# Related Structures

In this section functions for creating other structures from a permutation Coxeter group are briefly listed. See the appropriate chapters of the Handbook for more details.

## `CoxeterGroup(GrpFP, W): Cat, GrpPermCox -> GrpFPCox`

## `Presentation(W): GrpPermCox -> GrpFPCox`

The finitely presented group isomorphic to the permutation Coxeter group $W$. See Chapter [Finitely Presented Groups](../../FinitelyPresentedGroups/FPGroups/index-fpgroups.md#chapgrpfp).

## `ReflectionGroup(W): GrpPermCox -> GrpMat`

## `CoxeterGroup(GrpMat, W): Cat, GrpPermCox -> GrpFPCox`

The reflection group isomorphic to the Coxeter group $W$. See Chapter [Reflection Groups](../ReflectionGroups/index-reflection-groups.md#chapgrprfl).

## `LieAlgebra(W, R): GrpPermCox, Rng -> AlgLie`

The reductive Lie algebra over the ring $R$ with Weyl group $W$. If $W$ is noncrystallographic, an error is flagged. See Section [Almost Reductive Lie Algebras](../LieAlgebras/family-construct.md#subsectalglieconstrred).

## `GroupOfLieType(W, R): GrpPermCox, Rng -> GrpLie`

The group of Lie type over the ring $R$ with Weyl group $W$. The roots and coroots of $W$ must have integral components. See Chapter [Groups of Lie Type](../ChevalleyGroups/index-chevalley-groups.md#chapgrplie).