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

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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]
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 -- -- --