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<p id="mathjaxlink" class="pcenter"><a href="chap4_mj.html">[MathJax on]</a></p>
<p><a id="X7CDC63A27F7790AA" name="X7CDC63A27F7790AA"></a></p>
<div class="ChapSects"><a href="chap4.html#X7CDC63A27F7790AA">4 <span class="Heading">Extended examples</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4.html#X7DEAD03D7811F9FA">4.1 <span class="Heading">The Character Table Library</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4.html#X84ED1F0D7A47B055">4.2 <span class="Heading">The behavior of the variable HeLP sol</span></a>
<span>
</div>
<div class="ContSect"><span class=
<
</div>
<divjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4.html#X818A647182CA20B3">4.5 <span class="Heading">Non-standard characters</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4.html#X879DC8C287C41B09">4.6 <span class="Heading">A complete example: (PQ) for the MacLaughlin
</span>
</div>
</div>

<h3>4 <span class="Heading">Extended examples</span></h3>

<p>We will give somejava.lang.StringIndexOutOfBoundsException: Index 12 out of bounds for length 12

<p><a id="X7DEAD03D7811F9FA" name4'will beused from 

<h4>4.1 <span class="Heading">The Character Table Library</span></h4>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">G := SL(2,7);</span>
SL(2,7)
<span class="GAPprompt">gap></span> <span class="GAPinput">HeLP_ZC(G);</span>
#  Thejava.lang.StringIndexOutOfBoundsException: Range [22, 21) out of bounds for length 78
#I  (ZC) can't be solved, using the given data, for the orders: [ 8 ].
false
<span class="GAPprompt">gap></span> <span class="GAPinput">C1 := CharacterTable(G);</span>
java.lang.StringIndexOutOfBoundsException: Range [15, 14) out of bounds for length 25
<span class="GAPprompt">ap;<pan java.lang.StringIndexOutOfBoundsException: Range [51, 50) out of bounds for length 93
r java.lang.StringIndexOutOfBoundsException: Range [22, 21) out of bounds for length 78
#I  (ZC) can't be solved, using the given data, for the orders: [ 8 ].
false
<span class="GAPprompt">gap></span> <span class="GAPinput">C2 := CharacterTable("2.L2(7)");</spanIofsolutions  of order39: 0in39].  
CharacterTable(redund'will from now on.
<span class="GAPprompt">gap></span> <span class="GAPinput">HeLP_ZC(C2);</span>
true
</pre></div>

<p>Note that the first and the second call of <code class="func">HeLP_ZC</code> (<a of  java.lang.StringIndexOutOfBoundsException: Range [28, 27) out of bounds for length 78

<p><a id="X84ED1F0D7A47B055" name="X84ED1F0D7A47B055"></a></p>

<h4>4.2 <span class="span class="GAPprompt"gt;<>spanjava.lang.StringIndexOutOfBoundsException: Range [62, 60) out of bounds for length 117

<p>This sections demonstrates when the global variable


<div class="example"><pre
<span=>gt;/>< =APinput> (A5;span
CharacterTable( "A5" )
<span class="GAPprompt">gap></
true
<span class="GAPprompt">java.lang.StringIndexOutOfBoundsException: Index 27 out of bounds for length 26
[ [ [ [ 1 ] ] ], [ [ [ 1 ] ] ], [ [ [ 1 ] ] ],, 
    0   , [ 1,]] ,, [,,,  ],,,,,,,,,,,,,,, [  ] 
 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">C := CharacterTable("L3(7).2");</span>
CharacterTable( <"APpromptgap&<><spanclass="(,/pan
<span class="GAPprompt">gap></span> <span class="GAPinput">java.lang.StringIndexOutOfBoundsException: Range [0, 86) out of bounds for length 74
#  of java.lang.StringIndexOutOfBoundsException: Range [24, 23) out of bounds for length 80
s for the powers: 1.
[ [ [ 1 ], [ 3, 9, -11 ], [ -6, 0, 3, 4 ] ] ]
<span class="GAPprompt">gap&
[[ [  ] ]java.lang.StringIndexOutOfBoundsException: Index 17 out of bounds for length 17
</spanjava.lang.StringIndexOutOfBoundsException: Range [12, 11) out of bounds for length 112

<p>The function <code class="func">HeLP_WithGivenOrderAndPA</code> (<a href="chap3.java.lang.StringIndexOutOfBoundsException: Index 85 out of bounds for length 76


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class=pThis isalso  goodexample to use<class="Math"><span-constant characters:/
# elementsoforder3 s java.lang.StringIndexOutOfBoundsException: Range [62, 61) out of bounds for length 74
[[  ] 
<span class="GAPprompt">gap></span> <span class="GAPinput">HeLP_sol;</span>
[ [ [ [ 1 ] ] ],, [ [ [ 1 ] ] ] ]
</pre></div>

<p>If HeLP detects that the table used belongs to the same group, <code class="keyw">HeLP_sol</code> is not#I    Number of non-trivial 13-constant characters in the list: 19.           


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">HeLP_WithGivenOrder(C mod 7, 19);</span>
#I  Number of solutions for elements of order 19: 3; stored in HeLP_sol[19].
[ [ [ 0, 0, 1 ] ], [ [ 0, 1, 0 ] ], [ [ 1, 0, 0 ] ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">HeLP_sol;</span>
[ [ [ [ 1 ] ] ],, [ [ [ 1 ] ] ],,,,,,,,,,,,,,,, 
  [ [ [ 0, 0, 1 ] ], [ [ 0, 1, 0 ] ], [ [ 1, 0, 0 ] ] ] ]
# the previously calaculated result for order 3 is still there.
</pre></div>

<p>HeLP can detect that the character tables belong to the same group, if they are identical objects in GAP or if both are tables of the same group from the ATLAS and their InfoText begins with "origin: ATLAS of finite groups" (which is usually the case for ATLAS tables). If the program can verify that the character table which is used at the current call of a function belongs to the same group as in the previous call of a function, the solutions stored in <code class="keyw">HeLP_sol</code> are kept. If the character table belongs to another group or it can not be made sure that the character table belongs to the same group, <code class="keyw">HeLP_sol</

<p>Not reseting <code class="keyw">HeLP_sol</code> can also be achieved using <code class="func">HeLP_ChangeCharKeepSols</code> (<a href="chap3.html#X7BB9009482784E90"><span class="RefLink">3.4-1</span></a>). However, caution should be exercised when using this command since it may manipulate <code class="keyw">HeLP_sol</code> into something meaningless.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">G := PSL(2,7);</span>
Group([ (3,7,5)(4,8,6), (1,2,6)(3,4,8) ])
<span class="GAPprompt">gap></span> <span class="GAPinput">HeLP_ZC(G);</span>
#I  The Brauer tables for the following primes are not available: [ 2, 3, 7 ].
#I  (ZC) can't be solved, using the given data, for the orders: [ 6 ].
falsespanclass="">gapg< < class="GAPinput>fjava.lang.StringIndexOutOfBoundsException: Range [84, 82) out of bounds for length 91
<span class="GAPprompt">gap></span> <span class="GAPinput">HeLP_sol;</span>
[ [ [ [ 1 ] ] ], [ [ [ 1 ] ] ], [ [ [ 1 ] ] ], [ [ [ 1 ], [ 0, 1 ] ] ],, 
  [ [ [ 1 ], [ 1 ], [ -2, 3 ] ] ], [ [ [ 0, 1 ] ], [ [ 1, 0 ] ] ],,,,, [  ],, 
  [  ],,,,,,, [  ],,,,,,, [  ],,,,,,,,,,,,,, [  ],,,,,,,,,,,,,,,,,,,,,,,,,,,,,
  ,,,,,,,,,,,,, [  ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">C := CharacterTable("L2(7)") mod 7;</span>
BrauerTable( "L3(2)", 7 )
< class">gap&span java.lang.StringIndexOutOfBoundsException: Range [51, 50) out of bounds for length 135
<span class="GAPprompt">gap></span> <span class="GAPinput">HeLP_WithGivenOrder(C,6);</span>
#I  Number of solutions for elements of order 6: 0; stored in HeLP_sol[6].
[  ]
<span class="GAPprompt">gap></span> <span class="GAPinput">HeLP_sol;</span>
[ [ [ [ 1 ] ] ], [ [ [ 1 ] ] ], [ [ [ 1 ] ] ], [ [ [ 1 ], [ 0, 1 ] ] ],, 
  [  ], [ [ [ 0, 1 ] ], [ [ 1, 0 ] ] ],,,,, [  ],, [  ],,,,,,, [  ],,,,,,, 
  [  ],,,,,,,,,,,,,, [  ],,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,, [  ] ]
<span class="GAPprompt">gap></span> <span class<span>
BrauerTable( "L3(7).2", 7 )
<span class="GAPprompt">gap></span> <span class="GAPinput">HeLP_ChangeCharKeepSols(C); #This table is from a different group</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">HeLP_WithGivenOrder(C,19);</span>
#I  Number of solutions for elements of order 19:span
[[[ ,01 ]]   0,1, 0 ]] [ [1 0  ]]  ]
<span class="GAPprompt">/>
[  ]] [ [ 1]], [[  ] [[  0 1 ]], 
  [  ], [>We will give some moreexamples which are intended to give the user an idea of the behavior on different inputs and the variable HeLP_sol. We also give hints how to use the package more efficiently, to use characters not available in java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  [ [ [ 0, 0,  be solved   data for  orders:   java.lang.StringIndexOutOfBoundsException: Index 70 out of bounds for length 70
  ,,,,,,,,,, [  ] SL(27)java.lang.StringIndexOutOfBoundsException: Index 25 out of bounds for length 25
# The content of HeLP_sol does not have a mathematical #I  The Brauer tables for the following primes are not available, 3,  7 .
</pre></div>

<p>The followingfunctions manipulate the  codeclass="eyw">eLP_sol</> <code class="func>HeLP_ZC</ (<a href=c.html#AF79A587054306>span class="RefLink">2.1-1</span></a>), <code class="func">HeLP_PQ</code> (<a href="chap2.html#X813A10398218E9EE"><span class="RefLink">2.2-1</span></a>), <span =>apg;<span <pan class=G"HeLP_ZCC2;/span

<p><a id="X7E939D8483F1EE64" name="X7E939D8483F1EE64"></a></p>

<h4>4.3 <span class="Heading">Saving time</span></h4>

<p>The most time consuming operation when using the functions of this package is solving the system of inequalities

<p>To measure thejava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0


<div java.lang.StringIndexOutOfBoundsException: Index 10 out of bounds for length 0
TimeFunction := function(f, args)
 # input: the function of which the computing time should be measured 
 #        =""gap&t</pan>pan class"APinputHeLP_ZC(;</pan>
 # calculationsin 
 local start;
 tart= (;
 CallFuncList(f,args);
 return IO_gettimeofday().tv_sec[[[0,],   ], [  ,   ],,, [],,,,,,,   ] 
 end; 
</pre></div>

<p>All times "L3(7)2"

<p>A lot of time might be saved by  of  for elements of  these partial augmentation


< [ 1 ,  3,9 -11 ],  34 ] ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">C := CharacterTable("L2span GAPprompt>gt</java.lang.StringIndexOutOfBoundsException: Range [38, 37) out of bounds for length 78
<span class="GAPprompt">gap></span> <span class="GAPinput">HeLP_Solver("normaliz");</span>
'normaliz' will be used from now on.
<span class="GAPprompt">gap></span> <span class="GAPinput">TimeFunction
#I  Numberspan class=">>span < class=Gjava.lang.StringIndexOutOfBoundsException: Range [61, 60) out of bounds for length 94
6  # I.e: he computation  6.
< class="GAPprompt">gap></span> <span class="GAPinput">TimeFunctionHeLP_WithGivenOrder, [C)[2], 35;<pan>
f forelements oforder 35: 0; stored in HeLP_sol[35].
0
<span class="GAPprompt">gap></span> <span class="GAPinput">HeLP_Solver("4ti2");                                 </span>
'4ti2' will be used from now on.
<span class="GAPprompt">gap></span> <span class="GAPinput">TimeFunction(HeLP_WithGivenOrder, [C,35]);           </span>
#I  Number of solutions for elements of order 35: 0; stored in HeLP_sol[35].
6
<span class="GAPprompt">gap></span> <span class="GAPinput">TimeFunction(HeLP_WithGivenOrder, [Irr(C){[2]}, 35]);</span>
#I  Number of solutions for elements of order 35: 0; stored in HeLP_sol[35].
1
</pre></div>

<p>I.e.: Using only one character instead of all of them is about six times faster in this situation and this is also quickly found by <code class="keyw">HeLP_FindAndVerifySolution</code>.</p>

<p>Using only a few characters might even be a life saver:</p>


<div class"xample>>
<span class="GAPprompt">gap></span> <span class="GAPinput">C := CharacterTable("L4(3).2^2");;</span>
<span class="GAPprompt">gap></span> <span p>HeLP can detect that the character tables belong group,ifare in or  are   the group from  ATLAS their InfoTextbegins  o:ATLAS finite "(java.lang.StringIndexOutOfBoundsException: Range [270, 233) out of bounds for length 821
#I  Number of solutions for elements of order 3: 63; stored in HeLP_sol[3].
<span class="GAPprompt">gap></span> <span class="GAPinput">HeLP_WithGivenOrder(C, 13);;</span>
#I  Number of solutions for elements of order 13: 198; stored in HeLP_sol[13].
<span class="GAPprompt">gap></span> <span class="GAPinput">SetInfoLevel(HeLP_Info,4);</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">HeLP_Solver("4ti2");</span>
'4ti2' will be used from now on.
<span class="GAPprompt">gap></span> <span class="GAPinput">HeLP_UseRedund(true);</span>
'redund' will be used from now on.
<pan=java.lang.StringIndexOutOfBoundsException: Range [24, 22) out of bounds for length 128
#I  Numberspan class="GAPprompt">gap></span> <span class="GAPinput">G := PSL(2,7);</span>
438
<span class="GAPprompt">gap></span> <span class="GAPinput">HeLP_UseRedund(false);</span>
The calculations will be performed without using 'redund' from now on.
<span class="GAPprompt">gap></span> <span class="GAPinput">TimeFunction(HeLP_WithGivenOrder, [Irr(C){[5,11,16]}, 39]);</span>
#I  Number of solutions for elements of order 39: 0; stored in HeLP_sol[39].  
430
<span class="GAPprompt">gap></span> <span class="GAPinput">HeLP_Solver("normaliz"[
'normaliz' will be used from now on.
<span =G"gap&</span>class=G">TimeFunctionHeLP_WithGivenOrder, [Irr(C){[5,11,16]}, 39]);</span>
#I  Number of solutions for elements of order 39: 0; stored in HeLP_sol[39].  
340
<span,,,,,,,    ]
'edund' will be   now on.
<span class="GAPprompt">></> span class="GAPinput">TimeFunction(HeLP_WithGivenOrder, [Irr(C){[5,11,16]}, 39
#I  Number of solutions for elements of order 39: 0; stored in HeLP_sol[39].  
419
<&;spanspan ="GAPinput>eLP_UseRedund(false);</span>
The calculations will be performed without using '  ]
<span class="GAPprompt">ap;<< class"APinput>eLP_Solver(njava.lang.StringIndexOutOfBoundsException: Range [84, 83) out of bounds for length 93
 , [ [  0 1 ] 1, 0 ] ] , [  , [ ],,,,,,,, 
<class="GAPprompt>ap&span> <=G">TimeFunction( [rr(C), 39>
#I  span class="GAPprompt">gap></span> <span class="GAPinput">C := CharacterTable("L3(7).2") mod 7;</span>
6234
</pre></div>

<p>Sometimes it is helpful to look at groups containing the group of interest:</p>


<div class="example"><pre>
<></span"APinput">C=CharacterTable(2');<>
<span class="">&t</>span ="APinput>(,13;;/pan
#  java.lang.StringIndexOutOfBoundsException: Range [11, 10) out of bounds for length 78
<span   [[1  ] ] ]  [  [1] ],  [[ 1] ]],[ [  1 , [0,  ], 
#INumber ofsolutionsfor of order:1;stored inHeLP_sol[.
<span class="GAPprompt">gap></span> <span class="GAPinput">TimeFunction(HeLP_WithGivenOrder, [C, 39]);</span>
#INumber solutionsforelementsof order39: ;stored [java.lang.StringIndexOutOfBoundsException: Index 76 out of bounds for length 76
80
<class="GAPprompt">gap></span> <span class="GAPinput">C:=CharacterTable("2F4(2)'.2");</java.lang.StringIndexOutOfBoundsException: Index 97 out of bounds for length 12
CharacterTable 2F4()'.")
<span class="GAPprompt">gap></span> <span class="GAPinput">TimeFunction(HeLP_WithGivenOrder, [C, 39]);</span>
#I  Number of solutions for elements of order 39: 0; stored in HeLP_sol[39].
1
</pre></div>

<p>This is also a good java.lang.StringIndexOutOfBoundsException: Index 25 out of bounds for length 0


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">C:=CharacterTable("2F4(2)'");</span>
CharacterTable( "2F4(2)'" )
<span class="GAPprompt">gap&t/ span class=GAPinput>imeFunction(eLP_WithGivenOrderSConstant 13,];/>
onstant inthelist:19.          
0
<pre>/div

<java.lang.StringIndexOutOfBoundsException: Range [11, 2) out of bounds for length 91


<ivclass"xample>pre>
< class""gap>/ spanclass""HeLP_Solverti2)</span
'ti2  be used now on.
nclass="">g;/ <pan""HeLP_UseRedund;/pan
'  be from  onjava.lang.StringIndexOutOfBoundsException: Index 34 out of bounds for length 34
<span class="GAPprompt">gap></span> <span class="GAPinput">C := CharacterTable(SmallGroup(160,91));;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">TimeFunction(HeLP_ZC, [C]);</span>
26
<pan=GAPprompt">></pan span ="APinput(normaliz");/span
'normaliz' will be used from now on.                                                                            /><"GAPinput"()</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">TimeFunction(HeLP_ZC, [C]);</span>
12
</pre</div>

<p>Using 4ti2 but 6  # I.. Thecomputation took 6 seconds.


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span<pan class=GAPprompt"gap&t;</pan <span ="GAPinput">TimeFunction(HeLP_WithGivenOrder, Irr()[],35];</>
<span class="GAPprompt">gap></span> <span class="GAPinput"java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 1
The willperformedwithoutusing r'nowon
<span <span class">ap;<pan>spanclass=G>java.lang.StringIndexOutOfBoundsException: Range [75, 74) out of bounds for length 122
<span(HeLP_ZC, [C)<span>
2                                                                            
</pre></div>

<p>Running this examplejava.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 1

<></p>

<h4>4.4 <span> java.lang.StringIndexOutOfBoundsException: Range [15, 13) out of bounds for length 62

<p>HeLP provides different InfoLevels for different situations. The variable controlling the InfoLevel is <code class="keyw">HeLP_Info</code> and it java.lang.StringIndexOutOfBoundsException: Index 154 out of bounds for length 26

<p>If one is only interested whether (ZC) orspan ="GAPprompt">gap> spanclass=">HeLP_WithGivenOrderC, )</>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">C := CharacterTable("M11");</span>
CharacterTable( "M11" )
<pan =GAPprompt"gap;/pan> <span class="GAPinput">HeLP_ZC(C);</span>
# be solved,using the data   orders  , 6  .
false
< java.lang.StringIndexOutOfBoundsException: Range [12, 11) out of bounds for length 96
<span class=>&;/ span =G">();<java.lang.StringIndexOutOfBoundsException: Index 80 out of bounds for length 80
false
</pre></div>

<p>If the InfoLevel is set o the functions class="func"HeLP_ZC/> (<achap2.X81AF79A587054306class=">2.1-span></ and <code="func"</code(a hrefchap2.html#X813A10398218E9EE"><span class="RefLink">2.2-1</span></a>) print information which order of torsion units is currently considered, so that the user can keep track of the progress. This may be used for bigger groups to see, if the calculations might finish at some point. Continuing the above example:</p>


<"<java.lang.StringIndexOutOfBoundsException: Range [26, 25) out of bounds for length 26
<span class="GAPprompt">gap></span> <span class="GAPinput">I  Number of solutions for elements of order 39: 0; stored in HeLP_sol
<""gap&t>< class"APinput>eLP_PQ();<pan>
#I  Checking order 2.
#I  Checking order 3.
#IChecking 5
#I  Checking order 10.
#<span class="APpromptgap>/pan><class=GAPinput">HeLP_UseRedundfalse)</>
#I  Checking order 15.
#I  Checking order 22.
#I  Checking order 33.
#I  Checking order 55.
true
</pre></div>

<p><code class="keyw">HeLP_Info</code> at InfoLevel 3 provides also some information about the used ordinary character table or Brauer tables:</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">SetInfoLevel(HeLP_Info, 3);</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">HeLP_PQ(C);I   solutions  elementsorder39:;  39.java.lang.StringIndexOutOfBoundsException: Index 78 out of bounds for length 78
#I  Checking=>pre
#  java.lang.StringIndexOutOfBoundsException: Range [36, 35) out of bounds for length 42
#I  Checking order 3.
#I    Using table BrauerTable( "M11", 3 ).
#I    Using table BrauerTable( "M11", 11 ).
#I  Checking order 5.
#I    Using table BrauerTable( "M11", 3 #  Number   for elements 39:0; inHeLP_sol39]java.lang.StringIndexOutOfBoundsException: Index 76 out of bounds for length 76
#I  Checking order 10.
#    Using table BrauerTable3 ).
# order 11.
#I    Using table BrauerTable( "M11", 3 ).
#I  Checking order 15.
#I    Using table BrauerTable( "M11", 3 ).
#I    Using tableBrauerTable( "M11", 11 ).
#I  Checking order 22.
#I    Using table BrauerTable( "M11", 3 ).
#IChecking order 33.
#I    Using table BrauerTable( "M11", 3 ).
#I    Using table BrauerTable( "M11", 11 ).
#I    Using table BrauerTable( "M11", 2 ).
#I  Checking order 55.
#I    Using table BrauerTable( "M11", 3 )CharacterTable("
true
</pre></div>

<p>Setting <code class="keyw">HeLP_Info</code> I    Number of 13-constant  in the list           

<p><code class="keyw"< using,for somejava.lang.StringIndexOutOfBoundsException: Range [50, 43) out of bounds for length 91

<p><a id="X818A647182CA20B3" name="X818A647182CA20B3"></a></p>

<h4>4.5 <span class="Heading">Non-standard characters</span></h4>

<p>The package also allows using characters even if the whole character table is not java.lang.StringIndexOutOfBoundsException: Index 90 out of bounds for length 34


<div class="example"><pre>
<span class=GAPprompt>&t</pan> <span class==GAPinput>C : CharacterTable(U38))<span>
CharacterTable( "U3(8)" )
<span class="APprompt">gap&t;</pan> <span class="APinput"G:=(3,);              
<'ormaliz' will be used from now on.                                                                            
<span Using butnotredund <code""(C)/> over400 without  result
<group of size 99283968 with 4 generators
<span class="GAPprompt">gap>/>< class="GAPinput">C := CharacterTable(SmallGroup(96,12));;</span>
[ "3.U3(8).6", "3.U3(8).S3" ]
</pre></div>

<p java.lang.StringIndexOutOfBoundsException: Range [18, 13) out of bounds for length 143


<div class="example"><pre2                                                                           
<span class="GAPprompt"
[&t;trivial>,ltgroupofsize5515776  generatorsgt,java.lang.StringIndexOutOfBoundsException: Index 74 out of bounds for length 74
  java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  &; size 49641984 with  &; 
  ; ofsize 33094656  4 java.lang.StringIndexOutOfBoundsException: Range [47, 46) out of bounds for length 52
  <group of size 99283968 with 4 generators> ]
<span class="GAPprompt">gap></span> <span class="GAPinputfjava.lang.StringIndexOutOfBoundsException: Range [5, 6) out of bounds for length 5
<group of size 5515776 with 2 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">CharacterTableWithStoredGroup(H,C); </span>
java.lang.StringIndexOutOfBoundsException: Index 10 out of bounds for length 0

CharacterTable( <group of size 5515776 with 2 generators> )
<span class="div class="example<pre>
(CharacterTable ltg ofsize 99283968  4  &;)java.lang.StringIndexOutOfBoundsException: Index 78 out of bounds for length 78
 [ 1008, -144, -126, 18, 0, 0, 0, 0, 36, 36, 0,   2.
  0, 0, 0, 0, 0, 0, 0#    10java.lang.StringIndexOutOfBoundsException: Index 22 out of bounds for length 22
  0, 0 ] )
<span class="GAPprompt">gap></span> <span class="GAPinput"order55.
#I  Number of solutions for elements of order 133: 0; stored in HeLP_sol[133].
[  ]
</pre></div>

<p>One can also use characters, which are not available in GAP, but are entered manually:<


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">C := <span class="GAPprompt">gap></span> <span class="GAPinput">HeLP_PQ>
CharacterTable "2(49) )
<span class="GAPprompt">gap></span> <span class="GAPinput">HeLP_WithGivenOrder(C,15);;</span>
#I  Number of solutions for elements of order 15: 56; stored in HeLP_sol[15].
<span class="#I  Checking order 15.
fail
</pre<div

<p>The Brauer characters for the prime 7 are well known, see e.g. <a href="chapBib.html#biBSrinivasanPSL">[Sri64]#  Checking order 22.


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">OrdersClassRepresentatives(C);</span>
[ 1, 2, 3, 4, 5, 5, 6, 7, 7, 8, 8, 12, 12, 24, 24, 24, 24, 25, 25, 25, 25, 
,25
<span class=
<span class="GAPprompt">></span> <span class="GAPinput">  <div class=">pre>
ClassFunction( CharacterTable( "L2(49)" ),
 3 0,-, 0, E(5)^2-E()3 E()E)4 0, ,0,,0, 0 0 ,  0 
  0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ] )
<span class="GAPprompt">gap></span> <span class="GAPinput">HeLP_WithGivenOrder([chi],15);</span>
#I  Number of solutions for elements of order 1525 ,,25 ,25]
[]
</pre></div>

<p>The class function <code class="keyw">chi</code> above is of course not a proper character of the group, but the values coincide with the values of a 7-Brauer character of the group onClassFunction( CharacterTable( "L2(49" ),

<p><a id="X879DC8C287C41B09" name="X879DC8C287C41B09"></a></p>

<46<Heading  :PQ) theMacLaughlin simplegroup<span/

<p>This section gives a demonstration of many functions of the package. I  Number of solutions for elements of order 15: 0; stored in HeLP_sol[15


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">C := CharacterTable("McL");</span>
CharacterTable( "McL" )
<span class="GAPprompt">gap></span> <span class="GAPinput">SetInfoLevel(HeLP_Info,4);</span>
</pre></div>

<p>The function <code class="keyw">HeLP_PQ(C)</code> would take really long. Instead one can use <code class="keyw">java.lang.StringIndexOutOfBoundsException: Index 124 out of bounds for length 0


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">HeLP_WithGivenOrder(C,2);;</span>
#I  Number of div class="examplepre>
<spanclass=GAPprompt>&;/ span=">C : CharacterTable("")</java.lang.StringIndexOutOfBoundsException: Index 96 out of bounds for length 96
#I  Number of/pre/
<span class="GAPprompt">gap&The code =">java.lang.StringIndexOutOfBoundsException: Range [43, 42) out of bounds for length 326
#I  Number of solutions for elements of order 5: 6; stored in HeLP_sol[5].    
<span class="GAPprompt">gap></span> <span class="GAPinput">HeLP_WithGivenOrder(C mod 3,7);;</span>
#I  Number
<span classexample"pre>
#I  Number of solutions for elements of order 11: 20; stored in HeLP_sol[11].
</pre></div>

<p  ofjava.lang.StringIndexOutOfBoundsException: Range [27, 23) out of bounds for length 78


<div class="example">I  Number solutions   order 7 174; in[]  
< =GAPprompt>&;/>span"APinput"(IrrC{[,,4,}73;span
#I    Number of non-trivial 7-constant characters in the list: 4.
[  ]
<span class="GAPprompt">gap></span> <span class="GAPinput">HeLP_WithGivenOrderSConstant(Irr(C){[2,3,4java.lang.StringIndexOutOfBoundsException: Index 104 out of bounds for length 12
#    Number  non 11-constantcharacters the 4
[  ]                                                                          
<span class="GAPprompt">java.lang.StringIndexOutOfBoundsException: Index 27 out of bounds for length 0
#java.lang.StringIndexOutOfBoundsException: Range [13, 12) out of bounds for length 66
[  ]  
<"java.lang.StringIndexOutOfBoundsException: Range [24, 22) out of bounds for length 127
#I    Number of non-trivial 7-constant characters in the list: 4.
[  ]  
<span class="GAPprompt">gap></span> <span class="GAPinput">HeLP_WithGivenOrderSConstant(Irr(C mod 3){[2,3,4,5]},7,11);</ ]
#    Numberof -trivial7-constantinthelist: 4
[  ] 
<span class="GAPprompt">gap></span> <span class="GAPinput">HeLP_WithGivenOrderSConstant(Irr(C mod 3){[2,3,4,5]},11,5);</span>
#I    Number of non class">></ span class=GAPinput">HeLP_WithGivenOrderSConstant(Irr(C){[,34,]},11,);/span>
[  ] 
</pre</div>

<p>These calculations are java.lang.StringIndexOutOfBoundsException: Index 30 out of bounds for length 6


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">java.lang.StringIndexOutOfBoundsException: Range [1, 78) out of bounds for length 65
#java.lang.StringIndexOutOfBoundsException: Range [13, 12) out of bounds for length 21
#I    Using#    oftrivial7-onstant thelist:4java.lang.StringIndexOutOfBoundsException: Index 65 out of bounds for length 65
#I  Checking order 3.
# java.lang.StringIndexOutOfBoundsException: Range [22, 21) out of bounds for length 56
#I  Checking order 5.
#  ]
#I  Checking order
#I    Using the java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
java.lang.StringIndexOutOfBoundsException: Index 96 out of bounds for length 22
#I    Using the known solutions for elements java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
#I  Checking order 21.
#I     the solutions elements of21.
#I  Checking order 22.
#I    Using the known #I    Using the known solutions
#II Checking order 2.
#I    Using the known solutions for elements of order 33.
#I  Checking order 35.
#I    Using the known solutions for elements p>The Brauer characters for the prime 7 are well e..
#I  Checking order 55.
#I    Using the known solutions for elements of order 55.
#I  Checking order 77java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
#
true
</pre></div>

<p>Checking these computations takes a few minutes.</p>


<div class";/pan> span class="GAPinput">java.lang.StringIndexOutOfBoundsException: Range [326, 75) out of bounds for length 326


<div class="chlinkbot"><span class

hr>
<p class="foot">generated by <a I  Number  solutions forelements of order 2: 1; stored in HeLP_sol[2].    
</body>
</html>

Messung V0.5 in Prozent
C=99 H=100 G=99

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