Unitriangular matrix group:UT(3,Z8)

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VIEW DEFINITIONS USING THIS AS EXAMPLE: All terms |Group properties |Subgroup properties|
VIEW FACTS USING THIS AS EXAMPLE: All facts |Group property non-implications |Subgroup property non-implications |Group metaproperty dissatisfactionsSubgroup metaproperty dissatisfactions

Definition

As matrices

This group is defined as the unitriangular matrix group of degree three over ring:Z8. Explicitly, it is the group (under matrix multiplication) of upper-triangular $3\times 3$ unipotent matrices over the ring $\mathbb {Z} /8\mathbb {Z}$, i.e., matrices of the form:

$\left\{{\begin{pmatrix}1&a_{12}&a_{13}\\0&1&a_{23}\\0&0&1\\\end{pmatrix}}:a_{12},a_{13},a_{23}\in \mathbb {Z} /8\mathbb {Z} \right\}$

Arithmetic functions

Want to compare and contrast arithmetic function values with other groups of the same order? Check out groups of order 512#Arithmetic functions

Basic arithmetic functions

Function	Value	Similar groups	Explanation
underlying prime of p-group	2
order (number of elements, equivalently, cardinality or size of underlying set)	512	groups with same order	As $UT(n,R),n=3,|R|=8$: $|R|^{n(n-1)/2}=8^{3(2)/2}=8^{3}=512$
prime-base logarithm of order	9	groups with same prime-base logarithm of order
max-length of a group	9		max-length of a group equals prime-base logarithm of order for group of prime power order
chief length	9		chief length equals prime-base logarithm of order for group of prime power order
composition length	9		composition length equals prime-base logarithm of order for group of prime power order
nilpotency class	2	groups with same order and nilpotency class | groups with same prime-base logarithm of order and nilpotency class | groups with same nilpotency class	As $UT(n,R)$ for $R$ commutative: $n-1=3-1=2$
derived length	2	groups with same order and derived length | groups with same prime-base logarithm of order and derived length | groups with same derived length	As $UT(n,R)$ for $R$ commutative: smallest integer greater than or equal to $\log _{2}n$
minimum size of generating set	2	groups with same order and minimum size of generating set | groups with same prime-base logarithm of order and minimum size of generating set | groups with same minimum size of generating set