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GroupTheory

  

Suzuki2B2

 

Calling Sequence

Parameters

Description

Examples

Compatibility

Calling Sequence

Suzuki2B2( q )

Parameters

q

-

: {posint,algebraic} : an odd power of 2, or an expression

Description

• 

The Suzuki groups Szq , of type ²B₂q, for an odd power q of 2, are a series of (typically) simple groups of Lie type, first constructed by M. Suzuki. They are defined only for q=22e+1 an odd power of 2 (where, here, 0e).

• 

The groups Szq should not be confused with the "Suzuki group" of order 448345497600, one of the sporadic finite simple groups. (See GroupTheory[SuzukiGroup].)

• 

The Suzuki groups Szq are notable among the finite simple groups in that they are the only finite non-abelian simple groups whose order is not divisible by 3.

• 

The Suzuki2B2( q ) command constructs a permutation group isomorphic to Szq , for admissible values of q up to 512.

• 

If the argument q is not numeric, or if it is an odd power of 2 greater than 512, then a symbolic group representing Szq is returned.

  

(The Suzuki groups Sz8 and Sz32 are also available by using the ExceptionalGroup command.)

Examples

withGroupTheory:

The smallest of the Suzuki groups is a non-simple group of order 20 that is, in fact, a soluble Frobenius group.

GSuzuki2B22

GroupTheory:-PermutationGroupmodule...end module,module...end module,degree=5

(1)

GroupOrderG

20

(2)

IsSimpleG

false

(3)

IsSolubleandIsFrobeniusGroupG

true

(4)

csCompositionSeriesG

CompositionSeriesmodule...end module

(5)

seqGroupOrderS,S=cs

20,10,5,1

(6)

useplots,GraphTheoryindisplayArrayDrawGraphCayleyGraphG,DrawSubgroupLatticeG,'labels'='ids'end use

For values of q larger than 2, the group Szq is simple.

GSuzuki2B232

GroupTheory:-PermutationGroupmodule...end module,module...end module,degree=1025

(7)

GroupOrderG

32537600

(8)

IsSimpleG

true

(9)

IsCNGroupG

true

(10)

OrderClassPolynomialG,x

7936000x41+15744000x31+6507520x25+1301504x5+1016800x4+31775x2+x

(11)

DisplayCharacterTableSuzuki2B28

C

1a

2a

4a

4b

5a

7a

7b

7c

13a

13b

13c

|C|

1

455

1820

1820

5824

4160

4160

4160

2240

2240

2240

 

 

 

 

 

 

 

 

 

 

 

 

χ__1

1

1

1

1

1

1

1

1

1

1

1

χ__2

14

−2

2I

2I

−1

0

0

0

1

1

1

χ__3

14

−2

2I

2I

−1

0

0

0

1

1

1

χ__4

35

3

−1

−1

0

0

0

0

−1213−11013+−1313+−11113

−1413−1613+−1713+−1913

−1813−11213+−1113+−1513

χ__5

35

3

−1

−1

0

0

0

0

−1813−11213+−1113+−1513

−1213−11013+−1313+−11113

−1413−1613+−1713+−1913

χ__6

35

3

−1

−1

0

0

0

0

−1413−1613+−1713+−1913

−1813−11213+−1113+−1513

−1213−11013+−1313+−11113

χ__7

64

0

0

0

−1

1

1

1

−1

−1

−1

χ__8

65

1

1

1

0

−127−157

−147−137

−167−117

0

0

0

χ__9

65

1

1

1

0

−167−117

−127−157

−147−137

0

0

0

χ__10

65

1

1

1

0

−147−137

−167−117

−127−157

0

0

0

χ__11

91

−5

−1

−1

1

0

0

0

0

0

0

For non-numeric arguments, a symbolic group is returned.

GSuzuki2B2q

GroupTheory:-Suzuki2B2q

(12)

GroupOrderG

q2q2+1q1

(13)

IsSimpleGassuming2<q

true

(14)

IsCNGroupG

true

(15)

A symbolic group is also returned if the numeric argument q exceeds 512.

GSuzuki2B22101

GroupTheory:-Suzuki2B22535301200456458802993406410752

(16)

ifactorGroupOrderG

2202580994910600935218735279937600503817460697534250373638732486577432339208719341117531003194129

(17)

MinimumPermutationRepresentationDegreeG

6427752177035961102167848369364650410088811975131171341205505

(18)

IsSimpleG

true

(19)

Compatibility

• 

The GroupTheory[Suzuki2B2] command was introduced in Maple 2020.

• 

For more information on Maple 2020 changes, see Updates in Maple 2020.

See Also

GroupTheory

GroupTheory[ExceptionalGroup]

GroupTheory[GroupOrder]

GroupTheory[IsCNGroup]

GroupTheory[IsFrobenius]

GroupTheory[IsSimple]

GroupTheory[SuzukiGroup]