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

## Contents

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## 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 $p = 2$
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 $p = 2$
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 -- -- --