Unitriangular matrix group:UT(4,3)

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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

This group is defined in the following equivalent ways:

• It is the unitriangular matrix group of degree four over the field of three elements.
• It is a $3$-Sylow subgroup of general linear group:GL(4,3), and hence also of special linear group:SL(4,3), projective general linear group:PGL(4,3), and projective special linear group:PSL(4,3).

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