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\title{ Association scheme $@$rMxMQ$7$?BP>N72$K$*$1$k(J
			    Normalizer $@$N7W;;(J }
\author{ $@5\(J \quad $@K\(J \quad\quad $@@t(J\\
$@;3M|Bg3X9)3XIt(J }
\date{}
\begin{document}
\maketitle
\section{ association scheme $@$NDj5A(J}
$X=\{x_1,x_2,\cdots , x_n \}$$@$O=89g!"(J $|X|=n$ $@$H$9$k!#(J
$R_i \subset X\times X$, $0 \leq i \leq d$ $@$G$"$C$F!"(J
$$X\times X=R_0\cup R_1\cup\cdots\cup R_d,\qquad
\mbox{disjoint union}$$
$$R_0 = \{(x,x)\; | \; x \in X\; \} \qquad\qquad\qquad\qquad $$
$@$G$"$k$H$9$k!#(J$n\times n$$@9TNs(J$A_i$, $0 \leq i \leq d$ $@$r!"(J
$$A_i\mbox{$@$N(J}(j,k)\mbox{$@@.J,(J} = \left\{ \begin{array}{cc}
1\qquad&(x_j,x_k)\in R_i\mbox{$@$N$H$-(J}\\
0\qquad&\mbox{$@$=$&$G$J$$$H$-(J}
\end{array}\right.$$
$R_i$$@$r(J$i$$@HVL\$N(Jincidence$@4X78(J,
$A_i$$@$r(J$i$$@HVL\$N(Jadjacency$@9TNs(J
$@$H$$$&!#(J
$(X, \{R_i\}_{0\leq i\leq d})$$@$,(J association scheme $@$G$"$k$H$O!"(J
$@G$0U$N(J$A_i$$@$NE>CV9TNs$,$I$l$+$N(J$A_{i'}$$@$K$J$C$F$$$F!"(J
$A_0, A_1, \cdots , A_d$ $@$,!"$=$l$i$N@8@.$9$k(Jalgebra$@$N4pDl$K$J$C$F(J
$@$$$k$3$H$G$"$k!#$D$^$j!"G$0U$N(J$i,j$$@$K$D$$$F(J
$$A_iA_j=\sum_{k=0}^{d}p_{i,j}^kA_k$$
$@$,@.N)$9$k$3$H$G$"$k!#(J

$@Nc!'(J$G$$@$r(J$X$$@>e$N(Jtransitive$@$JCV4972$H$9$k!#(J
$G$$@$N(J$X\times X$$@$N(Jorbit$@$NA4BN$r!"(J$R_0, R_1, \cdots , R_d$
$@$H$9$l$P!"(Jassociation scheme$@$,$G$-$k!#(J

\section{ relation matrix $@$NDj5A(J}

$(X, \{R_i\}_{0\leq i\leq d})$$@$r(J association scheme $@$H$9$k!#(J
$A_0,\; A_1, \cdots , A_d$ $@$r$=$N(J adjacency matrix $@$H$9$k!#(J
$$A:=0A_0+1A_1+2A_2+\cdot \cdot \cdot +dA_d$$
$@$r(J association scheme $@$N(J {\it relation matrix} $@$H$$$&!#(J

$@0J2<!"(Jrelation matrix $@$,(J $A$ $@$H$J$k(J association scheme $@$r(J
$@4JC1$K(J association scheme $A$ $@$H$b$$$&$3$H$K$9$k!#(J


\section{ association scheme $@$NF17?@-(J}

association scheme $@$r=q$-I=$9$H$-!"(Jincidence $R_i$ $@$NHV9f$NIU$1(J
$@J}!"$D$^$j!"(Jadjacency matrix $A_1,\; A_2, \cdots , A_d$
 $@$NHV9fIU$1$H!"D:E@=89g(J
$X=\{x_1,\; x_2, \cdots , x_n\}$ $@$NHV9f$NIU$1J}$K<+M3EY$,$"$k!#(J
$@$3$N#2$D$NHV9fIU$1$rE,Ev$KJQ$($F(J association scheme $A$ $@$H(J $B$
$@$,0lCW$9$k$H$-!"$3$l$i$rF17?$H$$$&!#(J

$\{1,\;  2, \cdots , d\}$ $@>e$NCV49(J $g$ $@$H(J $@CV499TNs(J $P$ $@$rE,Ev$K(J
$@A*$Y$P(J
$$P^{-1}A^gP=B$$
$@$H$J$k$3$H$G$"$k!#$3$3$G(J $A^g$ $@$O(J $A$ $@$N3F@.J,$NCM(J ( incidence
$@HV9f(J) $i$ $@$r(J $i^g$ $@$KJQ$($?9TNs$G$"$k!#(J

$@Nc!'(J
$$A=\left(
\begin{array}{ccccc}
0&1&2&2&1\\
1&0&1&2&2\\
2&1&0&1&2\\
2&2&1&0&1\\
1&2&2&1&0
\end{array}\right),\quad\quad
B=\left(
\begin{array}{ccccc}
0&2&1&1&2\\
2&0&2&1&1\\
1&2&0&2&1\\
1&1&2&0&2\\
2&1&1&2&0
\end{array}\right)$$
$@F17?<LA|$=$N#1(J
$$g_1=(1,2),\qquad P_1=\mbox{$@C10L9TNs(J}\qquad\qquad$$
$@F17?<LA|$=$N#2(J
$$g_2=\mbox{$@91EyCV49(J}\quad
P_2=\left(
\begin{array}{ccccc}
1&0&0&0&0\\
0&0&1&0&0\\
0&1&0&0&0\\
0&0&0&0&1\\
0&0&0&1&0
\end{array}\right)
\begin{array}{c}
\mbox{$@D:E@HV9f$NCV49$H$7$F$O(J }\\
(2,3)(4,5) \end{array}$$

\section{ assocation scheme $@$N(J automorphism}

$P^{-1}AP=A$ $@$H$J$kCV499TNs(J $P$ $@$^$?$O$=$l$rM?$($kD:E@HV9f$NCV49$r(J
assocation scheme $@$N(J {\it automorphism} $@$H$$$&!#(J
$A$ $@$N(J automorphism$@A4BN$N$J$972$r(J $Aut(A)$ $@$GI=$9!#(J

\section{ Association scheme $@$rMxMQ$7$?BP>N72$K$*$1$k(Jnormalizer$@$N7W;;(J}

\hspace*{1cm} {\Large transitive $@$JCV4972$N>l9g(J }

$G$ $@$O(J $\Omega$$@>e$N(J transitive $@$JCV4972$H$9$k!#(J
$A$ $@$O(J $G$ $@$r(J $\Omega\times\Omega$ $@$K:nMQ$5$;$F!"$=$N(J$orbits$$@$+$i(J
association scheme $@$r:n$k!#(J
$A$$@$r!"$=$N(J relation matrix $@$H$9$k!#(J%$A$$@$K$O(Jincidence$@HV9f$NIU$1J}$N(J
%$@<+M3EY$,$"$k!#(J\\
$x$ $@$OBP>N72(J$S^{\Omega}$$@$N85$G(J$G$$@$r(Jnormalize$@$7$F$$$k$H$9$k!#(J
$O$$@$r(J$G$$@$N(J$\Omega\times\Omega$$@>e$N0l$D$N(Jorbit$@$H$9$k$H!"(J
$$(O^x)^G=(O^G)^x=O^x$$
$@$h$j(J$O^x$$@$b(J$G$$@$N(Jorbit$@$H$J$k$N$G!"(J$x$$@$O(J$G$$@$N(J$\Omega\times\Omega$$@$N(J$orbits$$@$N(J
$@CV49$r0z$-5/$3$9!#$7$?$,$C$F!"(J
$x$$@$G(J$A$$@$ND:E@HV9f$rCV49$7$F$b!"(J$A$$@$N(Jincidence$@HV9f$rJQ$($?9TNs$K$7$+(J
$@$J$i$J$$!#$^$?!"(J$Aut(A)$$@$O(J$G$$@$N(J$\Omega\times\Omega$$@$N(J$orbits$$@$NA4$F$r8GDj(J
$@$7$F$$$F!"(Jorbits$@$K:nMQ$9$k(J
$@85A4BN$N$N$J$+$G(Jkernel$@$K$J$k$N$G!"(J$x$$@$O(J$Aut(A)$$@$r(Jnormalize$@$9$k!#(J

$@$=$3$G!"(J$G$$@$N(J$S^{\Omega}$$@$K$*$1$k(Jnormalizer$@$r7W;;$9$k$N$K!"(J
$@$^$:!"(J association scheme $@$NF17?H=Dj$N%W%m%0%i%`$rMxMQ$7$F(J
$Aut(A)$$@$N(J$S^{\Omega}$$@$K$*$1$k(Jnormalizer $N(Aut(A))$$@$r7W;;$9$k!#(J
$@<!$K!"(J$N(Aut(A))$$@$K$*$1$k(J$G$$@$N(Jnormalizer$@$r(J {\sf GAP}$@$N%3%^%s%I(J
Normalizer$@$r;H$C$F7W;;$9$k$H$$$&J}K!$r$H$k!#(J

$@$3$l$K$h$C$F!"<!?t(J$|\Omega|$$@$KHf$Y$F0L?t$NBg$-$$(J
$@BP>N72$K$*$1$k(Jnormalizer$@$N7W;;$,!"$=$s$J$K0L?t$NBg$-$/$J$$(J
$N(Aut(A))$$@$K$*$1$k(Jnormalizer$@$N7W;;$G:Q$`$3$H$K$J$k!#(J\\
($@Cm(J) doubly transitive $@$N$H$-$O>e$G=R$Y$?;v$OL50UL#!#(J\vspace{1mm}
\par
\hspace*{1cm} {\large intransitive $@$JCV4972$N>l9g(J}\\

orbit$@Kh$K(Jassociation scheme$@$r:n$k!#(J
association scheme$@$NF17?H=Dj%W%m%0%i%`$G8_$$$KF17?$J(J
association scheme$@$r7W;;$7$FF17?$rM?$($kCV49$r0l$D5a$a$k!#(J
$@$=$l$>$l$N(Jorbit$@$N(Jassociation scheme$@$N(Jnormalizer$@$r(J
$@A0$N(Jtransitive$@$J>l9gJ}$NK!$G7W;;$9$k!#(J
$@N>J}$NCV49$G@8@.$5$l$k72$NCf$GM?$($i$l$?72$N(Jnormalizer$@$r(J{\sf GAP}$@$N%3%^%s%I(J
Normalizer$@$r;H$C$F7W;;$9$k!#(J

\section{ $@<B837k2L(J}

$@#2#2<!$N(J transitive$@$JCV4972#5#98D$r;H$C$F(J
{\sf GAP} $@$N(J Normalizer$@%3%^%s%I$H>e$G=R$Y$?J}K!$N(J
{\sf GAP} $@$N%W%m%0%i%`$H$rHf3S$7$?!#7W;;;~4V$r(J
Runtime() $@$G(J 600,000$@%_%jIC(J($@#1#0J,(J)$@$GBG$A@Z$C$F9T$C$?!#(J\\
\begin{tabular}{lr}
$@#2$D$NJ}K!6&7W;;2DG=(J &         $@#1#58D(J\\
$@>e$G=R$Y$?J}K!$N$_7W;;2DG=(J&    $@#2#48D(J\\
$@#2$D$NJ}K!6&7W;;IT2DG=(J&        $@#2#08D(J
\end{tabular}
\vspace{1mm}\\
Normalizer$@$G7W;;;~4V#1ICDxEY$^$G$N$H$-$O>e$G=R$Y$?J}K!$O(J
$@#5!A#1#5G\$N7W;;;~4V$,$+$+$C$?!#5U$K!"(JNormalizer$@$G$O7W;;$G$-$J$+$C$?$,(J
$@>e$NJ}K!$G7W;;$G$-$?$H$-$O!"$[$H$s$I$N>l9g?tIC$G7W;;$,=*$C$?!#D9$$>l9g(J
$@$G$b#1J,DxEY$@$C$?!#(J\vspace{2mm}\\
$@>e$NNc$N(J1$@E@8GDjItJ,72#5#78D(J(regular$@$J>l9g$O=|$/!#=EJ#$bM-$jF@$k(J)
$@$N#2#1<!BP>N72$NCf$G$N>l9g$NF1MM$J7W;;<B83$N7k2L!#(J\\
\begin{tabular}{lr}
$@#2$D$NJ}K!6&7W;;2DG=(J &         $@#2#68D(J\\
$@>e$G=R$Y$?J}K!$N$_7W;;2DG=(J&    $@#1#98D(J\\
$@#2$D$NJ}K!6&7W;;IT2DG=(J&        $@#1#28D(J
\end{tabular}

$@#2#0<!BP>N72$NCf$G!"(Jtransitive$@$J#2#1<!$NCV4972$N#1E@8GDjItJ,72$N>l9g!#(J\\
\begin{tabular}{lr}
$@#2$D$NJ}K!6&7W;;2DG=(J &         $@#9#48D(J\\
$@>e$G=R$Y$?J}K!$N$_7W;;2DG=(J&    $@#6#38D(J\\
$@#2$D$NJ}K!6&7W;;IT2DG=(J&          $@#58D(J
\end{tabular}
\vspace{2mm}\\

($@Cm(J) MAGMA$@$N(Jnormalizer$@7W;;$O0[MM$KB.$$>l9g$,$"$k!#(J
$A_5$$@$N#6#0<!$N(Jregular$@$JI=8=(J
$@$G$b?tIC$G#6#0<!$NBP>N72$NCf$G$N(Jnormalizer$@$r7W;;$7$F$7$^$&!#(J
$@#2#2<!$N(Jregular$@$J(Jcyclic$@72$H(Jdihedral$@72$O(J{\sf GAP}$@$N(JNormalizer$@$G$b(J
$@$3$3$K=R$Y$?J}K!$G$b7W;;$G$-$J$+$C$?$,!"(JMAGMA$@$O0l=V$G7W;;$7$F$7$^$C$?!#(J
$@$7$+$7!"(Jtransitive$@$G$O$J$$72$N$H$-$O5^$KCY$/$J$C$F(J{\sf GAP}$@$H$"$^$j(J
$@JQ$o$i$J$$MM$K;W$($?!#(J
$@#2#2<!$NCf$G!"0L?t#2#1$N=d2s72$N>l9g$G$b7W;;$G$-$J$+$C$?!#(J
$@#2#2<!$N(Jtransitive$@$J72A4$F$N(JMAGMA$@$G$N(Jnormalizer$@7W;;<B83$N7k2L$G!"(J
{\sf GAP}$@$KHf$Y$FHs>o$KB.$$>l9g$O(J
$@$*$*$`$ML>A0$N$D$$$F$$$k72$G$"$k$h$&$K;W$($?!#(J
$@0lHLE*$J>l9g$N(JMAGMA$@$H(J{\sf GAP}$@$N0c$$$O%W%m%0%i%`$N7ABV$N0c$$$,(J
$@K\<AE*$JItJ,$G$"$k$H;W$&!#(J

($@Cm(J) $@7+$jJV$7$F7W;;$7$?$H$-!"$=$N7W;;;~4V$OF10l$K$J$i$J$$!#(J
$@$=$l$O!"(Jstrong generator $@$J$I$r$D$/$k$H$-$K(J random element $@$r(J
$@;H$C$F$$$k$?$a$@$m$&$H;W$&!#(J
\vspace{2mm}\\

$@#2#2<!$N(J transitive$@$JCV4972#5#98D$N0lItJ,$N7W;;Hf3SI=(J\\
{\sf GAP}$@$H(JAssociation Scheme$@$N1~MQ$O(JSS20$@DxEY$N(Jwork station$@$G(J
$@7W;;;~4V(J3000$@IC$^$G$GBG@Z$j!"(JMAGMA$@$O(JSS5$@$N(Jwork station$@$G$N7W;;7k2L!#(J

\begin{tabular}{rrccc}
$@HV9f(J&$@0L?t(J&{\sf GAP}&Assoc.Scheme&MAGMA\\
1&22&$\times$&$\times$&1$@IC0J2<(J\\
2&22&$\times$&$\times$&1$@IC0J2<(J\\
13&1320&$\times$&13$@IC(J&336$@IC(J\\
14&1320&1221$@IC(J&5$@IC(J&2432$@IC(J\\
27&15840&$\times$&26$@IC(J&910$@IC(J\\
29&22528&1937$@IC(J&5$@IC(J&1$@IC0J2<(J\\
44&$\cdots$&370$@IC(J&285$@IC(J&4516$@IC(J\\
48&$\cdots$&$\times$&$\times$&25873$@IC(J\\
59&22$!$&1.9$@IC(J&$-\!\! -$&0.7$@IC(J
\end{tabular}


\end{document}
