSymplecticGroup (2, 64)$

$G ≔ PSL (2, 64)$

(1)

>	$Degree (G)$

$65$

(2)

>	$GroupOrder (G)$

$262080$

(3)

>	$IsTransitive (G)$

$true$

(4)

>	$AreIsomorphic (PSp (2, 2), Symm (3))$

$true$

(5)

>	$AreIsomorphic (PSp (2, 3), Alt (4))$

$true$

(6)

>	$GroupOrder (PSp (4, 3))$

$25920$

(7)

>	$IsSimple (PSp (4, 3))$

$true$

(8)

>	$Display (CharacterTable (PSp (4, 3)))$

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

$χ__1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
$χ__2$	$5$	$−3$	$1$	$\frac{1}{2} - \frac{3 I \sqrt{3}}{2}$	$\frac{1}{2} + \frac{3 I \sqrt{3}}{2}$	$−1$	$2$	$1$	$−1$	$0$	$- \frac{3}{2} + \frac{I \sqrt{3}}{2}$	$- \frac{3}{2} - \frac{I \sqrt{3}}{2}$	$I \sqrt{3}$	$−I \sqrt{3}$	$0$	$1$	$\frac{1}{2} + \frac{\sqrt{−3}}{2}$	$\frac{1}{2} - \frac{\sqrt{−3}}{2}$	$- \frac{1}{2} + \frac{I \sqrt{3}}{2}$	$- \frac{1}{2} - \frac{I \sqrt{3}}{2}$
$χ__3$	$5$	$−3$	$1$	$\frac{1}{2} + \frac{3 I \sqrt{3}}{2}$	$\frac{1}{2} - \frac{3 I \sqrt{3}}{2}$	$−1$	$2$	$1$	$−1$	$0$	$- \frac{3}{2} - \frac{I \sqrt{3}}{2}$	$- \frac{3}{2} + \frac{I \sqrt{3}}{2}$	$−I \sqrt{3}$	$I \sqrt{3}$	$0$	$1$	$\frac{1}{2} - \frac{\sqrt{−3}}{2}$	$\frac{1}{2} + \frac{\sqrt{−3}}{2}$	$- \frac{1}{2} - \frac{I \sqrt{3}}{2}$	$- \frac{1}{2} + \frac{I \sqrt{3}}{2}$
$χ__4$	$6$	$−2$	$2$	$−3$	$−3$	$3$	$0$	$2$	$0$	$1$	$1$	$1$	$1$	$1$	$−2$	$−1$	$0$	$0$	$−1$	$−1$
$χ__5$	$10$	$2$	$−2$	$- \frac{7}{2} - \frac{3 I \sqrt{3}}{2}$	$- \frac{7}{2} + \frac{3 I \sqrt{3}}{2}$	$1$	$1$	$2$	$0$	$0$	$\frac{1}{2} - \frac{3 I \sqrt{3}}{2}$	$\frac{1}{2} + \frac{3 I \sqrt{3}}{2}$	$−1$	$−1$	$−1$	$1$	$- \frac{1}{2} + \frac{\sqrt{−3}}{2}$	$- \frac{1}{2} - \frac{\sqrt{−3}}{2}$	$\frac{1}{2} - \frac{I \sqrt{3}}{2}$	$\frac{1}{2} + \frac{I \sqrt{3}}{2}$
$χ__6$	$10$	$2$	$−2$	$- \frac{7}{2} + \frac{3 I \sqrt{3}}{2}$	$- \frac{7}{2} - \frac{3 I \sqrt{3}}{2}$	$1$	$1$	$2$	$0$	$0$	$\frac{1}{2} + \frac{3 I \sqrt{3}}{2}$	$\frac{1}{2} - \frac{3 I \sqrt{3}}{2}$	$−1$	$−1$	$−1$	$1$	$- \frac{1}{2} - \frac{\sqrt{−3}}{2}$	$- \frac{1}{2} + \frac{\sqrt{−3}}{2}$	$\frac{1}{2} + \frac{I \sqrt{3}}{2}$	$\frac{1}{2} - \frac{I \sqrt{3}}{2}$
$χ__7$	$15$	$−1$	$−1$	$6$	$6$	$3$	$0$	$3$	$−1$	$0$	$2$	$2$	$−1$	$−1$	$2$	$−1$	$0$	$0$	$0$	$0$
$χ__8$	$15$	$7$	$3$	$−3$	$−3$	$0$	$3$	$−1$	$1$	$0$	$1$	$1$	$−2$	$−2$	$1$	$0$	$0$	$0$	$−1$	$−1$
$χ__9$	$20$	$4$	$4$	$2$	$2$	$5$	$−1$	$0$	$0$	$0$	$−2$	$−2$	$1$	$1$	$1$	$1$	$−1$	$−1$	$0$	$0$
$χ__10$	$24$	$8$	$0$	$6$	$6$	$0$	$3$	$0$	$0$	$−1$	$2$	$2$	$2$	$2$	$−1$	$0$	$0$	$0$	$0$	$0$
$χ__11$	$30$	$6$	$2$	$- \frac{3}{2} - \frac{9 I \sqrt{3}}{2}$	$- \frac{3}{2} + \frac{9 I \sqrt{3}}{2}$	$−3$	$0$	$2$	$0$	$0$	$- \frac{3}{2} - \frac{I \sqrt{3}}{2}$	$- \frac{3}{2} + \frac{I \sqrt{3}}{2}$	$−I \sqrt{3}$	$I \sqrt{3}$	$0$	$−1$	$0$	$0$	$\frac{1}{2} + \frac{I \sqrt{3}}{2}$	$\frac{1}{2} - \frac{I \sqrt{3}}{2}$
$χ__12$	$30$	$6$	$2$	$- \frac{3}{2} + \frac{9 I \sqrt{3}}{2}$	$- \frac{3}{2} - \frac{9 I \sqrt{3}}{2}$	$−3$	$0$	$2$	$0$	$0$	$- \frac{3}{2} + \frac{I \sqrt{3}}{2}$	$- \frac{3}{2} - \frac{I \sqrt{3}}{2}$	$I \sqrt{3}$	$−I \sqrt{3}$	$0$	$−1$	$0$	$0$	$\frac{1}{2} - \frac{I \sqrt{3}}{2}$	$\frac{1}{2} + \frac{I \sqrt{3}}{2}$
$χ__13$	$30$	$−10$	$2$	$3$	$3$	$3$	$3$	$−2$	$0$	$0$	$−1$	$−1$	$−1$	$−1$	$−1$	$−1$	$0$	$0$	$1$	$1$
$χ__14$	$40$	$−8$	$0$	$- 5 - 3 I \sqrt{3}$	$- 5 + 3 I \sqrt{3}$	$−2$	$1$	$0$	$0$	$0$	$1 - I \sqrt{3}$	$1 + I \sqrt{3}$	$1 + I \sqrt{3}$	$1 - I \sqrt{3}$	$1$	$0$	$- \frac{1}{2} - \frac{\sqrt{−3}}{2}$	$- \frac{1}{2} + \frac{\sqrt{−3}}{2}$	$0$	$0$
$χ__15$	$40$	$−8$	$0$	$- 5 + 3 I \sqrt{3}$	$- 5 - 3 I \sqrt{3}$	$−2$	$1$	$0$	$0$	$0$	$1 + I \sqrt{3}$	$1 - I \sqrt{3}$	$1 - I \sqrt{3}$	$1 + I \sqrt{3}$	$1$	$0$	$- \frac{1}{2} + \frac{\sqrt{−3}}{2}$	$- \frac{1}{2} - \frac{\sqrt{−3}}{2}$	$0$	$0$
$χ__16$	$45$	$−3$	$−3$	$\frac{9}{2} - \frac{9 I \sqrt{3}}{2}$	$\frac{9}{2} + \frac{9 I \sqrt{3}}{2}$	$0$	$0$	$1$	$1$	$0$	$- \frac{3}{2} + \frac{3 I \sqrt{3}}{2}$	$- \frac{3}{2} - \frac{3 I \sqrt{3}}{2}$	$0$	$0$	$0$	$0$	$0$	$0$	$- \frac{1}{2} - \frac{I \sqrt{3}}{2}$	$- \frac{1}{2} + \frac{I \sqrt{3}}{2}$
$χ__17$	$45$	$−3$	$−3$	$\frac{9}{2} + \frac{9 I \sqrt{3}}{2}$	$\frac{9}{2} - \frac{9 I \sqrt{3}}{2}$	$0$	$0$	$1$	$1$	$0$	$- \frac{3}{2} - \frac{3 I \sqrt{3}}{2}$	$- \frac{3}{2} + \frac{3 I \sqrt{3}}{2}$	$0$	$0$	$0$	$0$	$0$	$0$	$- \frac{1}{2} + \frac{I \sqrt{3}}{2}$	$- \frac{1}{2} - \frac{I \sqrt{3}}{2}$
$χ__18$	$60$	$−4$	$4$	$6$	$6$	$−3$	$−3$	$0$	$0$	$0$	$2$	$2$	$−1$	$−1$	$−1$	$1$	$0$	$0$	$0$	$0$
$χ__19$	$64$	$0$	$0$	$−8$	$−8$	$4$	$−2$	$0$	$0$	$−1$	$0$	$0$	$0$	$0$	$0$	$0$	$1$	$1$	$0$	$0$
$χ__20$	$81$	$9$	$−3$	$0$	$0$	$0$	$0$	$−3$	$−1$	$1$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$

>	$IsSimple (PSp (4, 2))$

$false$

(9)

>	$AreIsomorphic (PSp (4, 2), Symm (6))$

$true$

(10)

The smallest simple group whose order is a perfect square.

>	$G ≔ PSp (4, 7)$

$G ≔ PSp (4, 7)$

(11)

>	$IsSimple (G)$

$true$

(12)

>	$GroupOrder (G) = 11760^{2}$

$138297600 = 138297600$

(13)

>	$ClassifyFiniteSimpleGroup (PSp (2, 4))$

$⟨CFSG: Alternating Group A_{5}⟩$

(14)

>	$GroupOrder (PSp (4, q))$

$\frac{q^{4} (q^{2} - 1) (q^{4} - 1)}{igcd (2, q - 1)}$

(15)

>	$IsSimple (PSp (n, q))$

$\{\begin{array}{c} \{\begin{array}{c} false & q = 2 \\ false & q = 3 \\ true & otherwise \end{array} & n = 2 \\ \{\begin{array}{c} false & q = 2 \\ true & otherwise \end{array} & n = 4 \\ true & otherwise \end{array}$

(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.

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