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	$a_{12}=a_{23}=0$, $a_{13}\neq 0$
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	--	--	--