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Computations with the GAP Character Table Library

(Version 1.3.9 of CTblLib)

Thomas Breuer
Email: sam@math.rwth-aachen.de
Homepage: https://www.math.rwth-aachen.de/~Thomas.Breuer

Copyright

© 2013–2024 by Thomas Breuer

This manuscript may be distributed under the terms and conditions of the GNU Public License Version 3 or later, see http://www.gnu.org/licenses.

Contents

1 Maintenance Issues for the GAP Character Table Library
 1.1 Disproving Possible Character Tables (November 2006)

  1.1-1 A Perfect Pseudo Character Table (November 2006)
  1.1-2 An Error in the Character Table of \(E_6(2)\) (March 2016)
  1.1-3 An Error in a Power Map of the Character Table of \(2.F_4(2).2\) (November 2015)
  1.1-4 A Character Table with a Wrong Name (May 2017)
 1.2 Some finite factor groups of perfect space groups (February 2014)

  1.2-1 Constructing the space groups in question
  1.2-2 Constructing the factor groups in question
  1.2-3 Examples with point group \(A_5\)
  1.2-4 Examples with point group \(L_3(2)\)
  1.2-5 Example with point group SL\(_2(7)\)
  1.2-6 Example with point group \(2^3.L_3(2)\)
  1.2-7 Examples with point group \(A_6\)
  1.2-8 Examples with point group \(L_2(8)\)
  1.2-9 Example with point group \(M_{11}\)
  1.2-10 Example with point group \(U_3(3)\)
  1.2-11 Examples with point group \(U_4(2)\)
  1.2-12 A remark on one of the example groups
 1.3 Generality problems (December 2004/October 2015)

  1.3-1 Listing possible generality problems
  1.3-2 A generality problem concerning the group \(J_3\) (April 2015)
  1.3-3 A generality problem concerning the group \(HN\) (August 2022)
 1.4 Brauer Tables that can be derived from Known Tables

  1.4-1 Brauer Tables via Construction Information
  1.4-2 Liftable Brauer Characters (May 2017)
 1.5 Information about certain subgroups of the Monster group

  1.5-1 The Monster group does not contain subgroups of the type \(2.U_4(2)\) (August 2023)
  1.5-2 Perfect central extensions of \(L_3(4)\) (August 2023)
  1.5-3 The character table of \((2 \times O_8^+(3)).S_4 \leq 2.B\) (October 2023)
2 Using Table Automorphisms for Constructing Character Tables in GAP
 2.1 Overview
 2.2 Theoretical Background

  2.2-1 Character Table Automorphisms
  2.2-2 Permutation Equivalence of Character Tables
  2.2-3 Class Fusions
  2.2-4 Constructing Character Tables of Certain Isoclinic Groups
  2.2-5 Character Tables of Isoclinic Groups of the Structure \(p.G.p\) (October 2016)
  2.2-6 Isoclinic Double Covers of Almost Simple Groups
  2.2-7 Characters of Normal Subgroups
 2.3 The Constructions

  2.3-1 Character Tables of Groups of the Structure \(M.G.A\)
  2.3-2 Character Tables of Groups of the Structure \(G.S_3\)
  2.3-3 Character Tables of Groups of the Structure \(G.2^2\)
  2.3-4 Character Tables of Groups of the Structure \(2^2.G\) (August 2005)
  2.3-5 \(p\)-Modular Tables of Extensions by \(p\)-singular Automorphisms
  2.3-6 Character Tables of Subdirect Products of Index Two (July 2007)
 2.4 Examples for the Type \(M.G.A\)

  2.4-1 Character Tables of Dihedral Groups
  2.4-2 An \(M.G.A\) Type Example with \(M\) noncentral in \(M.G\) (May 2004)
  2.4-3 Atlas Tables of the Type \(M.G.A\)
  2.4-4 More Atlas Tables of the Type \(M.G.A\)
  2.4-5 The Character Tables of \(4_2.L_3(4).2_3\) and \(12_2.L_3(4).2_3\)
  2.4-6 The Character Tables of \(12_1.U_4(3).2_2'\) and \(12_2.U_4(3).2_3'\) (December 2015)
  2.4-7 Groups of the Structures \(3.U_3(8).3_1\) and \(3.U_3(8).6\) (February 2017)
  2.4-8 The Character Table of \((2^2 \times F_4(2)):2 < B\) (March 2003)
  2.4-9 The Character Table of \(2.(S_3 \times Fi_{22}.2) < 2.B\) (March 2003)
  2.4-10 The Character Table of \((2 \times 2.Fi_{22}):2 < Fi_{24}\) (November 2008)
  2.4-11 The Character Table of \(S_3 \times 2.U_4(3).2_2 \leq 2.Fi_{22}\) (September 2002)
  2.4-12 The Character Table of \(4.HS.2 \leq HN.2\) (May 2002)
  2.4-13 The Character Tables of \(4.A_6.2_3\), \(12.A_6.2_3\), and \(4.L_2(25).2_3\)
  2.4-14 The Character Table of \(4.L_2(49).2_3\) (December 2020)
  2.4-15 The Character Table of \(4.L_2(81).2_3\) (December 2020)
  2.4-16 The Character Table of \(9.U_3(8).3_3\) (March 2017)
  2.4-17 Pseudo Character Tables of the Type \(M.G.A\) (May 2004)
  2.4-18 Some Extra-ordinary \(p\)-Modular Tables of the Type \(M.G.A\) (September 2005)
 2.5 Examples for the Type \(G.S_3\)

  2.5-1 Small Examples
  2.5-2 Atlas Tables of the Type \(G.S_3\)
 2.6 Examples for the Type \(G.2^2\)

  2.6-1 The Character Table of \(A_6.2^2\)
  2.6-2 Atlas Tables of the Type \(G.2^2\) – Easy Cases
  2.6-3 The Character Table of \(S_4(9).2^2\) (September 2011)
  2.6-4 The Character Tables of Groups of the Type \(2.L_3(4).2^2\) (June 2010)
  2.6-5 The Character Tables of Groups of the Type \(6.L_3(4).2^2\) (October 2011)
  2.6-6 The Character Tables of Groups of the Type \(2.U_4(3).2^2\) (February 2012)
  2.6-7 The Character Tables of Groups of the Type \(4_1.L_3(4).2^2\) (October 2011)
  2.6-8 The Character Tables of Groups of the Type \(4_2.L_3(4).2^2\) (October 2011)
  2.6-9 The Character Table of Aut\((L_2(81))\)
  2.6-10 The Character Table of \(O_8^+(3).2^2_{111}\)
 2.7 Examples for the Type \(2^2.G\)

  2.7-1 The Character Table of \(2^2.Sz(8)\)
  2.7-2 Atlas Tables of the Type \(2^2.G\) (September 2005)
  2.7-3 The Character Table of \(2^2.O_8^+(3)\) (March 2009)
  2.7-4 The Character Table of the Schur Cover of \(L_3(4)\) (September 2005)
 2.8 Examples of Extensions by \(p\)-singular Automorphisms

  2.8-1 Some \(p\)-Modular Tables of Groups of the Type \(M.G.A\)
  2.8-2 Some \(p\)-Modular Tables of Groups of the Type \(G.S_3\)
  2.8-3 \(2\)-Modular Tables of Groups of the Type \(G.2^2\)
  2.8-4 The \(3\)-Modular Table of \(U_3(8).3^2\)
 2.9 Examples of Subdirect Products of Index Two

  2.9-1 Certain Dihedral Groups as Subdirect Products of Index Two
  2.9-2 The Character Table of \((D_{10} \times HN).2 < M\) (June 2008)
  2.9-3 A Counterexample (August 2015)
3 Constructing Character Tables of Central Extensions in GAP
 3.1 Coprime Central Extensions

  3.1-1 The Character Table Head
  3.1-2 The Irreducible Characters
  3.1-3 Ordering of Conjugacy Classes
  3.1-4 Compatibility with Smaller Factor Groups
 3.2 Examples

  3.2-1 Central Extensions of Simple Atlas Groups
  3.2-2 Central Extensions of Other Atlas Groups
  3.2-3 Compatible Central Extensions of Maximal Subgroups
  3.2-4 The 2B Centralizer in \(3.Fi_{24}'\) (January 2004)
4 GAP Computations Concerning Hamiltonian Cycles in the Generating Graphs of Finite Groups
 4.1 Overview
 4.2 Theoretical Background

  4.2-1 Character-Theoretic Lower Bounds for Vertex Degrees
  4.2-2 Checking the Criteria
 4.3 GAP Functions for the Computations

  4.3-1 Computing Vertex Degrees from the Group
  4.3-2 Computing Lower Bounds for Vertex Degrees
  4.3-3 Evaluating the (Lower Bounds for the) Vertex Degrees
 4.4 Character-Theoretic Computations

  4.4-1 Sporadic Simple Groups, except the Monster
  4.4-2 The Monster
  4.4-3 Nonsimple Automorphism Groups of Sporadic Simple Groups
  4.4-4 Alternating and Symmetric Groups \(A_n\), \(S_n\), for \(5 \leq n \leq 13\)
 4.5 Computations With Groups

  4.5-1 Nonabelian Simple Groups of Order up to \(10^7\)
  4.5-2 Nonsimple Groups with Nonsolvable Socle of Order at most \(10^6\)
 4.6 The Groups \(PSL(2,q)\)
5 GAP Computations with \(O_8^+(5).S_3\) and \(O_8^+(2).S_3\)
 5.1 Overview
 5.2 Constructing Representations of \(M.2\) and \(S.2\)

  5.2-1 A Matrix Representation of the Weyl Group of Type \(E_8\)
  5.2-2 Embedding the Weyl group of Type \(E_8\) into GO\({}^+(8,5)\)
  5.2-3 Compatible Generators of \(M\), \(M.2\), \(S\), and \(S.2\)
 5.3 Constructing Representations of \(M.3\) and \(S.3\)

  5.3-1 The Action of \(M.3\) on \(M\)
  5.3-2 The Action of \(S.3\) on \(S\)
 5.4 Constructing Compatible Generators of \(H\) and \(G\)
 5.5 Application: Regular Orbits of \(H\) on \(G/H\)
 5.6 Appendix: The Permutation Character \((1_H^G)_H\)
 5.7 Appendix: The Data File
6 Solvable Subgroups of Maximal Order in Sporadic Simple Groups
 6.1 The Result
 6.2 The Approach

  6.2-1 Use the Table of Marks
  6.2-2 Use Information from the Character Table Library
 6.3 Cases where the Table of Marks is available in GAP
 6.4 Cases where the Table of Marks is not available in GAP

  6.4-1 \(G = Ru\)
  6.4-2 \(G = Suz\)
  6.4-3 \(G = ON\)
  6.4-4 \(G = Co_2\)
  6.4-5 \(G = Fi_{22}\)
  6.4-6 \(G = HN\)
  6.4-7 \(G = Ly\)
  6.4-8 \(G = Th\)
  6.4-9 \(G = Fi_{23}\)
  6.4-10 \(G = Co_1\)
  6.4-11 \(G = J_4\)
  6.4-12 \(G = Fi_{24}^{\prime}\)
  6.4-13 \(G = B\)
  6.4-14 \(G = M\)
 6.5 Proof of the Corollary
7 Large Nilpotent Subgroups of Sporadic Simple Groups
 7.1 The Result
 7.2 The Proof
 7.3 Alternative: Use GAP's Tables of Marks
8 Permutation Characters in GAP
 8.1 Some Computations with \(M_{24}\)
 8.2 All Possible Permutation Characters of \(M_{11}\)
 8.3 The Action of \(U_6(2)\) on the Cosets of \(M_{22}\)
 8.4 Degree \(20\,736\) Permutation Characters of \(U_6(2)\)
 8.5 Degree \(57\,572\,775\) Permutation Characters of \(O_8^+(3)\)
 8.6 The Action of \(O_7(3).2\) on the Cosets of \(2^7.S_7\)
 8.7 The Action of \(O_8^+(3).2_1\) on the Cosets of \(2^7.A_8\)
 8.8 The Action of \(S_4(4).4\) on the Cosets of \(5^2.[2^5]\)
 8.9 The Action of \(Co_1\) on the Cosets of Involution Centralizers
 8.10 The Multiplicity Free Permutation Characters of \(G_2(3)\)
 8.11 Degree \(11\,200\) Permutation Characters of \(O_8^+(2)\)
 8.12 A Proof of Nonexistence of a Certain Subgroup
 8.13 A Permutation Character of the Lyons group
 8.14 Identifying two subgroups of Aut\((U_3(5))\) (October 2001)
 8.15 A Permutation Character of Aut\((O_8^+(2))\) (October 2001)
 8.16 Four Primitive Permutation Characters of the Monster Group

  8.16-1 The Subgroup \(2^2.2^{11}.2^{22}.(S_3 \times M_{24})\) (June 2009)
  8.16-2 The Subgroup \(2^3.2^6.2^{12}.2^{18}.(L_3(2) \times 3.S_6)\) (September 2009)
  8.16-3 The Subgroup \(2^5.2^{10}.2^{20}.(S_3 \times L_5(2))\) (October 2009)
  8.16-4 The Subgroup \(2^{{10+16}}.O_{10}^+(2)\) (November 2009)
 8.17 A permutation character of the Baby Monster (June 2012)
 8.18 A permutation character of \(2.B\) (October 2017)
 8.19 Generation of sporadic simple groups by \(\pi\)- and \(\pi'\)-subgroups (December 2021)
9 Ambiguous Class Fusions in the GAP Character Table Library
 9.1 Some GAP Utilities
 9.2 Fusions Determined by Factorization through Intermediate Subgroups

  9.2-1 \(Co_3N5 \rightarrow Co_3\) (September 2002)
  9.2-2 \(31:15 \rightarrow B\) (March 2003)
  9.2-3 \(SuzN3 \rightarrow Suz\) (September 2002)
  9.2-4 \(F_{{3+}}N5 \rightarrow F_{{3+}}\) (March 2002)
 9.3 Fusions Determined Using Commutative Diagrams Involving Smaller Subgroups

  9.3-1 \(BN7 \rightarrow B\) (March 2002)
  9.3-2 \((A_4 \times O_8^+(2).3).2 \rightarrow Fi_{24}^{\prime}\) (November 2002)
  9.3-3 \(A_6 \times L_2(8).3 \rightarrow Fi_{24}^{\prime}\) (November 2002)
  9.3-4 \((3^2:D_8 \times U_4(3).2^2).2 \rightarrow B\) (June 2007)
  9.3-5 \(7^{1+4}:(3 \times 2.S_7) \rightarrow M\) (May 2009)
  9.3-6 \(3^7.O_7(3):2 \rightarrow Fi_{24}\) (November 2010)
  9.3-7 \({}^2E_6(2)N3C \rightarrow {}^2E_6(2)\) (January 2019)
 9.4 Fusions Determined Using Commutative Diagrams Involving Factor Groups

  9.4-1 \(3.A_7 \rightarrow 3.Suz\) (December 2010)
  9.4-2 \(S_6 \rightarrow U_4(2)\) (September 2011)
 9.5 Fusions Determined Using Commutative Diagrams Involving Automorphic Extensions

  9.5-1 \(U_3(8).3_1 \rightarrow {}^2E_6(2)\) (December 2010)
  9.5-2 \(L_3(4).2_1 \rightarrow U_6(2)\) (December 2010)
 9.6 Conditions Imposed by Brauer Tables

  9.6-1 \(L_2(16).4 \rightarrow J_3.2\) (January 2004)
  9.6-2 \(L_2(17) \rightarrow S_8(2)\) (July 2004)
  9.6-3 \(L_2(19) \rightarrow J_3\) (April 2003)
 9.7 Fusions Determined by Information about the Groups

  9.7-1 \(U_3(3).2 \rightarrow Fi_{24}^{\prime}\) (November 2002)
  9.7-2 \(L_2(13).2 \rightarrow Fi_{24}^{\prime}\) (September 2002)
  9.7-3 \(M_{11} \rightarrow B\) (April 2009)
  9.7-4 \(L_2(11):2 \rightarrow B\) (April 2009)
  9.7-5 \(L_3(3) \rightarrow B\) (April 2009)
  9.7-6 \(L_2(17).2 \rightarrow B\) (March 2004)
  9.7-7 \(L_2(49).2_3 \rightarrow B\) (June 2006)
  9.7-8 \(2^3.L_3(2) \rightarrow G_2(5)\) (January 2004)
  9.7-9 \(5^{{1+4}}.2^{{1+4}}.A_5.4 \rightarrow B\) (April 2009)
  9.7-10 The fusion from the character table of \(7^2:2L_2(7).2\) into the table of marks (January 2004)
  9.7-11 \(3 \times U_4(2) \rightarrow 3_1.U_4(3)\) (March 2010)
  9.7-12 \(2.3^4.2^3.S_4 \rightarrow 2.A12\) (September 2011)
  9.7-13 \(127:7 \rightarrow L_7(2)\) (January 2012)
  9.7-14 \(L_2(59) \rightarrow M\) (May 2009)
  9.7-15 \(L_2(71) \rightarrow M\) (May 2009)
  9.7-16 \(L_2(41) \rightarrow M\) (April 2012)
10 GAP computations needed in the proof of [DNT13, Theorem 6.1 (ii)]
 10.1 \(G/N \cong Sz(8)\) and \(|N| = 2^{12}\)
 10.2 \(G/N \cong M_{22}\) and \(|N| = 2^{10}\)
 10.3 \(G/N \cong J_2\) and \(|N| = 2^{12}\)
 10.4 \(G/N \cong J_2\) and \(|N| = 5^{14}\)
 10.5 \(G/N \cong J_2\) and \(|N| = 2^{28}\)
 10.6 \(G/N \cong {}^3D_4(2)\) and \(|N| = 2^{26}\)
 10.7 \(G/N \cong {}^3D_4(2)\) and \(|N| = 3^{25}\)
11 GAP Computations Concerning Probabilistic Generation of Finite Simple Groups
 11.1 Overview
 11.2 Prerequisites

  11.2-1 Theoretical Background
  11.2-2 Computational Criteria
 11.3 GAP Functions for the Computations

  11.3-1 General Utilities
  11.3-2 Character-Theoretic Computations
  11.3-3 Computations with Groups
 11.4 Character-Theoretic Computations

  11.4-1 Sporadic Simple Groups
  11.4-2 Automorphism Groups of Sporadic Simple Groups
  11.4-3 Other Simple Groups – Easy Cases
  11.4-4 Automorphism Groups of other Simple Groups – Easy Cases
  11.4-5 \(O_8^-(3)\)
  11.4-6 \(O_{10}^+(2)\)
  11.4-7 \(O_{10}^-(2)\)
  11.4-8 \(O_{12}^+(2)\)
  11.4-9 \(O_{12}^-(2)\)
  11.4-10 \(S_6(4)\)
  11.4-11 \(\ast\) \(S_6(5)\)
  11.4-12 \(S_8(3)\)
  11.4-13 \(U_4(4)\)
  11.4-14 \(U_6(2)\)
 11.5 Computations using Groups

  11.5-1 \(A_{2m+1}\), \(2 \leq m \leq 11\)
  11.5-2 \(A_5\)
  11.5-3 \(A_6\)
  11.5-4 \(A_7\)
  11.5-5 \(L_d(q)\)
  11.5-6 \(\ast\) \(L_d(q)\) with prime \(d\)
  11.5-7 Automorphic Extensions of \(L_d(q)\)
  11.5-8 \(L_3(2)\)
  11.5-9 \(M_{11}\)
  11.5-10 \(M_{12}\)
  11.5-11 \(O_7(3)\)
  11.5-12 \(O_8^+(2)\)
  11.5-13 \(O_8^+(3)\)
  11.5-14 \(O^+_8(4)\)
  11.5-15 \(\ast\) \(O_9(3)\)
  11.5-16 \(O_{10}^-(3)\)
  11.5-17 \(O_{14}^-(2)\)
  11.5-18 \(O_{12}^+(3)\)
  11.5-19 \(\ast\) \(S_4(8)\)
  11.5-20 \(S_6(2)\)
  11.5-21 \(S_8(2)\)
  11.5-22 \(\ast\) \(S_{10}(2)\)
  11.5-23 \(U_4(2)\)
  11.5-24 \(U_4(3)\)
  11.5-25 \(U_6(3)\)
  11.5-26 \(U_8(2)\)
References
Index

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