# Unitriangular matrix group:UT(4,3)

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This article is about a particular group, i.e., a group unique upto isomorphism. View specific information (such as linear representation theory, subgroup structure) about this group
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## Definition

This group is defined in the following equivalent ways:

## Arithmetic functions

Want to compare and contrast arithmetic function values with other groups of the same order? Check out groups of order 729#Arithmetic functions

### Basic arithmetic functions

Function Value Similar groups Explanation for function value
order (number of elements, equivalently, cardinality or size of underlying set) 729 groups with same order As $UT(n,q), n = 4, q = 3$: $q^{n(n-1)/2} = 3^{4(3)/2} = 3^6 = 729$
prime-base logarithm of order 6 groups with same prime-base logarithm of order
max-length of a group 6 max-length of a group equals prime-base logarithm of order for group of prime power order
chief length 6 chief length equals prime-base logarithm of order for group of prime power order
composition length 6 composition length equals prime-base logarithm of order for group of prime power order
exponent of a group 9 groups with same order and exponent of a group | groups with same exponent of a group As $UT(4,q)$, characteristic three: 9
prime-base logarithm of exponent 2 groups with same order and prime-base logarithm of exponent | groups with same prime-base logarithm of order and prime-base logarithm of exponent | groups with same prime-base logarithm of exponent
nilpotency class 3 groups with same order and nilpotency class | groups with same prime-base logarithm of order and nilpotency class | groups with same nilpotency class
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
Frattini length 2 groups with same order and Frattini length | groups with same prime-base logarithm of order and Frattini length | groups with same Frattini length

## GAP implementation

### Group ID

This finite group has order 729 and has ID 307 among the groups of order 729 in GAP's SmallGroup library. For context, there are groups of order 729. It can thus be defined using GAP's SmallGroup function as:

SmallGroup(729,307)

For instance, we can use the following assignment in GAP to create the group and name it $G$:

gap> G := SmallGroup(729,307);

Conversely, to check whether a given group $G$ is in fact the group we want, we can use GAP's IdGroup function:

IdGroup(G) = [729,307]

or just do:

IdGroup(G)

to have GAP output the group ID, that we can then compare to what we want.

### Other descriptions

Description Functions used
SylowSubgroup(GL(4,3),3) SylowSubgroup, GL
SylowSubgroup(SL(4,3),3) SylowSubgroup, SL
SylowSubgroup(PGL(4,3),3) SylowSubgroup, PGL
SylowSubgroup(PSL(4,3),3) SylowSubgroup, PSL