# Unitriangular matrix group:UT(4,2)

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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 64#Arithmetic functions

### Basic arithmetic functions

Function Value Similar groups Explanation for function value
order (number of elements, equivalently, cardinality or size of underlying set) 64 groups with same order As $UT(n,q), n= 4, q = 2$: $q^{n(n-1)/2} = 2^{4(3)/2} = 2^6 = 64$
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 As $UT(4,q)$, characteristic two: 4
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

### Arithmetic functions of a counting nature

Function Value Similar groups Explanation for function value
number of conjugacy classes 16 groups with same order and number of conjugacy classes | groups with same number of conjugacy classes As $UT(4,q), q = 2$:
$2q^3 + q^2 - 2q = 16 + 4 - 4 = 16$
See element structure of unitriangular matrix group of degree four over a finite field
number of conjugacy classes of subgroups 99 groups with same order and number of conjugacy classes of subgroups | groups with same number of conjugacy classes of subgroups
number of subgroups 225 groups with same order and number of subgroups | groups with same number of subgroups

## Elements

Further information: element structure of unitriangular matrix group:UT(4,2)

### 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 $p = 2$
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 -- -- --

## Linear representation theory

Further information: linear representation theory of unitriangular matrix group:UT(4,2)

## GAP implementation

### Group ID

This finite group has order 64 and has ID 138 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(64,138)

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

gap> G := SmallGroup(64,138);

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

IdGroup(G) = [64,138]

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
UnitriangularPcpGroup(4,2) UnitriangularPcpGroup
SylowSubgroup(GL(4,2),2) SylowSubgroup, GL
SylowSubgroup(Holomorph(ElementaryAbelianGroup(8)),2) SylowSubgroup, Holomorph, ElementaryAbelianGroup
SylowSubgroup(Holomorph(DirectProduct(CyclicGroup(4),CyclicGroup(2))),2) SylowSubgroup, Holomorph, DirectProduct, CyclicGroup