tmstdrd.tom

GAP源码

#############################################################################
##
#W  tmstdrd.tom          GAP table of marks library           Thomas Merkwitz
#W                                                              Thomas Breuer
##
#H  @(#)$Id: tmstdrd.tom,v 1.2 2001/07/30 11:09:32 gap Exp $
##
#Y  Copyright (C)  1999,  Lehrstuhl D fuer Mathematik,  RWTH Aachen,  Germany
##
##  This file contains the descriptions of standard generators of tables of
##  marks in the {\GAP} library.
##
##  Each entry in the list `LIBTOMKNOWN.STDGEN' is a list of length $4$.
##  \beginlist
##  \item{-}
##      At the first position the *identifier* value of the table of marks in
##      the {\GAP} library is stored.
##  \item{-}
##      The value at the second position is one of the strings `\"Y\"',
##      `\"N\"', the former meaning that the standard generators coincide
##      with those defined in Rob Wilson's {\ATLAS} of Group Representations
##      (see~`http://www.mat.bham.ac.uk/atlas').
##      More precisely, this was true at least in March~1998.
##  \item{-}
##      The value at position three is a string of names of the standard
##      generators, for example `\"a, b\"' means that a pair of generators
##      with names `a' and `b' is dealt with.
##  \item{-}
##      Finally, the value at position four is a string that describes how
##      standard generators for the group $G$ in question are defined,
##      in terms of their names.
##      The following terms occur, separated by commas.
##      $|g|=q$ means that the element $g$ has order $q$,
##      $|C(g)|=n$ means that the centralizer of $g$ in $G$ has order $n$,
##      $|x|=q, x^m=g$ means that the $m$-th power of of any element of order
##      $q$ in $G$ is conjugate to $g$.
##      Additionally, if $G$ is a cyclic downward extension of another group
##      $H$ then the definition of standard generators of $G$ may include
##      that they are preimages of standard generators of $H$;
##      This is then mentioned in the string at position three.
##  \endlist
##
Revision.tmstdrd_tom :=
    "@(#)$Id: tmstdrd.tom,v 1.2 2001/07/30 11:09:32 gap Exp $";


LIBTOMKNOWN.STDGEN := [

["A5",         "Y", "a, b",
                    "|a|=2, |b|=3, |ab|=5" ],
["S5",         "Y", "c, d",
                    "|z|=6, z^3=c, |d|=4, |cd|=5" ],
["2.A5",       "Y", "e, f",
                    "|e|=4, |f|=3, |ef|=5" ],
["2.S5",       "N", "g, h",
                    "|g|=2, |C(g)|=12, |h|=5, |gh|=8" ],
["2.S5'",      "N", "i, j",
                    "|i|=10, |j|=12, |ij|=8, |ijj|=6" ],

["L2(7)",      "Y", "a, b",
                    "|a|=2, |b|=3, |ab|=7" ],
["L2(7).2",    "N", "c, d",
                    "|c|=2, |C(c)|=12, |d|=3, |cd|=8, |cdcdd|=7" ],

["A6",         "Y", "a, b",
                    "|a|=2, |b|=4, |ab|=5" ],
["S6",         "Y", "c, d",
                    "|z|=6, z^3=c, |d|=5, |cd|=6, |cdd|=6" ],
["A6.2_2",     "N", "e, f",
                    "|z|=10, z^5=e, |f|=3, |ef|=10" ],
["A6.2_3",     "N", "g, h",
                    "|g|=3, |h|=8, |gh|=4, |ghh|=4" ],
["A6.2^2",     "N", "i, j",
                    "|z|=6, z^3=i, |j|=10, |ij|=4" ],
["2.A6",       "N", "k, l",
                    "|k|=4, |l|=8, |kl|=10" ],
["2.S6",       "N", "preimages m, n of c, d",
                    "|m|=2, |n|=5" ],

["L2(8)",      "Y", "a, b",
                    "|a|=2, |b|=3, |ab|=7" ],
["L2(8).3",    "Y", "c, d",
                    "|c|=2, |d|=3, |cd|=9, |(cd)^3(dc)^2ddcdd|=7" ],

["L2(11)",     "Y", "a, b",
                    "|a|=2, |b|=3, |ab|=11" ],
["L2(11).2",   "N", "c, d",
                    "|z|=10, z^5=c, |d|=3, |cd|=10, |cdcdd|=11" ],

["L2(13)",     "Y", "a, b",
                    "|a|=2, |b|=3, |ab|=13" ],
["L2(13).2",   "N", "c, d",
                    "|z|=14, z^7=c, |d|=3, |cd|=14, |cdcdd|=6" ],

["L2(17)",     "Y", "a, b",
                    "|a|=2, |b|=3, |ab|=17" ],

["A7",         "Y", "a, b",
                    "|z|=6, z^2=a, |b|=5, |ab|=7" ],
["S7",         "N", "c, d",
                    "|z|=10, z^5=c, |d|=7, |cd|=6" ],
["2.A7",       "Y", "preimages e, f of a, b",
                    "|e|=3, |f|=5, |ef|=7" ],
["2.S7",       "N", "preimages g, h of c, d",
                    "|h|=7" ],

["L2(19)",     "Y", "a, b",
                    "|a|=2, |b|=3, |ab|=19" ],

["L2(16)",     "Y", "a, b",
                    "|a|=2, |b|=3, |ab|=15" ],
["L2(16).2",   "N", "c, d",
                    "|c|=3, |d|=4, |cd|=10, |cd^3|=6" ],
["L2(16).4",   "N", "e, f",
                    "|e|=3, |f|=8, |ef|=12, |effeefef|=2" ],

["L3(3)",      "Y", "a, b",
                    "|a|=2, |b|=3, |C(b)|=9, |ab|=13, |ababb|=4" ],
["L3(3).2",    "Y", "c, d",
                    "|c|=2, |C(c)|=48, |z|=12, z^3=d, |cd|=13, |(cd)^3d|=12" ],

["U3(3)",      "Y", "a, b",
                    "|a|=2, |b|=6, |ab|=7" ],
["U3(3).2",    "Y", "c, d",
                    "|y|=6, y^3=c, |z|=12, z^3=d, |cd|=7" ],

["L2(23)",     "Y", "a, b",
                    "|a|=2, |b|=3, |ab|=23" ],

["L2(25)",     "N", "a, b",
                    "|a|=2, |b|=3, |ab|=13, |ababb|=13" ],
["L2(25).2_2", "N", "c, d",
                    "|c|=4, |C(c)|=8, |d|=12, |cd|=4, |cdcdd|=3,\
 |cdc^3d^-1|=10" ],

["M11",        "Y", "a, b",
                    "|a|=2, |b|=4, |ab|=11, |(ab)^4(ba)^2bbabb|=4" ],

["L2(27)",     "N", "a, b",
                    "|a|=2, |b|=3, |ab|=14" ],

["L2(29)",     "Y", "a, b",
                    "|a|=2, |b|=3, |ab|=29" ],

["L2(31)",     "Y", "a, b",
                    "|a|=2, |b|=3, |ab|=31" ],

["A8",         "Y", "a, b",
                    "|z|=15, z^5=a, |b|=7, |ab|=6, |abb|=15" ],
["S8",         "Y", "c, d",
                    "|z|=10, z^5=c, |d|=7, |cd|=8" ],
["2.A8",       "N", "e, f",
                    "|z|=15, z^5=e, |f|=7, |ef|=6, |ef^3|=30" ],
["2.S8",       "N", "g, h",
                    "|g|=2, |C(g)|=96, |h|=15, |hghhgh^-1g|=20" ],

["L3(4)",      "Y", "a, b",
                    "|a|=2, |b|=4, |ab|=7, |abb|=5" ],
["L3(4).2_1",  "Y", "c, d",
                    "|z|=6, z^3=c, |d|=4, |C(d)|=16, |cd|=7" ],
["L3(4).2_2",  "Y", "e, f",
                    "|z|=14, z^7=e, |f|=5, |ef|=14, |eff|=8" ],
["L3(4).2_3",  "N", "g, h",
                    "|z|=8, z^4=g, |h|=10, |gh|=6, |ghh|=3" ],
["L3(4).3",    "N", "i, j",
                    "|i|=2, |j|=21, |ij|=21, |ij^3|=7" ],
["L3(4).2^2",  "N", "k, l",
                    "|z|=10, z^5=k, |l|=6, |C(l)|=36, |kl|=8, |kllklkll|=14" ],
["L3(4).6",     "Y", "m, n",
                    "|m|=2, |C(m)|=432, |n|=6, |C(n)|=36, |mn|=21, |mnmnn|=4" ],
["L3(4).3.2_2", "N", "p, q",
                    "|z|=14, z^7=p, |q|=21, |pq|=6, |pqqpqpq^3|=4" ],
["L3(4).3.2_3", "N", "r, s",
                    "|z|=10, z^5=r, |s|=21, |rs|=8, |rss|=8,\
 |srssrs^-1r|=8" ],
["L3(4).D12",       "N", "t, u",
                    "|z|=14, z^7=t, |u|=12, |tu|=8, |tuu|=14" ],
["3.L3(4)",         "Y", "preimages A, B of a, b",
                    "|A|=2, |B|=4" ],
["3.L3(4).2_3",     "N", "preimages C, D of g, h",
                    "|C|=2" ],
["2^2.L3(4)",       "N", "E, F",
                    "|z|=12, z^6=E, |F|=10, |EF|=6, |EFF|=3" ],
["2^2.L3(4).3",     "N", "G, H",
                    "|z|=15, z^5=G, |H|=7, |GH|=15, |GGHGHH|=6" ],
["2^2.L3(4).2_2",   "N", "I, J",
                    "|I|=3, |J|=28, |IJ|=8, |IJJ|=6, |IJ^3|=4" ],
["2^2.L3(4).3.2_2", "N", "K, L",
                    "|z|=15, z^5=K, |L|=28, |KL|=8, |KL^3|=28, |KL^4|=6,\
 |KLKKLL|=4" ],

["L2(37)",     "N", "a, b",
                    "|a|=2, |b|=3, |ab|=31" ],

["U4(2)",      "N", "a, b",
                    "|z|=12, z^6=a, |b|=9, |ab|=5" ],
["U4(2).2",    "N", "c, d",
                    "|z|=10, z^5=c, |d|=9, |cd|=10" ],
["2.U4(2)",    "N", "preimages e, f of a, b",
                    "|f|=18, |ef|=5" ],
["2.U4(2).2",  "N", "preimages g, h of c, d",
                    "|h|=9" ],

["Sz(8)",      "Y", "a, b",
                    "|a|=2, |b|=4, |abb|=7, |(ab)^2bbabb|=7" ],

["L2(32)",     "Y", "a, b",
                    "|a|=2, |b|=3, |ab|=31, |ababb|=11" ],

["L2(41)",     "N", "a, b",
                    "|a|=2, |b|=3, |ab|=41" ],

["L2(43)",     "N", "a, b",
                    "|a|=2, |b|=3, |ab|=43" ],

["L2(47)",     "N", "a, b",
                    "|a|=2, |b|=3, |ab|=47" ],

["L2(49)",     "N", "a, b",
                    "|a|=2, |b|=3, |ab|=47, |ababb|=8" ],

["U3(4)",      "Y", "a, b",
                    "|a|=2, |b|=3, |ab|=13" ],
["U3(4).2",    "N", "c, d",
                    "|z|=6, z^3=c, |d|=5, |C(d)|=50, |cd|=6, |cdd|=6" ],

["L2(53)",     "N", "a, b",
                    "|a|=2, |b|=3, |ab|=53" ],