Groups of order 16

Presentations
In these presentations, we use $$e$$ for the identity element.

Permutation representations
These give faithful permutation representations of the group:

Here is the GAP code verifying that these permutation representations work:

gap> IdGroup(Group([(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16)])); [ 16, 1 ] gap> IdGroup(Group([(1,2,3,4),(5,6,7,8)])); [ 16, 2 ] gap> IdGroup(Group([(1,2,3,4),(1,3)(5,6,7,8)])); [ 16, 4 ] gap> IdGroup(Group([(1,2,3,4,5,6,7,8),(9,10)])); [ 16, 5 ] gap> IdGroup(Group([(1,2,3,4,5,6,7,8),(1,5)(3,7)])); [ 16, 6 ] gap> IdGroup(Group([(1,2,3,4,5,6,7,8),(1,7)(3,5)(2,6)])); [ 16, 7 ] gap> IdGroup(Group([(1,2,3,4,5,6,7,8),(1,3)(5,7)(2,6)])); [ 16, 8 ] gap> IdGroup(Group([(1,2,3,4),(5,6),(7,8)])); [ 16, 10 ] gap> IdGroup(Group([(1,2,3,4),(1,3),(5,6)])); [ 16, 11 ] gap> IdGroup(Group([(1,2),(3,4),(5,6),(7,8)])); [ 16, 14 ]

Summary information
Here, the rows are arithmetic functions that take values between $$0$$ and $$4$$, and the columns give the possible values of these functions. The entry in each cell is the number of isomorphism classes of groups for which the row arithmetic function takes the column value. Note that all the row value sums must equal $$14$$.

Functions taking values between 0 and 4
These measure ranks of subgroups, lengths of series, or the prime-base logarithms of orders of certain subgroups.

Here now is the same table along with various measures of averages and deviations:

Same, with rows and columns interchanged:

Here are the correlations between these various arithmetic functions across the groups of order 16:

Arithmetic function values of a counting nature
Here are the GAP commands/code for these:

For everything except automorphism classes of subgroups, the following code works. Note that you may need to load the Sonata package to access the Subgroups function.

gap> F := List(AllSmallGroups(16),G -> List([ConjugacyClasses(G),Subgroups(G),ConjugacyClassesSubgroups(G), > NormalSubgroups(G),CharacteristicSubgroups(G)],Length));; gap> K := List([1..14],i -> [i,F[i]]);

GAP's output is:

[ [ 1, [ 16, 5, 5, 5, 5 ] ], [ 2, [ 16, 15, 15, 15, 3 ] ], [ 3, [ 10, 23, 17, 11, 5 ] ], [ 4, [ 10, 15, 13, 11, 7 ] ], [ 5, [ 16, 11, 11, 11, 7 ] ], [ 6, [ 10, 11, 10, 9, 7 ] ], [ 7, [ 7, 19, 11, 7, 5 ] ], [ 8, [ 7, 15, 10, 7, 7 ] ], [ 9, [ 7, 11, 9, 7, 5 ] ], [ 10, [ 16, 27, 27, 27, 4 ] ],  [ 11, [ 10, 35, 27, 19, 5 ] ], [ 12, [ 10, 19, 19, 19, 4 ] ], [ 13, [ 10, 23, 20, 17, 5 ] ], [ 14, [ 16, 67, 67, 67, 2 ] ] ]

Same, with rows and columns interchanged:

Numerical invariants
Here first is the table with the number of conjugacy classes of each size (for more detailed breakdown information, see element structure of groups of order 16). Note also that nilpotency class and order determine conjugacy class size statistics for groups up to prime-fourth order, so that there are only three types -- the abelian groups, the non-abelian groups of class two, and the groups of class exactly three:

Here are the degrees of irreducible representations. Note also that nilpotency class and order determine degrees of irreducible representations for groups up to prime-fourth order, so that there are only three types -- the abelian groups, the non-abelian groups of class two, and the groups of class exactly three:

Here is the lower central series information. Again, notice that the nilpotency class determines the information.

Here is the upper central series information. Again, notice that the nilpotency class determines the information.

Basic properties
The column headings represent the IDs of the groups:

Up to isoclinism
The equivalence classes up to being isoclinic were classified by Hall and Senior, and we call them Hall-Senior families.

Up to Hall-Senior genus
Two groups have the same Hall-Senior genus if they are isoclinic and their lattice of normal subgroups are isomorphic to each other.

Up to isologism for class two
Under the equivalence relation of being isologic groups with respect to the variety of groups of nilpotency class two, there are two equivalence classes:

Up to isologism for higher class
For any class equal to three or higher, there is a single equivalence class under isologism for that class for groups of order 16, because all groups of order 16 have class at most three and are hence isologic to the trivial group.

1-isomorphism
There are two pairs of 1-isomorphic groups: M16 (ID: 6) and direct product of Z8 and Z2 (ID: 5) are 1-isomorphic to each other, and central product of D8 and Z4 (ID: 13)and direct product of Z4 and V4 (ID: 10) are 1-isomorphic to each other.

Order statistics
There are many pairs of order statistics-equivalent groups. In addition to the 1-isomorphic pairs, we also have that direct product of Q8 and Z2 (ID: 12) and nontrivial semidirect product of Z4 and Z4 (ID:4) have the same order statistics as direct product of Z4 and Z4 (ID: 2). Also, the group SmallGroup(16,3) (ID: 3)is order statistics-equivalent to direct product of Z4 and V4, but not 1-isomorphic to it.

Conjugacy class structure
The sizes of conjugacy classes are completely determined by the nilpotency class of the group. Class 1 (abelian) means that all conjugacy classes are of size 1. Class exactly two means that there are four conjugacy classes of size 1 and six conjugacy classes of size 2. Class exactly three means two conjugacy classes of size 1, three of size 2, and two of size 4.

Central extensions with abelian groups both sides
We list here those groups arising as extensions with base group $$A$$ abelian, quotient group also abelian, and the action trivial.