# Unitriangular matrix group:UT(3,4)

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## Definition

### As a group of matrices

This group is the unitriangular matrix group of degree three (i.e., the group of $3 \times 3$ upper-triangular matrices with $1$s on the diagonal) over the field of four elements. It is isomorphic to the 2-Sylow subgroup of general linear group:GL(3,4), special linear group:SL(3,4), projective general linear group:PGL(3,4), projective special linear group:PSL(3,4).

The group can be described explicitly as: $\left \{ \begin{pmatrix} 1 & a_{12} & a_{13} \\ 0 & 1 & a_{23} \\ 0 & 0 & 1 \\\end{pmatrix} \mid a_{12},a_{13},a_{23} \in \mathbb{F}_4 \right \}$

The multiplication of matrices $A = (a_{ij})$ and $B = (b_{ij})$ gives the matrix $C = (c_{ij})$ where:

• $c_{12} = a_{12} + b_{12}$
• $c_{13} = a_{13} + b_{13} + a_{12}b_{23}$
• $c_{23} = a_{23} + b_{23}$

The identity element is the identity matrix.

The inverse of a matrix $A = (a_{ij})$ is the matrix $M = (m_{ij})$ where:

• $m_{12} = -a_{12}$
• $m_{13} = -a_{13} + a_{12}a_{23}$
• $m_{23} = -a_{23}$

Note that all addition and multiplication in these definitions is happening over the field $\mathbb{F}_4$.

## Arithmetic functions

Want to compare and contrast arithmetic function values with other groups of the same order? Check out groups of order 64#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) 64 groups with same order As $UT(n,q), q = 4, n = 3$: $q^{n(n-1)/2} = 4^{3(2)/2} = 64$
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 4 groups with same order and exponent of a group | groups with same exponent of a group
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
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
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 $UT(n,q), n = 3$: nilpotency class is $n - 1 = 3 - 1 = 2$
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 $UT(n,q), n = 3$: derived length is the smallest integer greater than or equal to $\log_2n$.
minimum size of generating set 4 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 As $UT(n,q), q = p^r, n = 3, r = 2$: $r(n - 1) = 2(3 - 1) = 2(2) = 4$
subgroup rank of a group 4 groups with same order and subgroup rank of a group | groups with same prime-base logarithm of order and subgroup rank of a group | groups with same subgroup rank of a group
rank of a p-group 4 groups with same order and rank of a p-group | groups with same prime-base logarithm of order and rank of a p-group | groups with same rank of a p-group
normal rank of a p-group 4 groups with same order and normal rank of a p-group | groups with same prime-base logarithm of order and normal rank of a p-group | groups with same normal rank of a p-group
characteristic rank of a p-group 4 groups with same order and characteristic rank of a p-group | groups with same prime-base logarithm of order and characteristic rank of a p-group | groups with same characteristic rank of a p-group

## Elements

Further information: element structure of unitriangular matrix group:UT(3,4)

### Conjugacy class structure

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 -- -- --

## GAP implementation

### Group ID

This finite group has order 64 and has ID 242 among the groups of order 64 in GAP's SmallGroup library. For context, there are groups of order 64. It can thus be defined using GAP's SmallGroup function as:

SmallGroup(64,242)

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

gap> G := SmallGroup(64,242);

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

IdGroup(G) = [64,242]

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(3,4),2) SylowSubgroup, GL
SylowSubgroup(SL(3,4),2) SylowSubgroup, SL
SylowSubgroup(PGL(3,4),2) SylowSubgroup, PGL
SylowSubgroup(PSL(3,4),2) SylowSubgroup, PSL