# Automorphism Groups

- [Introduction](introduction.md)

- [Creation of Automorphism Groups](creation.md)

  - [`AutomorphismGroup(G): Grp → GrpAuto`](creation.md#function-automorphismgroup-grp)

  - [`Example: Autogp Full`](creation.md#example-ex-1ac639)

  - [`AutomorphismGroup(G, Q, I): Grp, SeqEnum[GrpElt], SeqEnum[SeqEnum[GrpElt]] → GrpAuto`](creation.md#function-automorphismgroup-grp-seqenum-grpelt-seqenum-seqenum-grpelt)

- [Access Functions](access.md)

  - [`Group(A): GrpAuto → Grp`](access.md#function-group-grpauto)

  - [`NumberOfGenerators(A): GrpAuto → RngIntElt`](access.md#function-numberofgenerators-grpauto)

  - [`Ngens(A): GrpAuto → RngIntElt`](access.md#function-ngens-grpauto)

  - [`NumberOfPCGenerators(A): GrpAuto → RngIntElt`](access.md#function-numberofpcgenerators-grpauto)

  - [`NPCGenerators(A): GrpAuto → RngIntElt`](access.md#function-npcgenerators-grpauto)

  - [`NPCgens(A): GrpAuto → RngIntElt`](access.md#function-npcgens-grpauto)

  - [`Generators(A): GrpAuto → SetEnum`](access.md#function-generators-grpauto)

  - [`PCGenerators(A): GrpAuto → SetIndx`](access.md#function-pcgenerators-grpauto)

  - [`InnerGenerators(A): GrpAuto → SeqEnum`](access.md#function-innergenerators-grpauto)

  - [`CharacteristicSeries(A): GrpAuto → SeqEnum`](access.md#function-characteristicseries-grpauto)

  - [`IsSoluble(A): GrpAuto → BoolElt`](access.md#function-issoluble-grpauto)

  - [`IsSolvable(A): GrpAuto → BoolElt`](access.md#function-issolvable-grpauto)

  - [`IsSolubleAutomorphismGroupPGroup(A): GrpAuto → BoolElt`](access.md#function-issolubleautomorphismgrouppgroup-grpauto)

  - [`IsSolvableAutomorphismGroupPGroup(A): GrpAuto → BoolElt`](access.md#function-issolvableautomorphismgrouppgroup-grpauto)

- [Order Functions](order.md)

  - [`Order(A): GrpAuto → RngIntElt`](order.md#function-order-grpauto)

  - [`# A: GrpAuto → RngIntElt`](order.md#operation-operation-grpauto-rngintelt)

  - [`FactoredOrder(A): GrpAuto → [ <RngIntElt, RngIntElt> ]`](order.md#function-factoredorder-grpauto)

  - [`OuterOrder(A): GrpAuto → RngIntElt`](order.md#function-outerorder-grpauto)

  - [`Example: Autogp Order`](order.md#example-ex-829895)

- [Representations of an Automorphism Group](representations.md)

  - [`PermutationRepresentation(A): GrpAuto → Map, GrpPerm, SetIndx`](representations.md#function-permutationrepresentation-grpauto)

  - [`PermutationGroup(A): GrpAuto → GrpPerm`](representations.md#function-permutationgroup-grpauto)

  - [`PermutationSupport(A): GrpAuto → SetIndx`](representations.md#function-permutationsupport-grpauto)

  - [`PCGroupAutomorphismGroupPGroup(A): GrpAuto → BoolElt, Map, GrpPC`](representations.md#function-pcgroupautomorphismgrouppgroup-grpauto)

  - [`FPGroup(A): GrpAuto → GrpFP, Map`](representations.md#function-fpgroup-grpauto)

  - [`OuterFPGroup(A): GrpAuto → GrpFP, Map`](representations.md#function-outerfpgroup-grpauto)

  - [`Example: Autogp Rep1`](representations.md#example-ex-194f40)

  - [`Example: Autogp Rep2`](representations.md#example-ex-a8dd10)

- [Automorphisms](element.md)

  - [`A . i: GrpAuto, RngIntElt → GrpAutoElt`](element.md#operation-operation-grpauto-rngintelt-grpautoelt)

  - [`Identity(A): GrpAuto → GrpAutoElt`](element.md#function-identity-grpauto)

  - [`Id(A): GrpAuto → GrpAutoElt`](element.md#function-id-grpauto)

  - [`A ! 1: GrpAuto, RngIntElt → GrpAutoElt`](element.md#operation-op-grpauto-rngintelt)

  - [`A ! f: GrpAuto, Map → GrpAutoElt`](element.md#operation-op-grpauto-map)

  - [`Order(f): GrpAutoElt → RngIntElt`](element.md#function-order-grpautoelt)

  - [`f * g: GrpAutoElt, GrpAutoElt → GrpAutoElt`](element.md#operation-op-times-grpautoelt-grpautoelt)

  - [`f ^ n: GrpAutoElt, RngIntElt → GrpAutoElt`](element.md#operation-op-pow-grpautoelt-rngintelt)

  - [`(g₁, ..., gᵣ): GrpAutoElt, ..., GrpAutoElt → GrpAutoElt`](element.md#literal-literal-g1-gr-grpautoelt-grpautoelt-grpautoelt)

  - [`g eq h: GrpAutoElt, GrpAutoElt → BoolElt`](element.md#operation-op-eq-grpautoelt-grpautoelt)

  - [`g ne h: GrpAutoElt, GrpAutoElt → BoolElt`](element.md#operation-op-ne-grpautoelt-grpautoelt)

  - [`IsInner(f): GrpAutoElt → BoolElt, GrpElt`](element.md#function-isinner-grpautoelt)

  - [`Example: Autogp Elts`](element.md#example-ex-4eb08f)

  - [`Example: Characteristic Subgroups`](element.md#example-ex-4a4c20)

- [Stored Attributes of an Automorphism Group](attributes.md)

  - [`HasAttribute(A, s): GrpAuto, MonStgElt → BoolElt, .`](attributes.md#function-hasattribute-grpauto-monstgelt)

  - [`AssertAttribute(A, s, v): GrpAuto, MonStgElt, .`](attributes.md#function-assertattribute-grpauto-monstgelt)

  - [`Example: characteristicsubgps`](attributes.md#example-ex-0cc28f)

  - [`Example: Auto Maximals`](attributes.md#example-ex-531b03)

- [Holomorphs](holomorphs.md)

  - [`Holomorph(G): Grp → GrpPerm, HomGrp, HomGrp`](holomorphs.md#function-holomorph-grp)

  - [`Holomorph(GrpFP, G): Grp → Cat[GrpFP], GrpFP, HomGrp, HomGrp`](holomorphs.md#function-holomorph-grp-2)

  - [`Holomorph(G, A): Grp, GrpAuto → GrpPerm, HomGrp, HomGrp`](holomorphs.md#function-holomorph-grp-grpauto)

  - [`Holomorph(GrpFP, G, A): Cat[GrpFP], Grp, GrpAuto → GrpFP, HomGrp, HomGrp`](holomorphs.md#function-holomorph-cat-grpfp-grp-grpauto)

  - [`Example: holomorph`](holomorphs.md#example-ex-fba5ab)