Element structure of unitriangular matrix group:UT(4,2)

This article gives specific information, namely, element structure, about a particular group, namely: unitriangular matrix group:UT(4,2).
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This article describes the element structure of unitriangular matrix group:UT(4,2).

Summary

Item	Value
number of conjugacy classes	16 As $UT(4,q),q=2$: $2q^3+q^2-2q=2(2^3)+(2^2)-2(2)=16$
order	64 As $UT(n,q),q=2,n=4$: $q^{n(n-1)/2}=2^{4(3)/2}=2^6=64$
exponent	4 As $UT(4,q)$, $q$ a power of $p,p<5$: $p^2=2^2=4$
conjugacy class size statistics	size 1 (2 classes), size 2 (3 classes), size 4 (8 classes), size 8 (3 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 four

Compare with element structure of unitriangular matrix group of degree four over a finite field#Conjugacy class structure

Nature of conjugacy class	Jordan block size decomposition	Minimal polynomial	Size of conjugacy class (generic $q$)	Size of conjugacy class ($q=2$)	Number of such conjugacy classes (generic $q$)	Number of such conjugacy classes ($q=2$)	Total number of elements (generic $q$)	Total number of elements ($q=2$)	Order of elements in each such conjugacy class (generic $q$, power of prime $p$)	Order of elements in each such conjugacy class ($q=2$, so $p=2$)	Type of matrix (constraints on $a_{ij},i<j$)
identity element	1 + 1 + 1 + 1	$x-1$	1	1	1	1	1	1	1	1	all the $a_{ij},i<j$ are zero
non-identity element, but central (has Jordan blocks of size 1,1,2 respectively)	2 + 1 + 1	$(x-1{)}^2$	1	1	$q-1$	1	$q-1$	1	$p$	2	$a_{14}\ne 0$, all the others are zero
non-central but in derived subgroup, has Jordan blocks of size 1,1,2	2 + 1 + 1	$(x-1{)}^2$	$q$	2	$2(q-1)$	2	$2q(q-1)$	4	$p$	2	$a_{12}=a_{23}=a_{34}=0$ Among $a_{13}$ and $a_{24}$, exactly one of them is nonzero. $a_{14}$ may be zero or nonzero
non-central but in derived subgroup, Jordan blocks of size 2,2	2 + 2	$(x-1{)}^2$	$q$	2	$(q-1{)}^2$	1	$q(q-1{)}^2$	2	$p$	2	$a_{12}=a_{23}=a_{34}=0$ Both $a_{13}$ and $a_{24}$ are nonzero. $a_{14}$ may be zero or nonzero
outside derived subgroup, inside unique abelian subgroup of maximum order, with Jordan blocks of size 1,1,2	2 + 1 + 1	$(x-1{)}^2$	$q^2$	4	$q-1$	1	$q^2(q-1)$	4	$p$	2	$a_{12}=a_{34}=a_{14}=0$ $a_{23}$ is nonzero $a_{13}$ and $a_{24}$ are arbitrary
outside derived subgroup, inside unique abelian subgroup of maximum order, with Jordan blocks of size 2,2	2 + 2	$(x-1{)}^2$	$q^2$	4	$(q-1{)}^2$	1	$q^2(q-1{)}^2$	4	$p$	2	$a_{12}=a_{34}=0$ $a_{23}$ and $a_{14}$ are both nonzero $a_{13}$ and $a_{24}$ are arbitrary
outside abelian subgroup of maximum order, Jordan blocks of size 1,1,2	2 + 1 + 1	$(x-1{)}^2$	$q^2$	4	$2(q-1)$	2	$2q^2(q-1)$	8	$p$	2	Two subcases: Case 1: $a_{12}=a_{23}=a_{13}=0$, $a_{34}$ nonzero, $a_{14},a_{24}$ arbitrary Case 2: $a_{23}=a_{24}=a_{34}=0$, $a_{12}$ nonzero, $a_{13},a_{14}$ arbitrary
outside abelian subgroup of maximum order, Jordan blocks of size 2,2	2 + 2	$(x-1{)}^2$	$q^2$	4	$(q-1{)}^2$	1	$q^2(q-1{)}^2$	4	$p$	2	$a_{12},a_{34}$ both nonzero $a_{23}=0$ $a_{14},a_{24}$ arbitrary $a_{13}$ uniquely determined by other values
outside abelian subgroup of maximum order, Jordan blocks of size 1,3, with centralizer of order $q^4$	3 + 1	$(x-1{)}^3$	$q^2$	4	$(q-1{)}^2(q+1)$	3	$q^2(q-1{)}^2(q+1)$	12	$p$ if $p$ odd 4 if $p=2$	4	PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE] One of the people editing this page intended to fill in this information at a later stage, but hasn't gotten around to doing it yet. If you see this placeholder for a long time, file an error report at the error reporting page.
outside abelian subgroup of maximum order, Jordan blocks of size 1,3, with centralizer of order $q^3$	3 + 1	$(x-1{)}^3$	$q^3$	8	$2(q-1{)}^2$	2	$2q^3(q-1{)}^2$	16	$p$ if $p$ odd 4 if $p=2$	4	Two subcases: Case 1: $a_{12},a_{23}$ nonzero, $a_{34}=0$, other entries arbitrary Case 2: $a_{23},a_{34}$ nonzero, $a_{12}=0$, other entries arbitrary
Jordan block of size 4	4	$(x-1{)}^4$	$q^3$	8	$(q-1{)}^3$	1	$q^3(q-1{)}^3$	8	$p^2$ if $p<5$ $p$ if $p\ge 5$	4	$a_{12},a_{23},a_{34}$ nonzero $a_{13},a_{14},a_{24}$ arbitrary
Total (--)	--	--	--	--	$2q^3+q^2-2q$	16	$q^6$	64	--	--	--