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<a id="X7CDC63A27F7790AA" name="X7CDC63A27F7790AA"></a></java.lang.StringIndexOutOfBoundsException: Range [1, 61) out of bounds for length 56
<div class="ChapSects"><a href="chap4.html#X7CDC63A27F7790AA">4 Extended examples</a>
<div class="ContSect"> <a href="chap4.html#X7DEAD03D7811F9FA">4.1 The Character Table Library</a>

</div>
<div class=<pan ="GAPpromptgt; <class="GAPinput">HeLP_ChangeCharKeepSols(C); #This table belongs to the same group.
/
</div>
<div class="ContSect"> <a href="chap4.html#X7E939D8483F1EE64">4.3 
</div>
<div class="ContSect">
<
</div>
<div class="ContSect"> <a java.lang.StringIndexOutOfBoundsException: Index 82 out of bounds for length 27
<>
</div>
<div class="ContSect"> <a href="chap4.html [ 0 0, 1 ] ],[ [ 0 1 ] ], [ [[ ,0, 0 ] ]
</pan
</div>
</div>

<h3>4 <a id="X7DEAD03D7811F9FA" name="X7DEAD03D7811F9FA"></a>

<h4>4.1 The Character Table Library</h4>

<div class="example"><pre>
gap> G := SL(2,7);
SL(2,7)
gap> HeLP_ZC(G);
#I The Brauer tables for the following primes are not available: [ 2, 3, 7 ].
solved,using thegiven ,forthe [8 .
false
gap> C1 := CharacterTable(G);
CharacterTable( (2,))
gap> HeLP_ZC(C1);
# [ 2 ,]
#I (ZC) can't be solved, using the given data, for the orders: [ 8 ].
false
/ode> ""code>(a "hap2X81">java.lang.StringIndexOutOfBoundsException: Range [167, 166) out of bounds for length 1673
<spanclass=GAPprompt">ap> <pan"APinput"();<>
true
</pre></div>

Note that the first and the second call of <code class="func">HeLP_ZC</code> (<a href="chap2.html#X81AF79A587054306">2.1-1</a>) are equivalent -- the only difference being that in the first version the

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

<h4>4.2 The behavior of the variable HeLP sol</h4>

This sections demonstrates when the global variable <code

<div class="example"><pre>
gap> <java.lang.StringIndexOutOfBoundsException: Index 42 out of bounds for length 33
CharacterTable( "A5" )
g;/> <class=G">C)<pan>
true
gap& output: time needed for the inseconds
[ [ [ [ 1 ] ] ], [ [ s : IO_gettimeofday)java.lang.StringIndexOutOfBoundsException: Index 28 out of bounds for length 28
 [ [0, 1 ], [ [ 1,0 ] ] ] ],,,[ ],, [[ ,,,,,,,,,, [ ]java.lang.StringIndexOutOfBoundsException: Index 78 out of bounds for length 78
]
gap>
CharacterTable("L3(7). )
gap> HeLP_WithGivenOrderAndPA(Irr(C){[3,7,9,10]},21,[[1],[3,9,-11]]);
#INumber solutions order 21 withthese
s
[[1][ 3,,[-6, 0, 3, java.lang.StringIndexOutOfBoundsException: Index 45 out of bounds for length 45
<class="GAPprompt"gap> HeLP_sol;
[ [ [ [ 1 ] ] ] ]
</pre></div>

The function <code class="func">HeLP_WithGivenOrderAndPA</code> (<a href="chap3.html#X7CD0CEF283F13F7B"><pre>
<"GAPprompt">ap;</><panclass"APinput">HeLP_WithGivenOrder(C,3);
#I Number of solutions for # I.e.: The computation tooke. he computation took6 seconds.
[ [ [ 1 ] ] ]
gap> <spanspan( [rr(){2]] 35])</>
[ [ [ [ 1 ] ] ],, [ [ [ 1 ] ] ] ]
</pre></div>solutions of java.lang.StringIndexOutOfBoundsException: Range [48, 45) out of bounds for length 76

If HeLP detects that the table used belongs to the same group, <code class="keyw">HeLP_sol</code> is not reset:

<div class="example"><pre>
gap> HeLP_sol;
[ [ [ [ 1 ] ] ],, [ [ [ 1 ] ] ],,,,,,,,,,,,,,,,
 [ [ [ 0, 0, 1 ] ], [ [ 0, 1, 0 ] ], [ [ 1, 0, 0 ] ] ] ]
# the previously calaculated result for java.lang.StringIndexOutOfBoundsException: Index 42 out of bounds for length 0
</div=e">pre

< to thesame they identical objects GAP ifboth tablesof same from theATLAS and with "rigin of 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</code> is reset to the initial value <code class="keyw">[ [ [1] ] ]</code> representing the trivial solution for units of order 1.

< class=GAPprompt">gap> TimeFunction(HeLP_WithGivenOrder, [Irr(C){[5,11,16]}, 39]);

<div class="example"><pre>
<java.lang.StringIndexOutOfBoundsException: Range [6, 5) out of bounds for length 83
Group([ (3,7,5)(4,8,6), (1,2,6)(3,4,8) ])
gap> HeLP_ZC(G);
#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: [
false
gap> HeLP_sol;
[ [ [ [ 1 ] ] ], [ [ [ 1 ] ] ], [ [ [ 1 ] ] ], [ [ [ 1 ], [ 0, 1 ] ] ],,
 [ [ [ 1 ], [class"APprompt>gt; <span class"APinput(java.lang.StringIndexOutOfBoundsException: Range [96, 94) out of bounds for length 128
 [ ],,,,,,, [ ],,,,,,, [ ],,,,,,,,,,,,,, [ ],,,,,,,,,,,,,,,,,,,
 ,,,,,,,[ ] ]
<span class="' be usedfromgap&<pan<java.lang.StringIndexOutOfBoundsException: Range [45, 44) out of bounds for length 128
java.lang.StringIndexOutOfBoundsException: Range [118, 11) out of bounds for length 25
gap> HeLP_ChangeCharKeepSols(C); #This table belongs to the same group.
gap> HeLP_WithGivenOrder(C,6);
#I Number of solutions span class="GAPprompt">gapgt; < class"GAPinput">java.lang.StringIndexOutOfBoundsException: Range [77, 76) out of bounds for length 91
[java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4
> <pan =G">eLP_Solver"ormaliz");
[ [ [ [ 1 ] ] ], [ [ [ 1 ] ] ], [ [ [ 1 ] ] ], [ [ [ 1 ], [ 0, 1 ] ] ],,
 ][ ,1 ], [ [ ],,,, [], [,,,, [ ],,,java.lang.StringIndexOutOfBoundsException: Index 75 out of bounds for length 75
 [ ],,,,,,,,,,,,,, [ ],,,,,,,span class=">apgt;</ <pan class"APinputHeLP_WithGivenOrder,[]);gap> HeLP_ChangeCharKeepSols(C); #This table is from a different group
gap> gap> apg;span><spanclass=G">eLP_WithGivenOrderC,13);/>
gap>#Number of solutions for elements of order 13: 316; stored in HeLP_sol[13].
[ [[ 1 ] ], [ [ ] ]] [ [ [[ 1 ] [ ,1 ]]],
 [ ], [ [ [ 0, 1 ] ], [ [ 1, 0 ] ] ],,,,, [ ],, [ ],,I of elements oforder 3: in []java.lang.StringIndexOutOfBoundsException: Index 74 out of bounds for length 74
 [ [ [ 0, 0, 1 ] ], [ [ 0, 1, 0 # of for elements of order 39 0;stored in HeLP_sol[9].
 ,,,,,,,,,, [ ],,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,, [ ] ]
# The content of HeLP_sol<pan java.lang.StringIndexOutOfBoundsException: Range [13, 11) out of bounds for length 100
</pre></div>

The following functions manipulate the variable <code class="keyw">HeLP_sol</code>: ("22 java.lang.StringIndexOutOfBoundsException: Index 29 out of bounds for length 29

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

<h4>4.3 Saving time</h4>

The most time consuming operation when using the functions of this

To measure the time a function needs we use <code class="keyw">IO_gettimeofday</code> from the IO-package

<div class="example"><pre>
TimeFunctionspanGAPprompt&;< class=">imeFunctionH, [C, 13,3]
# input: the function of which onstant characters the list .
# and the list of arguments for this function<pre<div>
# p>If using 4ti2, for some groups switching redund on and off gives major improvements.
local start;
start := IO_gettimeofday(div =e"<>
span =GAPprompt>< =GAPinput>("4";/>
return IO_gettimeofday().tv_sec - start''willbe fromon
 end class"APprompt>gap&t <pan class=GAPinput>(true)</pan>
</redund'will used now .

All times will be given in seconds. The computations were perfomed on an a machine with four 2,6 GHz kernels.

A lot of time might be saved by testing only a few characters instead of a whole character table:

<div class="example"><pre>
gap> C s class=>ap;/><spanclass"">HeLP_Solver"<>
gap></pan <pan class="HeLP_Solver"normaliz";
'normaliz' will be used from now on.
gap> TimeFunction(HeLP_WithGivenOrder, [C,35]); 
#I Number of solutions for elements of order 35: 0; stored in HeLP_sol[/>div
6 I.: tookjava.lang.StringIndexOutOfBoundsException: Range [42, 41) out of bounds for length 42
<=>&t/><class C){[2]}, 35]{[},35)</pan>
#I Number of solutions for elements of order 35: 0; stored in HeLP_sol[35].
0
gap> HeLP_Solver("4ti2"); 
'4ti2' will calculationswill be performed without using 'edund from now .
<="APprompt">></>imeFunction(HeLP_WithGivenOrder, [C,35]); 
#I Number of solutions for elements of order 35: 0; stored in class="GAPprompt">gap> TimeFunction])/span
6
gap> TimeFunction(HeLP_WithGivenOrder, [Irr(C){[2]}, 35]);
#I Number of solutions for elements of order 35: 0; stored
1
</pre></div>

I.e.: Using only one character instead of all of p><a id="X8242093A82FE41FA" name="X8242093A82FE41FA"></a

<pUsingonly a few characters might even be a life saver:

<div class="example"><pre>
gap> C := CharacterTable("L4(3).2^2");;
<classgt< =GAPinput(C 3);/pan
#I Number of solutions for elements of order 3: 63; stored in HeLP_sol[3].
gap></java.lang.StringIndexOutOfBoundsException: Index 37 out of bounds for length 26
#I Number of solutions for elements of order 13: 198; stored in HeLP_sol[13].
</java.lang.StringIndexOutOfBoundsException: Range [38, 37) out of bounds for length 80
gap> SetInfoLevel(HeLP_Info, 0);
ap&t< class"APinputHeLP_ZCC)
'redund' will be used from now java.lang.StringIndexOutOfBoundsException: Range [0, 33) out of bounds for length 5
gap> TimeFunction(HeLP_WithGivenOrder, [Irr(C){[5,11,2, <code ></ode href="html#">) class>HeLP_PQ<> <="java.lang.StringIndexOutOfBoundsException: Range [215, 214) out of bounds for length 524
#I Number of solutions for elements of order 39: 0; stored in HeLP_sol[39].
438
gap> HeLP_UseRedund(false);
The calculations will be performed without using 'redund' from now on.
gap> TimeFunction(HeLP_WithGivenOrder, [Irr(C){[5,11,16]}, 39]);
#I Number of solutions for elements of order 39: 0; stored in HeLP_sol[39].
430
gap> HeLP_Solver("normaliz");
'normaliz' div class="example>pre>
gap> TimeFunction(HeLP_WithGivenOrder, [Irr(C){[5,11,16]}, 39]);
#[39].
340
gap> HeLP_UseRedund(true); 
'span class="APprompt>&; span =GAPinput">eLP_PQC);</pan>
gap> gap> HeLP_Solver("normaliz");
'normaliz' will be used from now on.
gap> </div>

Sometimes it is helpful to look at groups containing the group of interest:

<div class=example"<pre>
gap> <# Using tableBrauerTable("M11", 3 ).
gap> HeLP_WithGivenOrder(C, 13);;
#I Number of solutions for elements of order 13: 316; stored in HeLP_sol[13].
gap> HeLP_WithGivenOrder(C, 3);;
#I Number of solutions for elements of order 3: 1; stored in HeLP_sol[3].
gap> TimeFunction(HeLP_WithGivenOrder, [C, 39]);
# of solutions of order39 0 stored [.
80
gap> C:=CharacterTable("2F4(2)'.2");
CharacterTable( "2F4(2)'.2" )
gap&IUsing( "M11", 3
#I Number of solutions for elements of order 39: 0; stored#I Checking order
1
</pre></div>

This is java.lang.StringIndexOutOfBoundsException: Range [32, 29) out of bounds for length 43
 .

<div class="example"><pre>
gap> C:=java.lang.StringIndexOutOfBoundsException: Index 76 out of bounds for length 42
2F4(2)'" )
gap> TimeFunction(HeLP_WithGivenOrderSConstant, [C, 13, 3]);
# non-trivial13-characters : 19.
0
</pre></div>

<>Ifusing 4ti2 groups switching redund on and off gives major improvements.

<div class="example"><pre>
gap> HeLP_Solver("4ti2");
'4ti2' will be used from now on.
gap> gap> C := CharacterTable(SmallGroup(160,91));;

26
gap> = PSU(3,;
'java.lang.StringIndexOutOfBoundsException: Range [10, 9) out of bounds for length 112
gap> TimeFunction(HeLP_ZC, [C]);
12
</pre></div>

 4ti2 not redundcode class=keyw>HeLP_ZC</ode>ran for 400 hourswithouta.

<div class="example"><pre>
gap>C := CharacterTable(SmallGroup(96,12));;
gap> HeLP_UseRedund(false);
The
gap></span<Thismeans: The character table of the automorphism group A of PSU(3,8) is not available in GAP. However one can use induced characters:
gap> TimeFunction(HeLP_ZC, [C])java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

</pre></div>

Running this example using redund the computations does not proceed for elements of order 12.

<a id="X8242093A82FE41FA" name="X8242093A82FE41FA"></a>

<h4>4.4 Using InfoLevels</h4>

HeLP provides different InfoLevels for different situations. The variable controlling the l; group;,&;group 5515776 with 2 generators&;

If one is only interested whether (ZC) or (PQ) can be solved using the HeLP method, one can set <code class="keyw">HeLP_Info</code> to 0:

<div class="example ltgroup of size 49641984 4generators&t,
<span<roupof 33094656withgenerators>,
CharacterTable( "M11" )
gap> HeLP_ZC(C);
#I ZC can't be solved, using the given data, for the orders: [ 4, 6, 8 ].
alse
gap> SetInfoLevel(HeLP_Info, 0);
gap> HeLP_ZC(C);
false
</pre></div>

If the InfoLevel is set to 2, the functions <code class="func">HeLP_ZC</code> (<a href="chap2.html

<">pre
gap> <Character((&;roup 99283968with4generatorsgt ,
gap> HeLP_PQ(C);
IChecking order 2.
#I Checking order 3.
#I Checking order 5.
# Checkingorder10.
#I Checking order 11.
#I Checking order 15.
#I Checking order 22.
#I Checking order 33.
#I Checking order 55..
true
</pre></div>

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

<div class="example"><pre>
gap> <span java.lang.StringIndexOutOfBoundsException: Index 48 out of bounds for length 0
<C);</span
#I Checking order 2.
#I Using("249"java.lang.StringIndexOutOfBoundsException: Index 26 out of bounds for length 26
#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 ).
#I Checking order 10.
#I Using table BrauerTable( "M11", 3 ).
#I Checking order 11.
#I Using table BrauerTable( "M11", 3 ).
java.lang.StringIndexOutOfBoundsException: Range [13, 12) out of bounds for length 22
#I Using table>/>
#I Using table BrauerTable( "M11", 11 ).
#Ijava.lang.StringIndexOutOfBoundsException: Range [13, 12) out of bounds for length 22
#I Using table BrauerTable( "M11", 3 ).
#I Checking 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 ).
true
</pre></div>

Setting <code class="keyw">want changestrategy.

<code class="keyw">HeLP_Info</code> at level 5 informs the user on all changes of the used character table. Using it makes sense, if you work with the command <code class="func">HeLP_ChangeCharKeepSols</code> (<a href="chap3.html#X7BB9009482784E90">3.4-1</a>).

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

<h4>4.5 Non-standardp>package alsoallows java.lang.StringIndexOutOfBoundsException: Range [44, 43) out of bounds for length 124

The package also allows using characters span java.lang.StringIndexOutOfBoundsException: Range [12, 11) out of bounds for length 98

<div span class=G">> :=3,) <pan

CharacterTable( "U3(8)" )
gap> G := PSU(3,8); 
<permutation group of size 5515776 with 2 generators>
gap> A := AutomorphismGroup(G);
<group of size 99283968 with 4 generators>
gap> AllCharacterTableNames(Size, "3U3(8)6, 3U(8."]
[".(6,"3.()S"]
</pre></div>

< group A of PSU(3)is in GAP one can use java.lang.StringIndexOutOfBoundsException: Range [140, 138) out of bounds for length 143

<div class="example"><pre>
<spanclass=G
[ <trivial group>, <group of size 5515776 with 2 generators>,
 <group with 3 generators>, <group of size 16547328 with 3 generators>,
 <group of size 49641984 with 4 generators>s="GAPinput">NN := NormalSubgroups(A);
 <group of size 33094656 with 4 generators>,
 & of 99283968 with4 generators> ]
gap> H := NN[2]; #Subgroup of A isomorphic to G </java.lang.StringIndexOutOfBoundsException: Range [0, 129) out of bounds for length 52
<group of size 5515776 with 2 generators>
gap> CharacterTableWithStoredGroup(H,C); 
CharacterTable( <group of size 5515776 with 2 generators> )
gap> D := CharacterTable(H); 
ble(<t;group of size 5515776 with 2 generators> )
gap> chi := InducedClassFunction(Irr(D)[2],A);
Character( CharacterTable( <group of size 99283968 with 4 generators> ),
[1008,, -,-126 18,0, 0 0, 0, 6, 36,0,0 0, 0, 0,0 0, 0,0,0,0
 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -18, 0, 0,
 0, 0 ] )
gap> HeLP_WithGivenOrder([chi],7*19);
#I Number of solutions for elements of order 133: 0; stored in HeLP_sol[133].
[ ]
</pre></div>

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

<spanI 2java.lang.StringIndexOutOfBoundsException: Index 21 out of bounds for length 21
Ijava.lang.StringIndexOutOfBoundsException: Range [15, 11) out of bounds for length 56
gap> <span classIjava.lang.StringIndexOutOfBoundsException: Range [36, 11) out of bounds for length 56
#I Number of I Using the known java.lang.StringIndexOutOfBoundsException: Range [32, 31) out of bounds for length 57
gap>Ijava.lang.StringIndexOutOfBoundsException: Range [15, 11) out of bounds for length 57
fail
</pre></div>

< known, seeg

<div class="example"><pre>
gap>I 55
[ 1, 2, 3, 4, 5, 5, 6, 7, 7, 8, 8, 12, 12, 24, 24, 24, 24, 25, 25, 25, 25,
 25, 25, 25, 25, 25, 25 ]
gap> <span class="java.lang.StringIndexOutOfBoundsException: Index 59 out of bounds for length 22
<pan theknown 77.
ClassFunction( CharacterTable( "L2(49)" ),
[ 3, 0, -1, 0, -E(5)^2-E(5)^3, -E(
 ,0 , 0, ,,0, 0 ,0 ])
gap> HeLP_WithGivenOrder([chi],15);
#I Number of solutions for java.lang.StringIndexOutOfBoundsException: Index 31 out of bounds for length 4
[ ]
</pre></div>

<java.lang.StringIndexOutOfBoundsException: Range [93, 2) out of bounds for length 534

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

<h4>4.6 A complete example: (PQ) for the MacLaughlin simple group</h4>

This section gives a demonstration of many functions java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0

<div class="example"><pre>
gap> C := CharacterTable("McL");
CharacterTable( "McL" )
</><java.lang.StringIndexOutOfBoundsException: Range [62, 44) out of bounds for length 95
</pre></div>

The function <code class="keyw">HeLP_PQ(C)</code> would take really long. Instead one can use <code class="keyw">HeLP_AllOrdersPQ(C)</code> several times on a high InfoLevel. Any time you see the function needs long, just try some manual calculations. Compute first the partial augmentations of elements of prime order:

<div class="example"><pre>

#of java.lang.StringIndexOutOfBoundsException: Range [37, 36) out of bounds for length 78
gap> HeLP_WithGivenOrder(C mod 3,5);;
#I Number of solutions for elements of order 5: 6; stored in HeLP_sol[5].
gap> HeLP_WithGivenOrder(C mod 3,7);;
#I Number of solutions for elements of order 7: 174; stored in HeLP_sol[7].
gap> HeLP_WithGivenOrder(C mod 3,11);;
#I Number of solutions for elements of order 11: 20; stored in HeLP_sol[11].
</pre></div>

For mixed order in most situations p-constant characters are interesting. Check the tables for such characters of small degree.

<div class="example"><pre>
gap> HeLP_WithGivenOrderSConstant(Irr(C){[2,3,4,5]},7,3);
#I Number of non-trivial 7-constant characters in the list: 4.
[ ]
gap> HeLP_WithGivenOrderSConstant(Irr(C){[2,3,4,5]},11,2);
#I Number of non-trivial 11-constant characters in the list: 4.
[ ]
gap> HeLP_WithGivenOrderSConstant(Irr(C){[2,3,4,5]},11,3);
#I Number of non-trivial 11-constant characters in the list: 4.
[ ]
gap> HeLP_WithGivenOrderSConstant(Irr(C mod 3){[2,3,4,5]},7,5);
#I Number of non-trivial 7-constant characters in the list: 4.
[ ]
gap> HeLP_WithGivenOrderSConstant(Irr(C mod 3){[2,3,4,5]},7,11);
#I Number of non-trivial 7-constant characters in the list: 4.
[ ]
gap> HeLP_WithGivenOrderSConstant(Irr(C mod 3){[2,3,4,5]},11,5);
#I Number of non-trivial 11-constant characters in the list: 2.
[ ]
</pre></div>

These calculations are enough to obtain an affirmative answer to the Prime Graph Question:

<div class="example"><pre>
gap> HeLP_AllOrdersPQ(C);
#I Checking order 2.
#I Using the known solutions for elements of order 2.
#I Checking order 3.
#I Using the known solutions for elements of order 3.
#I Checking order 5.
#I Using the known solutions for elements of order 5.
#I Checking order 7.
#I Using the known solutions for elements of order 7.
#I Checking order 11.
#I Using the known solutions for elements of order 11.
#I Checking order 21.
#I Using the known solutions for elements of order 21.
#I Checking order 22.
#I Using the known solutions for elements of order 22.
#I Checking order 33.
#I Using the known solutions for elements of order 33.
#I Checking order 35.
#I Using the known solutions for elements of order 35.
#I Checking order 55.
#I Using the known solutions for elements of order 55.
#I Checking order 77.
#I Using the known solutions for elements of order 77.
true
</pre></div>

Checking these computations takes a few minutes.

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