# Projective special linear group:PSL(2,13)

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## Definition

This group is defined as the projective special linear group of degree two over field:F13, the field with 13 elements.

## Arithmetic functions

### Basic arithmetic functions

Function Value Similar groups Explanation
order (number of elements, equivalently, cardinality or size of underlying set) 1092 groups with same order As $PSL(2,q)$, $q = 13$, $(q^3 - q)/2 = q(q+1)(q-1)/2 = 13(14)(12)/2 = 1092$
See order formulas for linear groups of degree two
exponent of a group 546 groups with same order and exponent of a group | groups with same exponent of a group As $PSL(2,q), q = 13$, underlying prime $p = 13$ (odd): $p(q^2 - 1)/4 = 13(13^2 - 1)/4 = 546$
Frattini length 1 groups with same order and Frattini length | groups with same Frattini length the group is a simple non-abelian group
chief length 1 groups with same order and chief length | groups with same chief length the group is a simple non-abelian group
composition length 1 groups with same order and composition length | groups with same composition length the group is a simple non-abelian group

### Arithmetic functions of a counting nature

Function Value Similar groups Explanation
number of conjugacy classes 9 groups with same order and number of conjugacy classes | groups with same number of conjugacy classes As $PSL(2,q)$, $q = 13$ ( $q$ odd), $(q + 5)/2 = (13 + 5)/2 = 9$
See element structure of projective special linear group of degree two over a finite field, element structure of projective special linear group:PSL(2,13)
number of equivalence classes under rational conjugacy 6 groups with same order and number of equivalence classes under rational conjugacy | groups with same number of equivalence classes under rational conjugacy See element structure of projective special linear group:PSL(2,13)
number of conjugacy classes of subgroups 16 groups with same order and number of conjugacy classes of subgroups | groups with same number of conjugacy classes of subgroups See subgroup structure of projective special linear group:PSL(2,13)
number of subgroups 942 groups with same order and number of subgroups | groups with same number of subgroups See subgroup structure of projective special linear group:PSL(2,13)

## Group properties

Property Satisfied? Explanation
abelian group No
nilpotent group No
solvable group No
simple group, simple non-abelian group Yes projective special linear group is simple (with a couple of exceptions, but this isn't one of them)
minimal simple group Yes See classification of finite minimal simple groups: Any group of the form $PSL(2,p)$ with $p> 3$, $p$ prime, and $5$ dividing $p^2 + 1$

## Subgroups

Further information: subgroup structure of projective special linear group:PSL(2,13)

### Quick summary

Item Value
Number of subgroups 942
Number of conjugacy classes of subgroups 16
Number of automorphism classes of subgroups  ?
Isomorphism classes of Sylow subgroups and the corresponding fusion systems and Sylow numbers 2-Sylow: Klein four-group, fusion system is simple fusion system for Klein four-group, Sylow number is 91
3-Sylow: cyclic group:Z3, Sylow number is 91
7-Sylow: cyclic group:Z7, Sylow number is 78
13-Sylow: cyclic group:Z13, Sylow number is 14
Hall subgroups Other than the whole group, the trivial subgroup, and the Sylow subgroups, there are $\{ 2,3 \}$-Hall subgroups (of order 12) and $\{ 2,3,5 \}$-Hall subgroups (of order 60) and $\{ 3, 13\}$-Hall subgroups (of order 39)
maximal subgroups maximal subgroups have orders 12, 14, 78.
normal subgroups only the whole group and the trivial subgroup, because the group is simple. See projective special linear group is simple (with a couple of small exceptions, but this isn't one of them)
subgroups that are simple non-abelian groups (apart from the whole group itself) None. The group is a minimal simple group, because it is of the form $PSL(2,p)$, $p$ prime, $p > 3$, and $5 \mid p^2 + 1$. See classification of finite minimal simple groups.

## GAP implementation

### Group ID

This finite group has order 1092 and has ID 25 among the groups of order 1092 in GAP's SmallGroup library. For context, there are 77 groups of order 1092. It can thus be defined using GAP's SmallGroup function as:

SmallGroup(1092,25)

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

gap> G := SmallGroup(1092,25);

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

IdGroup(G) = [1092,25]

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
PSL(2,13) PSL