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.

	Users can see what groups come with Group Explorer,
	As the group library grows, users can selectively download groups they want, and
	Users 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:

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.

	Abelian groups not yet in the library (e.g. Z3 x Z3 x Z3)
	Some more interesting semidirect products (i.e. where the multiplicands aren't abelian)
	Some nonabelian pq groups (if any fit inside S20; e.g. does the nonabelian group of order 21 embed in S<20?)
	S5, if you can make a decent Cayley diagram
	How many of the groups of order less than or equal to 30 embed inside S20 or smaller?
	Anything else you can think of that's not so enormous as to be unwieldy