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Group Library

Here we describe the library of groups that comes with Group Explorer.

Firstly, we do this for three reasons related to the Group Explorer project.

bulletUsers can see what groups come with Group Explorer,
bulletAs the group library grows, users can selectively download groups they want, and
bulletUsers who wish to contribute new .gp files to the project will get them posted here.

Secondly, and more importantly, we do this to introduce a different flavor of small group classification reference to the web.  Although there are many excellent (and more informative) classifications of small finite groups available online (see our Links page), none of them seems aimed at a beginning audience.  This classification is a bit more accessible due to its graphical component, and the links to definitions and mentions of families of groups.

Groups that come with Group Explorer

Group Explorer comes with all groups of order twenty or less (except the trivial group), and a few groups of higher orders.  We classify them here by their order, and you can click on the filename to download each individually.  Refer also to our Links Page for other locations on the web that classify small finite groups.  The following websites deserve credit for being the targets of many links below: MathWorld, PlanetMath, and Wikipedia.

To jump down to the groups of a certain order, click the order here:

2   3   4   5   6   7   8   9   10   11   12   13   14   15   16   17   18   19   20   higher

Groups of order 2

Name

Description

Cyclic group of order 2
Also C2

Cayley diagram

Filename

Z_2.gp

Presentation

Notes

Part of the family of cyclic groups, Cn or Zn, for integers n > 2.
All cyclic groups are abelian.
All groups of prime order are cyclic.

Groups of order 3

Name

Description

Cyclic group of order 3
Also C3

Cayley diagram

Filename

Z_3.gp

Presentation

Notes

Part of the family of cyclic groups, Cn or Zn, for integers n > 2.
All cyclic groups are abelian.
All groups of prime order are cyclic.

Groups of order 4

Name

Description

Cyclic group of order 4
Also C4

Cayley diagram

Filename

Z_4.gp

Presentation

Notes

Part of the family of cyclic groups, Cn or Zn, for integers n > 2.
All cyclic groups are abelian.


Name

Description

Klein 4-group
Also D2

Cayley diagram

Filename

V_4.gp

Presentation

Notes

This group is abelian, and is therefore expressible as a product of cyclic groups (Z2 x Z2).  This is called the Fundamental Theorem of Abelian Groups, which is mentioned in our Abelian example page.  Read about the direct product operation here.
This group is the group of symmetries of a non-square rectangle.
The Klein 4-group is technically also a member of the family of dihedral groups; it is isomorphic to D2.  The dihedral group Dn is the group of symmetries of an n-gon in R3.

Groups of order 5

Name

Description

Cyclic group of order 5
Also C5

Cayley diagram

Filename

Z_5.gp

Presentation

Notes

Part of the family of cyclic groups, Cn or Zn, for integers n > 2.
All cyclic groups are abelian.
All groups of prime order are cyclic.

Groups of order 6

Name

Description

Cyclic group of order 6
Also C6

Cayley diagram

Filename

Z_6.gp

Presentation

Notes

Part of the family of cyclic groups, Cn or Zn, for integers n > 2.
All cyclic groups are abelian.


Name

Description

Symmetric group on 3 letters

Cayley diagram

Filename

S_3.gp

Presentation

Notes

The symmetric groups Sn are all permutations of n things.
S3
is also a member of the family of dihedral groups; it is isomorphic to D3.  The dihedral group Dn is the group of symmetries of an n-gon in R3.

Groups of order 7

Name

Description

Cyclic group of order 7
Also C7

Cayley diagram

Filename

Z_7.gp

Presentation

Notes

Part of the family of cyclic groups, Cn or Zn, for integers n > 2.
All cyclic groups are abelian.
All groups of prime order are cyclic.

Groups of order 8

Name

Description

Cyclic group of order 8
Also C8

Cayley diagram

Filename

Z_8.gp

Presentation

Notes

Part of the family of cyclic groups, Cn or Zn, for integers n > 2.
All cyclic groups are abelian.


Name

Description

Direct product of Z2 with Z4

Cayley diagram

Filename

Z_2 x Z_4.gp

Presentation

Notes

This group is abelian, and is therefore expressible as a product of cyclic groups.  This is called the Fundamental Theorem of Abelian Groups, which is mentioned in our Abelian example page.
Read about the direct product operation here.


Name

Description

Third power of Z2 by direct product

Cayley diagram

Filename

Z_2 x Z_2 x Z_2.gp

Presentation

Notes

This group is abelian, and is therefore expressible as a product of cyclic groups.  This is called the Fundamental Theorem of Abelian Groups, which is mentioned in our Abelian example page.
Read about the direct product operation here.


Name

Description

Dihedral group on 4 vertices, or the octic group

Cayley diagram

Filename

D_4.gp

Presentation

Notes

A VRML view of this group is available (requires a plug-in).
This group is a member of the family of dihedral groups.  The dihedral group Dn is the group of symmetries of an n-gon in R3.


Name

Description

Quaternion group of order 8

Cayley diagram

Filename

Q_4.gp

Presentation

Notes

The 8-element quaternion group shows up in many applications, including aspects of physics.
Sometimes the 8-element quaternion group is denoted Q8 and the 16-element quaternion group is denoted Q16.

Groups of order 9

Name

Description

Cyclic group of order 9
Also C9

Cayley diagram

Filename

Z_9.gp

Presentation

Notes

Part of the family of cyclic groups, Cn or Zn, for integers n > 2.
All cyclic groups are abelian.


Name

Description

Direct product of Z3 with itself

Cayley diagram

Filename

Z_3 x Z_3.gp

Presentation

Notes

This group is abelian, and is therefore expressible as a product of cyclic groups.  This is called the Fundamental Theorem of Abelian Groups, which is mentioned in our Abelian example page.
Read about the direct product operation here.

Groups of order 10

Name

Description

Cyclic group of order 10
Also C10

Cayley diagram

Filename

Z_10.gp

Presentation

Notes

Part of the family of cyclic groups, Cn or Zn, for integers n > 2.
All cyclic groups are abelian.


Name

Description

Dihedral group on 5 vertices

Cayley diagram

Filename

D_5.gp

Presentation

Notes

This group is a member of the family of dihedral groups.  The dihedral group Dn is the group of symmetries of an n-gon in R3.

Groups of order 11

Name

Description

Cyclic group of order 11
Also C11

Cayley diagram

Filename

Z_11.gp

Presentation

Notes

A VRML view of this group is available (requires a plug-in).
Part of the family of cyclic groups, Cn or Zn, for integers n > 2.
All cyclic groups are abelian.
All groups of prime order are cyclic.

Groups of order 12

Name

Description

Cyclic group of order 12
Also C12

Cayley diagram

Filename

Z_12.gp

Presentation

Notes

Part of the family of cyclic groups, Cn or Zn, for integers n > 2.
All cyclic groups are abelian.


Name

Description

Direct product of Z2 with Z6

Cayley diagram

Filename

Z_2 x Z_6.gp

Presentation

Notes

This group is abelian, and is therefore expressible as a product of cyclic groups.  This is called the Fundamental Theorem of Abelian Groups, which is mentioned in our Abelian example page.
Read about the direct product operation here.


Name

Description

Semidirect product of Z3 with Z4

Cayley diagram

Filename

Z_3 sdp Z_4.gp

Presentation

Notes

Read about the semidirect product operation here.


Name

Description

Alternating group on 4 letters

Cayley diagram

Filename

A_4.gp

Presentation

Notes

A VRML view of this group is available (requires a plug-in).
The alternating groups An are all even permutations of n things.
This group is the group of symmetries of a tetrahedron.  The tetrahedron is a regular polyhedron.


Name

Description

Dihedral group on 6 vertices

Cayley diagram

Filename

D_6.gp

Presentation

Notes

This group is a member of the family of dihedral groups.  The dihedral group Dn is the group of symmetries of an n-gon in R3.

Groups of order 13

Name

Description

Cyclic group of order 13
Also C13

Cayley diagram

Filename

Z_13.gp

Presentation

Notes

Part of the family of cyclic groups, Cn or Zn, for integers n > 2.
All cyclic groups are abelian.
All groups of prime order are cyclic.

Groups of order 14

Name

Description

Cyclic group of order 14
Also C14

Cayley diagram

Filename

Z_14.gp

Presentation

Notes

Part of the family of cyclic groups, Cn or Zn, for integers n > 2.
All cyclic groups are abelian.


Name

Description

Dihedral group on 7 vertices

Cayley diagram

Filename

D_7.gp

Presentation

Notes

This group is a member of the family of dihedral groups.  The dihedral group Dn is the group of symmetries of an n-gon in R3.

Groups of order 15

Name

Description

Cyclic group of order 15
Also C15

Cayley diagram

Filename

Z_15.gp

Presentation

Notes

Part of the family of cyclic groups, Cn or Zn, for integers n > 2.
All cyclic groups are abelian.

Groups of order 16

Name

Description

Cyclic group of order 16
Also C16

Cayley diagram

Filename

Z_16.gp

Presentation

Notes

Part of the family of cyclic groups, Cn or Zn, for integers n > 2.
All cyclic groups are abelian.


Name

Description

Fourth power of Z2 by direct product

Cayley diagram

Filename

Z_2 x Z_2 x Z_2 x Z_2.gp

Presentation

Notes

This group is abelian, and is therefore expressible as a product of cyclic groups.  This is called the Fundamental Theorem of Abelian Groups, which is mentioned in our Abelian example page.
Read about the direct product operation here.


Name

Description

Direct product of Z2, Z4, and Z2

Cayley diagram

Filename

Z_2 x Z_4 x Z_2.gp

Presentation

Notes

This group is abelian, and is therefore expressible as a product of cyclic groups.  This is called the Fundamental Theorem of Abelian Groups, which is mentioned in our Abelian example page.
Read about the direct product operation here.


Name

Description

Direct product of Z2 with Z8

Cayley diagram

Filename

Z_2 x Z_8.gp

Presentation

Notes

This group is abelian, and is therefore expressible as a product of cyclic groups.  This is called the Fundamental Theorem of Abelian Groups, which is mentioned in our Abelian example page.
Read about the direct product operation here.


Name

Description

Direct product of Z4 with itself

Cayley diagram

Filename

Z_4 x Z_4.gp

Presentation

Notes

A VRML view of this group is available (requires a plug-in).
This group is abelian, and is therefore expressible as a product of cyclic groups.  This is called the Fundamental Theorem of Abelian Groups, which is mentioned in our Abelian example page.
Read about the direct product operation here.


Name

Description

Direct product of D4 with Z2

Cayley diagram

Filename

D_4 x Z_2.gp

Presentation

Notes

A VRML view of this group is available (requires a plug-in).
Read about the direct product operation here.


Name

Description

Dihedral group on 8 vertices

Cayley diagram

Filename

D_8.gp

Presentation

Notes

This group is a member of the family of dihedral groups.  The dihedral group Dn is the group of symmetries of an n-gon in R3.


Name

Description

G4,4

Cayley diagram

Filename

G_4,4.gp

Presentation

Notes

Very little information was available about this group; please feel free to investigate the online classifications of small finite groups listed on our Links page, or to do your own search.  Submit any interesting information about this group to ncarter@bentley.edu.


Name

Modular-16

Description

Modular group of order 16

Cayley diagram

Filename

Modular_16.gp

Presentation

Notes

A VRML view of this group is available (requires a plug-in).
Very little information was available about this group; please feel free to investigate the online classifications of small finite groups listed on our Links page, or to do your own search.  Submit any interesting information about this group to ncarter@bentley.edu.


Name

Quasihedral-16

Description

Quasihedral group of order 16

Cayley diagram

Filename

Quasihedral_16.gp

Presentation

Notes

Very little information was available about this group; please feel free to investigate the online classifications of small finite groups listed on our Links page, or to do your own search.  Submit any interesting information about this group to ncarter@bentley.edu.


Name

Description

Direct product of Q4 with Z2

Cayley diagram

Filename

Q_4 x Z_2.gp

Presentation

Notes

A VRML view of this group is available (requires a plug-in).
Read about the direct product operation here.


Name

Description

Quaternion group of order 16

Cayley diagram

Filename

Q_8.gp

Presentation

Notes

Sometimes the 8-element quaternion group is denoted Q8 and the 16-element quaternion group is denoted Q16.


Name

--

Description

 

Cayley diagram

Filename

Unnamed1_16.gp

Presentation

Notes

Very little information was available about this group; please feel free to investigate the online classifications of small finite groups listed on our Links page, or to do your own search.  Submit any interesting information about this group to ncarter@bentley.edu.


Name

--

Description

 

Cayley diagram

Filename

Unnamed2_16.gp

Presentation

Notes

Very little information was available about this group; please feel free to investigate the online classifications of small finite groups listed on our Links page, or to do your own search.  Submit any interesting information about this group to ncarter@bentley.edu.

Groups of order 17

Name

Description

Cyclic group of order 17
Also C17

Cayley diagram

Filename

Z_17.gp

Presentation

Notes

Part of the family of cyclic groups, Cn or Zn, for integers n > 2.
All cyclic groups are abelian.
All groups of prime order are cyclic.

Groups of order 18

Name

Description

Cyclic group of order 18
Also C18

Cayley diagram

Filename

Z_18.gp

Presentation

Notes

Part of the family of cyclic groups, Cn or Zn, for integers n > 2.
All cyclic groups are abelian.


Name

Description

Direct product of Z3 with Z6

Cayley diagram

Filename

Z_3 x Z_6.gp

Presentation

Notes

This group is abelian, and is therefore expressible as a product of cyclic groups.  This is called the Fundamental Theorem of Abelian Groups, which is mentioned in our Abelian example page.
Read about the direct product operation here.


Name

Description

Dihedral group on 9 vertices

Cayley diagram

Filename

D_9.gp

Presentation

Notes

A VRML view of this group is available (requires a plug-in).
This group is a member of the family of dihedral groups.  The dihedral group Dn is the group of symmetries of an n-gon in R3.


Name

Description

Direct product of S3 with Z3

Cayley diagram

Filename

S_3 x Z_3.gp

Presentation

Notes

A VRML view of this group is available (requires a plug-in).
Read about the direct product operation here.


Name

Description

Semidirect product of Z3 cross Z3 with Z2

Cayley diagram

Filename

Z_3 x Z_3 sdp Z_2.gp

Presentation

Notes

This group involves first taking a direct product of two cyclic groups, and then taking the semidirect product of the resulting group with yet a third cyclic group.  Read about the direct product operation here.  Read about the semidirect product operation here.

Groups of order 19

Name

Description

Cyclic group of order 19
Also C19

Cayley diagram

Filename

Z_19.gp

Presentation

Notes

Part of the family of cyclic groups, Cn or Zn, for integers n > 2.
All cyclic groups are abelian.
All groups of prime order are cyclic.

Groups of order 20

Name

Description

Cyclic group of order 20
Also C20

Cayley diagram

Filename

Z_20.gp

Presentation

Notes

Part of the family of cyclic groups, Cn or Zn, for integers n > 2.
All cyclic groups are abelian.


Name

Description

Direct product of Z2 with Z10

Cayley diagram

Filename

Z_2 x Z_10.gp

Presentation

Notes

This group is abelian, and is therefore expressible as a product of cyclic groups.  This is called the Fundamental Theorem of Abelian Groups, which is mentioned in our Abelian example page.
Read about the direct product operation here.


Name

Description

Dihedral group on 10 vertices

Cayley diagram

Filename

D_10.gp

Presentation

Notes

This group is a member of the family of dihedral groups.  The dihedral group Dn is the group of symmetries of an n-gon in R3.


Name

Description

Frobenius group of order 20

Cayley diagram

Filename

Fr_20.gp

Presentation

Notes

A VRML view of this group is available (requires a plug-in).
This group is a member of the family of Frobenius groups.


Name

Description

Semidirect product of Z4 with Z5

Cayley diagram

Filename

Z_4 sdp Z_5.gp

Presentation

Notes

Read about the semidirect product operation here.

Groups of higher orders

Name

Description

Symmetric group on 4 letters

Cayley diagram

Filename

S_4.gp

Presentation

Notes

Several VRML views of this group are available: #1 #2 #3 #4 (requires a plug-in).
The symmetric groups Sn are all permutations of n things.
This group is the group of symmetries of a cube, and also the group of symmetries of an octagon.  The cube and the octagon are regular polyhedra.


Name

Description

Direct product of Z2, Z2, Z2, and Z3

Cayley diagram

Filename

Z_2 x Z_2 x Z_2 x Z_3.gp

Presentation

Notes

A VRML view of this group is available (requires a plug-in).
This group is abelian, and is therefore expressible as a product of cyclic groups.  This is called the Fundamental Theorem of Abelian Groups, which is mentioned in our Abelian example page.
Read about the direct product operation here.


Name

Description

Alternating group on 5 letters

Cayley diagram

Filename

A_5.gp

Presentation

Notes

Two VRML views of this group are available: #1 #2 (requires a plug-in).
The alternating groups An are all even permutations of n things.
This group is the group of symmetries of a dodecahedron, and also the group of symmetries of an icosahedron.  The dodecahedron and the icosahedron are regular polyhedra.
The fact that A5 is a simple group is important in Galois theory, which was invented to demonstrate the unsolvability of quintic polynomials by radicals.


Name

Description

Direct product of Z2, Z3, Z3, and Z4

Cayley diagram

Filename

Z_2 x Z_3 x Z_3 x Z_4.gp

Presentation

Notes

This group is abelian, and is therefore expressible as a product of cyclic groups.  This is called the Fundamental Theorem of Abelian Groups, which is mentioned in our Abelian example page.
Read about the direct product operation here.

Groups built by users

So far we have no user-contributed groups to offer.  As Group Explorer gets used more and the user base grows, we expect to have more to offer.  Contribute one by emailing Nathan!  Wondering which ones it would be good to have added?  Here are some ideas, keeping in mind the restriction mentioned on the Group Authoring page that all groups need to embed inside S20.

bulletAbelian groups not yet in the library (e.g. Z3 x Z3 x Z3)
bulletSome more interesting semidirect products (i.e. where the multiplicands aren't abelian)
bulletSome nonabelian pq groups (if any fit inside S20; e.g. does the nonabelian group of order 21 embed in S<20?)
bulletS5, if you can make a decent Cayley diagram
bulletHow many of the groups of order less than or equal to 30 embed inside S20 or smaller?
bulletAnything else you can think of that's not so enormous as to be unwieldy