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

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This article gives specific information, namely, element structure, about a particular group, namely: unitriangular matrix group:UT(3,4).
View element structure of particular groups | View other specific information about unitriangular matrix group:UT(3,4)

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 -- -- --