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GroupTheory

  

ProjectiveSymplecticGroup

  

construct a permutation group isomorphic to a projective symplectic group

 

Calling Sequence

Parameters

Description

Examples

Compatibility

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 PSpn,q  is the quotient of the symplectic group Spn,q  by its center.

• 

The groups PSpn,q  are simple except for the groupPSp2,2  , which is isomorphic to S3 , the group mPSP( 2, 3 ), isomorphic to A4 , and the group PSp4,2  which is isomorphic to S6 .

• 

Note that for n=2 the groups PSpn,q  and PSLn,q  are isomorphic.

• 

The integer n must be even.

• 

The ProjectiveSymplecticGroup( n, q ) command returns a permutation group isomorphic to the projective symplectic group PSpn,q  for values of the parameters n and q in the implemented ranges.

• 

The implemented ranges for n and q are as follows:

n=2

q241

n=4

q20

n=6

q5

n=8

q3

n=10

q=2

• 

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

withGroupTheory:

GProjectiveSymplecticGroup2,64

GroupTheory:-PSL2,64

(1)

DegreeG

65

(2)

GroupOrderG

262080

(3)

IsTransitiveG

true

(4)

AreIsomorphicPSp2,2,Symm3

true

(5)

AreIsomorphicPSp2,3,Alt4

true

(6)

GroupOrderPSp4,3

25920

(7)

IsSimplePSp4,3

true

(8)

DisplayCharacterTablePSp4,3

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

123I32

12+3I32

−1

2

1

−1

0

32+I32

32I32

−I3

I3

0

1

12−32

12+−32

12+I32

12I32

χ__3

5

−3

1

12+3I32

123I32

−1

2

1

−1

0

32I32

32+I32

I3

−I3

0

1

12+−32

12−32

12I32

12+I32

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

723I32

72+3I32

1

1

2

0

0

123I32

12+3I32

−1

−1

−1

1

12−32

12+−32

12I32

12+I32

χ__6

10

2

−2

72+3I32

723I32

1

1

2

0

0

12+3I32

123I32

−1

−1

−1

1

12+−32

12−32

12+I32

12I32

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

329I32

32+9I32

−3

0

2

0

0

32I32

32+I32

I3

−I3

0

−1

0

0

12+I32

12I32

χ__12

30

6

2

32+9I32

329I32

−3

0

2

0

0

32+I32