Element structure of unitriangular matrix group:UT(3,p)
This article gives specific information, namely, element structure, about a family of groups, namely: unitriangular matrix group:UT(3,p).
View element structure of group families  View other specific information about unitriangular matrix group:UT(3,p)
This article describes in detail the element structure of the unitriangular matrix group:UT(3,p) for a prime number . This is a special case of unitriangular matrix group of degree three over a finite field. For the element structure of this larger family, see element structure of unitriangular matrix group of degree three over a finite field.
The case , which gives dihedral group:D8, behaves somewhat differently from the case odd.
Summary
Item  Value 

number of conjugacy classes  
order  Agrees with general order formula for : 
conjugacy class size statistics  size 1 ( times), size ( times) 
orbits under automorphism group  Case : size 1 (1 conjugacy class of size 1), size 1 (1 conjugacy class of size 1), size 2 (1 conjugacy class of size 2), size 4 (2 conjugacy classes of size 2 each) Case odd : size 1 (1 conjugacy class of size 1), size ( conjugacy classes of size 1 each), size ( conjugacy classes of size each) 
number of orbits under automorphism group  4 if 3 if is odd 
order statistics  Case : order 1 (1 element), order 2 (5 elements), order 4 (2 elements) Case odd: order 1 (1 element), order ( elements) 
exponent  4 if if odd 
Conjugacy class structure
Number of conjugacy classes
The general theory says that number of conjugacy classes in unitriangular matrix group of fixed degree over a finite field is polynomial function of field size, where the degree of the polynomial is one less than the degree of matrices. Thus, we expect that the number of conjugacy classes is a polynomial function of the field size (which here equals the prime ) of degree 3  1 = 2. Indeed, this is the case, and the explicit polynomial is .
Conjugacy class structure in the unitriangular matrix group
Note that the characteristic polynomial of all elements in this group is , hence we do not devote a column to the characteristic polynomial.
For reference, we consider matrices of the form:
Nature of conjugacy class  Jordan block size decomposition  Minimal polynomial  Size of conjugacy class  Number of such conjugacy classes  Total number of elements  Order of elements in each such conjugacy class  Type of matrix 

identity element  1 + 1 + 1 + 1  1  1  1  1  
nonidentity element, but central (has Jordan blocks of size one and two respectively)  2 + 1  1  ,  
noncentral, has Jordan blocks of size one and two respectively  2 + 1  , but not both and are zero  
noncentral, has Jordan block of size three  3  if odd 4 if 
both and are nonzero  
Total ()           
Grouping by conjugacy class sizes
Conjugacy class size  Total number of conjugacy classes of this size  Total number of elements  Cumulative number of conjugacy classes  Cumulative number of elements 

1  
(total)  (total) 
Conjugacy classes with respect to the general linear group
If we consider the action of the general linear group by conjugation, then there is considerable fusion of conjugacy classes. Specifically, there are only three equivalence classes, corresponding to the set of unordered integer partitions of 3 describing the possible Jordan block decompositions.
Below is a summary of the information:
Jordan block size decomposition (partition of 3)  Number of conjugacy classes of size 1 Number of elements in these 
Number of conjugacy classes of size Number of elements in these 
Total (number of conjugacy classes, elements) 

1 + 1 + 1  1 1 
0 0 
1 1 
2 + 1  


3  0 0 


Total ()  

