Element structure of unitriangular matrix group:UT(3,4)

This article gives specific information, namely, element structure, about a particular group, namely: unitriangular matrix group:UT(3,4).
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Summary

Item Value
number of conjugacy classes 19
As $UT(3,q), q = 4$: $q^2 + q - 1 = 4^2 + 4 - 1 = 19$
order 64
As $UT(n,q), q = 4, n = 3$: $q^{n(n-1)/2} = 4^{3(2)/2} = 4^3 = 64$
exponent 4
As $UT(3,q), q = 4$ is a power of 2: 4
conjugacy class size statistics size 1 (4 classes), size 4 (15 classes)
order statistics order 1 (1 element), order 2 (27 elements), order 4 (36 elements)

Conjugacy class structure

Interpretation as unitriangular matrix group of degree three

Compare with element structure of unitriangular matrix group of degree three over a finite field
Nature of conjugacy class Jordan block size decomposition Minimal polynomial Size of conjugacy class (generic $q$) Size of conjugacy class ( $q = 4$) Number of such conjugacy classes (generic $q$) Number of such conjugacy classes ( $q = 4$) Total number of elements (generic $q$) Total number of elements ( $q = 4$) Order of elements in each such conjugacy class (generic $q$) Order of elements in each such conjugacy class ( $q = 4$, so $p = 2$) Type of matrix
identity element 1 + 1 + 1 + 1 $x - 1$ 1 1 1 1 1 1 1 1 $a_{12} = a_{13} = a_{23} = 0$
non-identity element, but central (has Jordan blocks of size one and two respectively) 2 + 1 $(x - 1)^2$ 1 1 $q - 1$ 3 $q - 1$ 3 $p$ 2 $a_{12} = a_{23} = 0$, $a_{13} \ne 0$
non-central, has Jordan blocks of size one and two respectively 2 + 1 $(x - 1)^2$ $q$ 4 $2(q - 1)$ 6 $2q(q - 1)$ 24 $p$ 2 $a_{12}a_{23} = 0$, but not both $a_{12}$ and $a_{23}$ are zero
non-central, has Jordan block of size three 3 $(x - 1)^3$ $q$ 4 $(q - 1)^2$ 9 $q(q - 1)^2$ 36 $p$ if $p$ odd
4 if $p = 2$
4 both $a_{12}$ and $a_{23}$ are nonzero
Total (--) -- -- -- -- $q^2 + q - 1$ 19 $q^3$ 64 -- -- --