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

Description Cyclic group of order 2
Also C_{2} 
Cayley diagram

Filename Z_2.gp

Presentation

Notes Part of the family of
cyclic groups, C_{n} or Z_{n},
for integers n > 2.
All cyclic groups are
abelian.
All groups of prime order are cyclic. 
Name

Description Cyclic group of order 3
Also C_{3} 
Cayley diagram

Filename Z_3.gp

Presentation

Notes Part of the family of
cyclic groups, C_{n} or Z_{n},
for integers n > 2.
All cyclic groups are
abelian.
All groups of prime order are cyclic. 
Name

Description Cyclic group of order 4
Also C_{4} 
Cayley diagram

Filename Z_4.gp

Presentation

Notes Part of the family of
cyclic groups, C_{n} or Z_{n},
for integers n > 2.
All cyclic groups are
abelian. 
Name

Description Cyclic group of order 5
Also C_{5} 
Cayley diagram

Filename Z_5.gp

Presentation

Notes Part of the family of
cyclic groups, C_{n} or Z_{n},
for integers n > 2.
All cyclic groups are
abelian.
All groups of prime order are cyclic. 
Name

Description Cyclic group of order 6
Also C_{6} 
Cayley diagram

Filename Z_6.gp

Presentation

Notes Part of the family of
cyclic groups, C_{n} or Z_{n},
for integers n > 2.
All cyclic groups are
abelian. 
Name

Description Cyclic group of order 7
Also C_{7} 
Cayley diagram

Filename Z_7.gp

Presentation

Notes Part of the family of
cyclic groups, C_{n} or Z_{n},
for integers n > 2.
All cyclic groups are
abelian.
All groups of prime order are cyclic. 
Name

Description Cyclic group of order 8
Also C_{8} 
Cayley diagram

Filename Z_8.gp

Presentation

Notes Part of the family of
cyclic groups, C_{n} or Z_{n},
for integers n > 2.
All cyclic groups are
abelian. 
Name

Description Quaternion group of order 8 
Cayley diagram

Filename Q_4.gp

Presentation

Notes The 8element
quaternion
group shows up in many applications, including aspects of physics.
Sometimes the 8element
quaternion
group is denoted Q_{8} and the 16element
quaternion
group is denoted Q_{16}. 
Name

Description Cyclic group of order 9
Also C_{9} 
Cayley diagram

Filename Z_9.gp

Presentation

Notes Part of the family of
cyclic groups, C_{n} or Z_{n},
for integers n > 2.
All cyclic groups are
abelian. 
Name

Description Cyclic group of order 10
Also C_{10} 
Cayley diagram

Filename Z_10.gp

Presentation

Notes Part of the family of
cyclic groups, C_{n} or Z_{n},
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 D_{n}
is the group of symmetries of an ngon in R^{3}. 
Name

Description Cyclic group of order 11
Also C_{11} 
Cayley diagram

Filename Z_11.gp

Presentation

Notes A
VRML view of this group is
available (requires a
plugin).
Part of the family of
cyclic groups, C_{n} or Z_{n},
for integers n > 2.
All cyclic groups are
abelian.
All groups of prime order are cyclic. 
Name

Description Cyclic group of order 12
Also C_{12} 
Cayley diagram

Filename Z_12.gp

Presentation

Notes Part of the family of
cyclic groups, C_{n} or Z_{n},
for integers n > 2.
All cyclic groups are
abelian. 
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 D_{n}
is the group of symmetries of an ngon in R^{3}. 
Name

Description Cyclic group of order 13
Also C_{13} 
Cayley diagram

Filename Z_13.gp

Presentation

Notes Part of the family of
cyclic groups, C_{n} or Z_{n},
for integers n > 2.
All cyclic groups are
abelian.
All groups of prime order are cyclic. 
Name

Description Cyclic group of order 14
Also C_{14} 
Cayley diagram

Filename Z_14.gp

Presentation

Notes Part of the family of
cyclic groups, C_{n} or Z_{n},
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 D_{n}
is the group of symmetries of an ngon in R^{3}. 
Name

Description Cyclic group of order 15
Also C_{15} 
Cayley diagram

Filename Z_15.gp

Presentation

Notes Part of the family of
cyclic groups, C_{n} or Z_{n},
for integers n > 2.
All cyclic groups are
abelian. 
Name

Description Cyclic group of order 16
Also C_{16} 
Cayley diagram

Filename Z_16.gp

Presentation

Notes Part of the family of
cyclic groups, C_{n} or Z_{n},
for integers n > 2.
All cyclic groups are
abelian. 
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 D_{n}
is the group of symmetries of an ngon in R^{3}. 
Name

Description G_{4,4} 
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 Modular16 
Description Modular group of order 16 
Cayley diagram

Filename Modular_16.gp

Presentation

Notes A
VRML view of this group is
available (requires a
plugin).
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 Quasihedral16 
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 Quaternion group of order 16 
Cayley diagram

Filename Q_8.gp

Presentation

Notes Sometimes the 8element
quaternion
group is denoted Q_{8} and the 16element
quaternion
group is denoted Q_{16}. 
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. 
Name

Description Cyclic group of order 17
Also C_{17} 
Cayley diagram

Filename Z_17.gp

Presentation

Notes Part of the family of
cyclic groups, C_{n} or Z_{n},
for integers n > 2.
All cyclic groups are
abelian.
All groups of prime order are cyclic. 
Name

Description Cyclic group of order 18
Also C_{18} 
Cayley diagram

Filename Z_18.gp

Presentation

Notes Part of the family of
cyclic groups, C_{n} or Z_{n},
for integers n > 2.
All cyclic groups are
abelian. 
Name

Description Cyclic group of order 19
Also C_{19} 
Cayley diagram

Filename Z_19.gp

Presentation

Notes Part of the family of
cyclic groups, C_{n} or Z_{n},
for integers n > 2.
All cyclic groups are
abelian.
All groups of prime order are cyclic. 
Name

Description Cyclic group of order 20
Also C_{20} 
Cayley diagram

Filename Z_20.gp

Presentation

Notes Part of the family of
cyclic groups, C_{n} or Z_{n},
for integers n > 2.
All cyclic groups are
abelian. 
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 D_{n}
is the group of symmetries of an ngon in R^{3}. 
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
plugin).
The symmetric groups S_{n} 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 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
plugin).
The
alternating groups A_{n} 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 A_{5} is a simple group is important in
Galois theory,
which was invented to demonstrate the unsolvability of quintic
polynomials by radicals. 
Groups built by users
So far we have no usercontributed 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 S_{20}.
 Abelian groups not yet in the library (e.g. Z_{3}
x Z_{3} x Z_{3}) 
 Some more interesting semidirect products (i.e. where the
multiplicands aren't abelian) 
 Some nonabelian pq groups (if any fit inside S_{20};
e.g. does the nonabelian group of order 21 embed in S_{<20}?) 
 S_{5}, if you can make a decent Cayley diagram 
 How many of the groups of order less than or equal to 30 embed
inside S_{20} or smaller? 
 Anything else you can think of that's not so enormous as to be
unwieldy 
