[GAP Forum] Re: P/G/L/V/ext names in Character Table Library

Thomas Breuer thomas.breuer at math.rwth-aachen.de
Fri Aug 27 16:33:24 BST 2004

Dear GAP Forum,

Mary Schaps asked for an interpretation of character table names
such as `P21/G2/L1/V1/ext2'.

Most of the tables with such names are contained (in CAS format,
with the same names)
in the microfiches that are a part of the book ``Perfect Groups'',
by D. Holt and W. Plesken.
The underlying groups are factor groups of space groups,
specifically, they are downward extensions of finite integral matrix groups
(point groups) $G$
with abelian groups of the form $L/n L$,
where $L$ is a sublattice of the natural Z-lattice on which $G$ acts,
and $n$ is a positive integer.

Each name is of the form `P<j>/G<k>/L<l>/V<m>/ext<n>', where
  <j> is the number of the isomorphism type of the point group $G$, say,
      in the list in Section 6.2 of the book
      (so `P21' means the alternating group on six points),
  <k> distinguishes the integral matrix representations of $G$,
  <l> and <m> distinguish the sublattices $L$ and the vector systems $V$
      (see the tables in Chapter 6 of the book), and
  <n> is the positive integer $n$ such that the character table is that of
      the extension of $G$ by the reduction mod $n$ of the sublattice.
The appendix of the book contains a list of the tables on the microfiches,
for example the table with the name `P21/G2/L1/V1/ext2' occurs in line -6
on p. 358 of the book.

(By the way, recently I found an error in the table with the name
`P12/G1/L2/V1/ext2'; this will be corrected with the next release
of the GAP Character Table Library.)

Mary Schaps also asked for the character table names of the forms
`P<n>L<m>' and `mo81'.
These names stem from the CAS library of character tables.
Names of the first form denote certain parabolic subgroups of linear
For example,

    'P1L62' denotes 2^5:L5(2) < L6(2),
    'P1L72' denotes 2^6:L6(2) < L7(2), and
    'P1L82' denotes 2^7:L7(2) < L8(2).

The `InfoText' value of the table `mo81' says

    origin: CAS library,
    names:=mo81; m8[1]
     order: 2^13.3^2.5.7 = 2,580,480
     number of classes: 64
     source:dye, r.h.
            the classes and characters of
            certain maximal and other subgroups
            of o 2n+2(2)
            ann.mat.pura appl.(4) 107
            (1975), 13-47
     comments:semidirect product of an elementary
              abelian group of order 2^6 and o6,
              table blown up using cas-system
    tests: 1.o.r., pow[2,3,5,7]

In ATLAS notation, one would call this table `2^6:S8',
the group is a maximal subgroup of `O8+(2).2',
see p. 85 of the ATLAS of Finite Groups.

I hope this helps,

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