Computations with the GAP Character Table Library
  
  
                           (Version 1.3.4 of CTblLib)
  
  
                                 Thomas Breuer
  
  
  
  Thomas Breuer
      Email:    mailto:sam@math.rwth-aachen.de
      Homepage: https://www.math.rwth-aachen.de/~Thomas.Breuer
  
  -------------------------------------------------------
  Copyright
  © 2013–2022 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 (CTblLibXpls)
  
  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)
    1.4 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)
  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 × F_4(2)):2 < B (March 2003)
      2.4-9 The Character Table of 2.(S_3 × Fi_22.2) < 2.B (March 2003)
      2.4-10 The Character Table of (2 × 2.Fi_22):2 < Fi_24 (November 2008)
      2.4-11 The Character Table of S_3 × 2.U_4(3).2_2 ≤ 2.Fi_22 (September
      2002)
      2.4-12 The Character Table of 4.HS.2 ≤ 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 × 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 ≤ n ≤ 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^'
      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 20736 Permutation Characters of U_6(2)
    8.5 Degree 57572775 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 11200 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 × M_24) (June 2009)
      8.16-2 The Subgroup 2^3.2^6.2^12.2^18.(L_3(2) × 3.S_6) (September 2009)
      8.16-3 The Subgroup 2^5.2^10.2^20.(S_3 × 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 π- and π'-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 → Co_3 (September 2002)
      9.2-2 31:15 → B (March 2003)
      9.2-3 SuzN3 → Suz (September 2002)
      9.2-4 F_{3+}N5 → F_{3+} (March 2002)
    9.3 Fusions Determined Using Commutative Diagrams Involving Smaller
    Subgroups
      9.3-1 BN7 → B (March 2002)
      9.3-2 (A_4 × O_8^+(2).3).2 → Fi_24^' (November 2002)
      9.3-3 A_6 × L_2(8).3 → Fi_24^' (November 2002)
      9.3-4 (3^2:D_8 × U_4(3).2^2).2 → B (June 2007)
      9.3-5 7^1+4:(3 × 2.S_7) → M (May 2009)
      9.3-6 3^7.O_7(3):2 → Fi_24 (November 2010)
      9.3-7 ^2E_6(2)N3C → ^2E_6(2) (January 2019)
    9.4 Fusions Determined Using Commutative Diagrams Involving Factor Groups
      9.4-1 3.A_7 → 3.Suz (December 2010)
      9.4-2 S_6 → U_4(2) (September 2011)
    9.5 Fusions Determined Using Commutative Diagrams Involving Automorphic
    Extensions
      9.5-1 U_3(8).3_1 → ^2E_6(2) (December 2010)
      9.5-2 L_3(4).2_1 → U_6(2) (December 2010)
    9.6 Conditions Imposed by Brauer Tables
      9.6-1 L_2(16).4 → J_3.2 (January 2004)
      9.6-2 L_2(17) → S_8(2) (July 2004)
      9.6-3 L_2(19) → J_3 (April 2003)
    9.7 Fusions Determined by Information about the Groups
      9.7-1 U_3(3).2 → Fi_24^' (November 2002)
      9.7-2 L_2(13).2 → Fi_24^' (September 2002)
      9.7-3 M_11 → B (April 2009)
      9.7-4 L_2(11):2 → B (April 2009)
      9.7-5 L_3(3) → B (April 2009)
      9.7-6 L_2(17).2 → B (March 2004)
      9.7-7 L_2(49).2_3 → B (June 2006)
      9.7-8 2^3.L_3(2) → G_2(5) (January 2004)
      9.7-9 5^{1+4}.2^{1+4}.A_5.4 → 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 × U_4(2) → 3_1.U_4(3) (March 2010)
      9.7-12 2.3^4.2^3.S_4 → 2.A12 (September 2011)
      9.7-13 127:7 → L_7(2) (January 2012)
      9.7-14 L_2(59) → M (May 2009)
      9.7-15 L_2(71) → M (May 2009)
      9.7-16 L_2(41) → M (April 2012)
  10 GAP computations needed in the proof of [?DNT?]
    10.1 G/N ≅ Sz(8) and |N| = 2^12
    10.2 G/N ≅ M_22 and |N| = 2^10
    10.3 G/N ≅ J_2 and |N| = 2^12
    10.4 G/N ≅ J_2 and |N| = 5^14
    10.5 G/N ≅ J_2 and |N| = 2^28
    10.6 G/N ≅ ^3D_4(2) and |N| = 2^26
    10.7 G/N ≅ ^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 ∗ 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 ≤ m ≤ 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 ∗ 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 ∗ O_9(3)
      11.5-16 O_10^-(3)
      11.5-17 O_14^-(2)
      11.5-18 O_12^+(3)
      11.5-19 ∗ S_4(8)
      11.5-20 S_6(2)
      11.5-21 S_8(2)
      11.5-22 ∗ 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)