# Polycyclic Groups

- [Introduction](introduction.md)

- [Polycyclic Groups and Polycyclic Presentations](polycyclic-groups.md)

  - [Introduction](polycyclic-groups.md#introduction)

  - [Specification of Elements](polycyclic-groups.md#specification-of-elements)

    - [`G ! Q: GrpGPC, [RngIntElt] → GrpGPCElt`](polycyclic-groups.md#operation-op-grpgpc-rngintelt)

    - [`Identity(G): GrpGPC → GrpGPCElt`](polycyclic-groups.md#function-identity-grpgpc)

    - [`Id(G): GrpGPC → GrpGPCElt`](polycyclic-groups.md#function-id-grpgpc)

    - [`G ! 1: GrpGPC, RngIntElt → GrpGPCElt`](polycyclic-groups.md#operation-op-grpgpc-rngintelt-2)

  - [Access Functions for Elements](polycyclic-groups.md#access-functions-for-elements)

    - [`ElementToSequence(x): GrpGPCElt → [RngIntElt]`](polycyclic-groups.md#function-elementtosequence-grpgpcelt)

    - [`Eltseq(x): GrpGPCElt → [RngIntElt]`](polycyclic-groups.md#function-eltseq-grpgpcelt)

    - [`LeadingTerm(x): GrpGPCElt → GrpGPCElt`](polycyclic-groups.md#function-leadingterm-grpgpcelt)

    - [`LeadingGenerator(x): GrpGPCElt → GrpGPCElt`](polycyclic-groups.md#function-leadinggenerator-grpgpcelt)

    - [`LeadingExponent(x): GrpGPCElt → RngIntElt`](polycyclic-groups.md#function-leadingexponent-grpgpcelt)

    - [`Depth(x): GrpGPCElt → RngIntElt`](polycyclic-groups.md#function-depth-grpgpcelt)

  - [Arithmetic Operations on Elements](polycyclic-groups.md#arithmetic-operations-on-elements)

    - [`g * h: GrpGPCElt, GrpGPCElt → GrpGPCElt`](polycyclic-groups.md#operation-op-times-grpgpcelt-grpgpcelt)

    - [`g *:= h: GrpGPCElt, GrpGPCElt`](polycyclic-groups.md#operation-operation-grpgpcelt-grpgpcelt)

    - [`g ^ n: GrpGPCElt, RngIntElt → GrpGPCElt`](polycyclic-groups.md#operation-op-pow-grpgpcelt-rngintelt)

    - [`g ^:= n: GrpGPCElt, RngIntElt`](polycyclic-groups.md#operation-operation-grpgpcelt-rngintelt)

    - [`g / h: GrpGPCElt, GrpGPCElt → GrpGPCElt`](polycyclic-groups.md#operation-op-div-grpgpcelt-grpgpcelt)

    - [`g /:= h: GrpGPCElt, GrpGPCElt`](polycyclic-groups.md#operation-operation-grpgpcelt-grpgpcelt-2)

    - [`g ^ h: GrpGPCElt, GrpGPCElt → GrpGPCElt`](polycyclic-groups.md#operation-op-pow-grpgpcelt-grpgpcelt)

    - [`g ^:= h: GrpGPCElt, GrpGPCElt`](polycyclic-groups.md#operation-operation-grpgpcelt-grpgpcelt-3)

    - [`(g₁, ..., gₙ): List(GrpGPCElt) → GrpGPCElt`](polycyclic-groups.md#literal-literal-g1-gn-list-grpgpcelt-grpgpcelt)

  - [Operators for Elements](polycyclic-groups.md#operators-for-elements)

    - [`Order(x): GrpGPCElt → RngIntElt`](polycyclic-groups.md#function-order-grpgpcelt)

    - [`IsFinite(x): GrpGPCElt → BoolElt`](polycyclic-groups.md#function-isfinite-grpgpcelt)

    - [`Parent(x): GrpGPCElt → GrpGPC`](polycyclic-groups.md#function-parent-grpgpcelt)

  - [Comparison Operators for Elements](polycyclic-groups.md#comparison-operators-for-elements)

    - [`g eq h: GrpGPCElt, GrpGPCElt → BoolElt`](polycyclic-groups.md#operation-op-eq-grpgpcelt-grpgpcelt)

    - [`g ne h: GrpGPCElt, GrpGPCElt → BoolElt`](polycyclic-groups.md#operation-op-ne-grpgpcelt-grpgpcelt)

    - [`IsIdentity(g): GrpGPCElt → BoolElt`](polycyclic-groups.md#function-isidentity-grpgpcelt)

    - [`IsId(g): GrpGPCElt → BoolElt`](polycyclic-groups.md#function-isid-grpgpcelt)

  - [Specification of a Polycyclic Presentation](polycyclic-groups.md#specification-of-a-polycyclic-presentation)

    - [`quo< GrpGPC : F | R : parameters >: GrpFP, List(GrpFPRel) → GrpGPC, Map`](polycyclic-groups.md#constructor-grpgpc-quo-def)

    - [`PolycyclicGroup< x₁, ..., xₙ | R : parameters >: List(Identifiers), List(GrpFPRel) → GrpGPC, Map`](polycyclic-groups.md#constructor-grpgpc-polycyclicgroup)

    - [`Example: Constructor`](polycyclic-groups.md#example-ex-600748)

    - [`Example: Polycyclic Group`](polycyclic-groups.md#example-ex-a6668b)

  - [Properties of a Polycyclic Presentation](polycyclic-groups.md#properties-of-a-polycyclic-presentation)

    - [`IsConsistent(G): GrpGPC → BoolElt`](polycyclic-groups.md#function-grpgpc-isconsistent)

    - [`IsIdenticalPresentation(G, H): GrpGPC, GrpGPC → BoolElt`](polycyclic-groups.md#function-isidenticalpresentation-grpgpc-grpgpc)

    - [`PresentationIsSmall(G): GrpGPC → BoolElt`](polycyclic-groups.md#function-grpgpc-presentationissmall)

- [Subgroups, Quotient Groups, Homomorphisms and Extensions](subgroup-quotient-homomorphism-extension.md)

  - [Construction of Subgroups](subgroup-quotient-homomorphism-extension.md#construction-of-subgroups)

    - [`sub<G | L>: GrpGPC, List → GrpGPC, Map`](subgroup-quotient-homomorphism-extension.md#constructor-grpgpc-sub)

    - [`ncl<G | L>: GrpGPC, List → GrpGPC, Map`](subgroup-quotient-homomorphism-extension.md#constructor-constructor-ncl-grpgpc-list-grpgpc-map)

  - [Coercions Between Groups and Subgroups](subgroup-quotient-homomorphism-extension.md#coercions-between-groups-and-subgroups)

    - [`G ! g: GrpGPC, GrpGPCElt → GrpGPCElt`](subgroup-quotient-homomorphism-extension.md#operation-op-grpgpc-grpgpcelt)

    - [`H ! g: GrpGPC, GrpGPCElt → GrpGPCElt`](subgroup-quotient-homomorphism-extension.md#operation-op-grpgpc-grpgpcelt-2)

    - [`K ! g: GrpGPC, GrpGPCElt → GrpGPCElt`](subgroup-quotient-homomorphism-extension.md#operation-op-grpgpc-grpgpcelt-3)

    - [`InclusionMap(G, H): GrpGPC, GrpGPC → Map`](subgroup-quotient-homomorphism-extension.md#function-inclusionmap-grpgpc-grpgpc)

    - [`Example: Subgroup`](subgroup-quotient-homomorphism-extension.md#example-ex-49b53c)

  - [Construction of Quotient Groups](subgroup-quotient-homomorphism-extension.md#construction-of-quotient-groups)

    - [`quo<G | L>: GrpGPC, List → GrpGPC, Map`](subgroup-quotient-homomorphism-extension.md#constructor-constructor-quo-grpgpc-list-grpgpc-map)

    - [`G / N: GrpGPC, GrpGPC → GrpGPC`](subgroup-quotient-homomorphism-extension.md#operation-op-div-grpgpc-grpgpc)

  - [Homomorphisms](subgroup-quotient-homomorphism-extension.md#homomorphisms)

    - [General remarks](subgroup-quotient-homomorphism-extension.md#general-remarks)

    - [Construction of Homomorphisms](subgroup-quotient-homomorphism-extension.md#construction-of-homomorphisms)

      - [`hom< P -> G | S : parameters>: Struct, Struct → Map`](subgroup-quotient-homomorphism-extension.md#constructor-constructor-hom-struct-struct-map)

  - [Construction of Extensions](subgroup-quotient-homomorphism-extension.md#construction-of-extensions)

    - [`DirectProduct(G, H): GrpGPC, GrpGPC → GrpGPC, [Map], [Map]`](subgroup-quotient-homomorphism-extension.md#function-grpgpc-directproduct)

  - [Construction of Standard Groups](subgroup-quotient-homomorphism-extension.md#construction-of-standard-groups)

    - [`AbelianGroup(GrpGPC, Q): Cat, [RngIntElt] → GrpGPC`](subgroup-quotient-homomorphism-extension.md#function-abeliangroup-cat-rngintelt)

    - [`CyclicGroup(GrpGPC, n): Cat, RngIntElt → GrpGPC`](subgroup-quotient-homomorphism-extension.md#function-cyclicgroup-cat-rngintelt)

    - [`DihedralGroup(GrpGPC, n): Cat, RngIntElt → GrpGPC`](subgroup-quotient-homomorphism-extension.md#function-dihedralgroup-cat-rngintelt)

    - [`ElementaryAbelianGroup(GrpGPC, p, n): Cat, RngIntElt, RngIntElt → GrpGPC`](subgroup-quotient-homomorphism-extension.md#function-elementaryabeliangroup-cat-rngintelt-rngintelt)

    - [`ExtraSpecialGroup(GrpGPC, p, n : parameters): Cat, RngIntElt, RngIntElt → GrpGPC`](subgroup-quotient-homomorphism-extension.md#function-extraspecialgroup-cat-rngintelt-rngintelt)

    - [`FreeAbelianGroup(GrpGPC, n): Cat, RngIntElt → GrpGPC`](subgroup-quotient-homomorphism-extension.md#function-freeabeliangroup-cat-rngintelt)

    - [`FreeNilpotentGroup(r, e): RngIntElt, RngIntElt → GrpGPC`](subgroup-quotient-homomorphism-extension.md#function-freenilpotentgroup-rngintelt-rngintelt)

    - [`Example: Homomorphism`](subgroup-quotient-homomorphism-extension.md#example-ex-5a225f)

    - [`Example: Symmetric2`](subgroup-quotient-homomorphism-extension.md#example-ex-f4c433)

- [Conversion between Categories](conversion.md)

  - [`AbelianGroup(G): GrpGPC → GrpAb, Map`](conversion.md#function-abeliangroup-grpgpc)

  - [`FPGroup(G): GrpGPC → GrpFP, Map`](conversion.md#function-grpgpc-fpgroup)

  - [`PCGroup(G): GrpGPC → GrpPC, Map`](conversion.md#function-grpgpc-pcgroup)

  - [`GPCGroup(G): GrpPerm → GrpGPC, Map`](conversion.md#function-gpcgroup-grpperm)

  - [`GPCGroup(G): GrpMat → GrpGPC, Map`](conversion.md#function-gpcgroup-grpmat)

  - [`GPCGroup(G): GrpAb → GrpGPC, Map`](conversion.md#function-gpcgroup-grpab)

  - [`GPCGroup(G): GrpPC → GrpGPC, Map`](conversion.md#function-gpcgroup-grppc)

  - [`Example: Subgroups Quotients Transfer`](conversion.md#example-ex-f0ed02)

- [Access Functions for Groups](access.md)

  - [`G . i: GrpGPC, RngIntElt → GrpGPCElt`](access.md#operation-operation-grpgpc-rngintelt-grpgpcelt)

  - [`Generators(G): GrpGPC → { @ GrpGPCElt @}`](access.md#function-generators-grpgpc)

  - [`PCGenerators(G): GrpGPC → { @ GrpGPCElt  @}`](access.md#function-grpgpc-pcgenerators)

  - [`Generators(H, G): GrpGPC, GrpGPC → { @ GrpGPCElt @}`](access.md#function-generators-grpgpc-grpgpc)

  - [`PCGenerators(H, G): GrpGPC, GrpGPC → { @ GrpGPCElt  @}`](access.md#function-grpgpc-pcgenerators-2)

  - [`NumberOfGenerators(G): GrpGPC → RngIntElt`](access.md#function-numberofgenerators-grpgpc)

  - [`Ngens(G): GrpGPC → RngIntElt`](access.md#function-ngens-grpgpc)

  - [`NumberOfPCGenerators(G): GrpGPC → RngIntElt`](access.md#function-numberofpcgenerators-grpgpc)

  - [`NPCgens(G): GrpGPC → RngIntElt`](access.md#function-npcgens-grpgpc)

  - [`NPCGenerators(G): GrpGPC → RngIntElt`](access.md#function-npcgenerators-grpgpc)

  - [`PCExponents(G): GrpGPC → [RngIntElt]`](access.md#function-pcexponents-grpgpc)

  - [`HirschNumber(G): GrpGPC → RngIntElt`](access.md#function-hirschnumber-grpgpc)

- [Set-Theoretic Operations in a Group](set.md)

  - [Functions Relating to Group Order](set.md#functions-relating-to-group-order)

    - [`FactoredIndex(G, H): GrpGPC, GrpGPC → [<RngIntElt, RngIntElt>]`](set.md#function-factoredindex-grpgpc-grpgpc)

    - [`FactoredOrder(G): GrpGPC → [<RngIntElt, RngIntElt>]`](set.md#function-factoredorder-grpgpc)

    - [`Index(G, H): GrpGPC, GrpGPC → RngIntElt`](set.md#function-index-grpgpc-grpgpc)

    - [`Order(G): GrpGPC → RngIntElt`](set.md#function-order-grpgpc)

    - [`# G: GrpGPC → RngIntElt`](set.md#operation-operation-grpgpc-rngintelt)

  - [Membership and Equality](set.md#membership-and-equality)

    - [`g in G: GrpGPCElt, GrpGPC → BoolElt`](set.md#operation-op-in-grpgpcelt-grpgpc)

    - [`g notin G: GrpGPCElt, GrpGPC → BoolElt`](set.md#operation-op-notin-grpgpcelt-grpgpc)

    - [`S subset G: { GrpGPCElt}, GrpGPC → BoolElt`](set.md#operation-op-subset-grpgpcelt-grpgpc)

    - [`S notsubset G: { GrpGPCElt}, GrpGPC → BoolElt`](set.md#operation-operation-notsubset-grpgpcelt-grpgpc-boolelt)

    - [`H subset G: GrpGPC, GrpGPC → BoolElt`](set.md#operation-op-subset-grpgpc-grpgpc)

    - [`IsSubgroup(H,G): GrpGPC, GrpGPC → BoolElt`](set.md#function-issubgroup-grpgpc-grpgpc)

    - [`H notsubset G: GrpGPC, GrpGPC → BoolElt`](set.md#operation-operation-notsubset-grpgpc-grpgpc-boolelt)

    - [`G eq H: GrpGPC, GrpGPC → BoolElt`](set.md#operation-op-eq-grpgpc-grpgpc)

    - [`G ne H: GrpGPC, GrpGPC → BoolElt`](set.md#operation-op-ne-grpgpc-grpgpc)

  - [Set Operations](set.md#set-operations)

    - [`Representative(G): GrpGPC → GrpGPCElt`](set.md#function-representative-grpgpc)

    - [`Rep(G): GrpGPC → GrpGPCElt`](set.md#function-rep-grpgpc)

    - [`RandomProcess(G): GrpGPC → Process`](set.md#function-grpgpc-randomprocess)

    - [`Random(P): Process → GrpGPCElt`](set.md#function-grpgpc-random)

    - [`Random(G): GrpGPC → GrpGPCElt`](set.md#function-random-grpgpc)

    - [`Random(G, max): GrpGPC, RngIntElt → GrpGPCElt`](set.md#function-random-grpgpc-rngintelt)

- [Coset Spaces](coset-space.md)

  - [`CosetTable(G, H): GrpGPC, GrpGPC → Map`](coset-space.md#function-grpgpc-cosettable)

  - [`Transversal(G, H): GrpGPC, GrpGPC → { @ GrpGPCElt  @}, Map`](coset-space.md#function-grpgpc-transversal)

  - [`RightTransversal(G, H): GrpGPC, GrpGPC → { @ GrpGPCElt  @}, Map`](coset-space.md#function-grpgpc-righttransversal)

  - [`Example: Coset Table`](coset-space.md#example-ex-fcfe92)

  - [`CosetAction(G, H): GrpGPC, GrpGPC → Map, GrpPerm, GrpGPC`](coset-space.md#function-grpgpc-cosetaction)

  - [`CosetImage(G, H): GrpGPC, GrpGPC → GrpPerm`](coset-space.md#function-grpgpc-cosetimage)

  - [`CosetKernel(G, H): GrpGPC, GrpGPC → GrpGPC`](coset-space.md#function-cosetkernel-grpgpc-grpgpc)

  - [`Example: Coset Action`](coset-space.md#example-ex-bf9acf)

- [The Subgroup Structure](subgroup-structure.md)

  - [General Subgroup Constructions](subgroup-structure.md#general-subgroup-constructions)

    - [`H ^ g: GrpGPC, GrpGPCElt → GrpGPC`](subgroup-structure.md#operation-op-pow-grpgpc-grpgpcelt)

    - [`Conjugate(H, g): GrpGPC, GrpGPCElt → GrpGPC`](subgroup-structure.md#function-conjugate-grpgpc-grpgpcelt)

    - [`H ^ G: GrpGPC, GrpGPC → GrpGPC`](subgroup-structure.md#operation-op-pow-grpgpc-grpgpc)

    - [`ncl< G | H >: GrpGPC, GrpGPC → GrpGPC`](subgroup-structure.md#constructor-constructor-ncl-grpgpc-grpgpc-grpgpc)

    - [`NormalClosure(G, H): GrpGPC, GrpGPC → GrpGPC`](subgroup-structure.md#function-normalclosure-grpgpc-grpgpc)

    - [`CommutatorSubgroup(G, H, K): GrpGPC, GrpGPC, GrpGPC → GrpGPC`](subgroup-structure.md#function-commutatorsubgroup-grpgpc-grpgpc-grpgpc)

    - [`CommutatorSubgroup(H, K): GrpGPC, GrpGPC → GrpGPC`](subgroup-structure.md#function-commutatorsubgroup-grpgpc-grpgpc)

  - [Subgroup Constructions Requiring a Nilpotent Covering Group](subgroup-structure.md#subgroup-constructions-requiring-a-nilpotent-covering-group)

    - [`H meet K: GrpGPC, GrpGPC → GrpGPC`](subgroup-structure.md#operation-op-meet-grpgpc-grpgpc)

    - [`H meet:= K: GrpGPC, GrpGPC → GrpGPC`](subgroup-structure.md#operation-operation-meet-grpgpc-grpgpc-grpgpc)

    - [`Centraliser(G, g): GrpGPC, GrpGPCElt → GrpGPC`](subgroup-structure.md#function-centraliser-grpgpc-grpgpcelt)

    - [`Centralizer(G, g): GrpGPC, GrpGPCElt → GrpGPC`](subgroup-structure.md#function-centralizer-grpgpc-grpgpcelt)

    - [`Centraliser(G, H): GrpGPC, GrpGPC → GrpGPC`](subgroup-structure.md#function-centraliser-grpgpc-grpgpc)

    - [`Centralizer(G, H): GrpGPC, GrpGPC → GrpGPC`](subgroup-structure.md#function-centralizer-grpgpc-grpgpc)

    - [`Core(G, H): GrpGPC, GrpGPC → GrpGPC`](subgroup-structure.md#function-core-grpgpc-grpgpc)

    - [`Normaliser(G, H): GrpGPC, GrpGPC → GrpGPC`](subgroup-structure.md#function-normaliser-grpgpc-grpgpc)

    - [`Normalizer(G, H): GrpGPC, GrpGPC → GrpGPC`](subgroup-structure.md#function-normalizer-grpgpc-grpgpc)

- [General Group Properties](group-boolean.md)

  - [`IsAbelian(G): GrpGPC → BoolElt`](group-boolean.md#function-isabelian-grpgpc)

  - [`IsCyclic(G): GrpGPC → BoolElt`](group-boolean.md#function-iscyclic-grpgpc)

  - [`IsElementaryAbelian(G): GrpGPC → BoolElt`](group-boolean.md#function-iselementaryabelian-grpgpc)

  - [`IsFinite(G): GrpGPC → BoolElt`](group-boolean.md#function-isfinite-grpgpc)

  - [`IsNilpotent(G): GrpGPC → BoolElt`](group-boolean.md#function-isnilpotent-grpgpc)

  - [`IsPerfect(G): GrpGPC → BoolElt`](group-boolean.md#function-isperfect-grpgpc)

  - [`IsSimple(G): GrpGPC → BoolElt`](group-boolean.md#function-issimple-grpgpc)

  - [`IsSoluble(G): GrpGPC → BoolElt`](group-boolean.md#function-issoluble-grpgpc)

  - [`IsSolvable(G): GrpGPC → BoolElt`](group-boolean.md#function-issolvable-grpgpc)

  - [General Properties of Subgroups](group-boolean.md#general-properties-of-subgroups)

    - [`IsCentral(G, H): GrpGPC, GrpGPC → BoolElt`](group-boolean.md#function-iscentral-grpgpc-grpgpc)

    - [`IsNormal(G, H): GrpGPC, GrpGPC → BoolElt`](group-boolean.md#function-isnormal-grpgpc-grpgpc)

  - [Properties of Subgroups Requiring a Nilpotent Covering Group](group-boolean.md#properties-of-subgroups-requiring-a-nilpotent-covering-group)

    - [`IsConjugate(G, H, K): GrpGPC, GrpGPC, GrpGPC → BoolElt, GrpGPCElt`](group-boolean.md#function-isconjugate-grpgpc-grpgpc-grpgpc)

    - [`IsSelfNormalising(G, H): GrpGPC, GrpGPC → BoolElt`](group-boolean.md#function-isselfnormalising-grpgpc-grpgpc)

    - [`IsSelfNormalizing(G, H): GrpGPC, GrpGPC → BoolElt`](group-boolean.md#function-isselfnormalizing-grpgpc-grpgpc)

    - [`Example: Subgroup Structure`](group-boolean.md#example-ex-905fd9)

    - [`Example: Subgroup Structure2`](group-boolean.md#example-ex-bc5c0c)

- [Normal Structure and Characteristic Subgroups](characteristic-subgroup-normal-structure.md)

  - [Characteristic Subgroups and Subgroup Series](characteristic-subgroup-normal-structure.md#characteristic-subgroups-and-subgroup-series)

    - [`Centre(G): GrpGPC → GrpGPC`](characteristic-subgroup-normal-structure.md#function-centre-grpgpc)

    - [`Center(G): GrpGPC → GrpGPC`](characteristic-subgroup-normal-structure.md#function-center-grpgpc)

    - [`DerivedLength(G): GrpGPC → RngIntElt`](characteristic-subgroup-normal-structure.md#function-derivedlength-grpgpc)

    - [`DerivedSeries(G): GrpGPC → [GrpGPC]`](characteristic-subgroup-normal-structure.md#function-derivedseries-grpgpc)

    - [`DerivedSubgroup(G): GrpGPC → GrpGPC`](characteristic-subgroup-normal-structure.md#function-derivedsubgroup-grpgpc)

    - [`DerivedGroup(G): GrpGPC → GrpGPC`](characteristic-subgroup-normal-structure.md#function-derivedgroup-grpgpc)

    - [`EFASeries(G): GrpGPC → [GrpGPC]`](characteristic-subgroup-normal-structure.md#function-grpgpc-efaseries)

    - [`FittingLength(G): GrpGPC → RngIntElt`](characteristic-subgroup-normal-structure.md#function-fittinglength-grpgpc)

    - [`FittingSeries(G): GrpGPC → [GrpGPC]`](characteristic-subgroup-normal-structure.md#function-fittingseries-grpgpc)

    - [`FittingSubgroup(G): GrpGPC → GrpGPC`](characteristic-subgroup-normal-structure.md#function-grpgpc-fittingsubgroup)

    - [`FittingGroup(G): GrpGPC → GrpGPC`](characteristic-subgroup-normal-structure.md#function-fittinggroup-grpgpc)

    - [`HasComputableLCS(G): GrpGPC → BoolElt`](characteristic-subgroup-normal-structure.md#function-grpgpc-hascomputablelcs)

    - [`LowerCentralSeries(G): GrpGPC → [GrpGPC]`](characteristic-subgroup-normal-structure.md#function-grpgpc-lowercentralseries)

    - [`NilpotencyClass(G): GrpGPC → RngIntElt`](characteristic-subgroup-normal-structure.md#function-grpgpc-nilpotencyclass)

    - [`NilpotentPresentation(G): GrpGPC → GrpGPC, Map`](characteristic-subgroup-normal-structure.md#function-nilpotentpresentation-grpgpc)

    - [`SemisimpleEFASeries(G): GrpGPC → [GrpGPC]`](characteristic-subgroup-normal-structure.md#function-grpgpc-semisimpleefaseries)

    - [`UpperCentralSeries(G): GrpGPC → [GrpGPC]`](characteristic-subgroup-normal-structure.md#function-uppercentralseries-grpgpc)

    - [`Example: Normal Structure`](characteristic-subgroup-normal-structure.md#example-ex-c7b9de)

  - [The Abelian Quotient Structure of a Group](characteristic-subgroup-normal-structure.md#the-abelian-quotient-structure-of-a-group)

    - [`AbelianQuotient(G): GrpGPC → GrpAb, Map`](characteristic-subgroup-normal-structure.md#function-abelianquotient-grpgpc)

    - [`AbelianQuotientInvariants(G): GrpGPC → [ RngIntElt ]`](characteristic-subgroup-normal-structure.md#function-abelianquotientinvariants-grpgpc)

    - [`AQInvariants(G): GrpGPC → [ RngIntElt ]`](characteristic-subgroup-normal-structure.md#function-aqinvariants-grpgpc)

    - [`ElementaryAbelianQuotient(G, p): GrpGPC, RngIntElt → GrpAb, Map`](characteristic-subgroup-normal-structure.md#function-elementaryabelianquotient-grpgpc-rngintelt)

    - [`FreeAbelianQuotient(G): GrpGPC → GrpAb, Map`](characteristic-subgroup-normal-structure.md#function-freeabelianquotient-grpgpc)

- [Conjugacy](conjugate.md)

  - [`IsConjugate(G, g, h): GrpGPC, GrpGPCElt, GrpGPCElt → BoolElt, GrpGPCElt`](conjugate.md#function-isconjugate-grpgpc-grpgpcelt-grpgpcelt)

  - [`IsConjugate(G, H, K): GrpGPC, GrpGPC, GrpGPC → BoolElt, GrpGPCElt`](conjugate.md#function-isconjugate-grpgpc-grpgpc-grpgpc-2)

  - [`Example: Conjugacy`](conjugate.md#example-ex-f79f1e)

- [Representation Theory](representation-theory.md)

  - [`EFAModuleMaps(G): GrpGPC → [ModGrp]`](representation-theory.md#function-grpgpc-efamodulemaps)

  - [`EFAModules(G): GrpGPC → [ModGrp]`](representation-theory.md#function-grpgpc-efamodules)

  - [`GModule(G, A, p): GrpGPC, GrpGPC, RngIntElt → ModGrp, Map`](representation-theory.md#function-gmodule-grpgpc-grpgpc-rngintelt)

  - [`GModule(G, A): GrpGPC, GrpGPC → ModGrp, Map`](representation-theory.md#function-gmodule-grpgpc-grpgpc)

  - [`GModule(G, A, B, p): GrpGPC, GrpGPC, GrpGPC, RngIntElt → ModGrp, Map`](representation-theory.md#function-gmodule-grpgpc-grpgpc-grpgpc-rngintelt)

  - [`GModule(G, A, B): GrpGPC, GrpGPC, GrpGPC → ModGrp, Map`](representation-theory.md#function-gmodule-grpgpc-grpgpc-grpgpc)

  - [`GModulePrimes(G, A): GrpGPC, GrpGPC → SetMulti`](representation-theory.md#function-grpgpc-gmoduleprimes)

  - [`GModulePrimes(G, A, B): GrpGPC, GrpGPC, GrpGPC → SetMulti`](representation-theory.md#function-gmoduleprimes-grpgpc-grpgpc-grpgpc)

  - [`SemisimpleEFAModuleMaps(G): GrpGPC → [ModGrp]`](representation-theory.md#function-grpgpc-semisimpleefamodulemaps)

  - [`SemisimpleEFAModules(G): GrpGPC → [ModGrp]`](representation-theory.md#function-grpgpc-semisimpleefamodules)

  - [`Example: Representation Theory`](representation-theory.md#example-ex-e27464)

  - [`Example: gmoduleprimes`](representation-theory.md#example-ex-e34f40)

  - [`Example: Fitting Subgroup`](representation-theory.md#example-ex-407bf7)

  - [`Example: Module Maps`](representation-theory.md#example-ex-5b5ab4)

- [Power Groups](power-group.md)

  - [`Parent(G): GrpGPC → PowStr`](power-group.md#function-grpgpc-parent)

  - [`PowerGroup(G): GrpPC → PowerGroup`](power-group.md#function-powergroup-grppc)