[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
groups.
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,
Thomas
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