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{\Large\textbf{Coloured Plane Groups}}
\footnote{0567-7394/80/060884-05\$01.00\quad \copyright 1980 International Union of Crystallography} \end{center} \begin{center} \textsc{By J.D. Jarratt} \\ \textit{Pure Mathematics Department, University of Sydney, Sydney, Australia}\\ \ \\ \textsc{and R.R.E. Schwarzenberger} \ \\ \textit{Science Education Department, University of Warwick, Coventry, England} \\ \ \\ \textit{(Received 9 August 1979; accepted 4 June 1980)} \end{center} \section*{Abstract} The 46 black and white plane groups are well known. The corresponding colour groups with more than two colours are extremely numerous. We give a listing of the 935 groups with$N$colours for$N$lying between 2 and 15 inclusive. \section{Introduction} Consider an$n$-dimensional space group$G$whose elements permute$N$colours transitively and let$G_1$be the subgroup keeping the first colour fixed. Then the index of$G_1$in$G$is$N$and the colours correspond naturally to the cosets. The effect of any member of$G$under group multiplication on the cosets is the same as its effect on the colours. For these reasons a \textit{coloured space group with$N$colours} is defined to be a pair$G\supset
G_1$consisting of a space group$G$and a subgroup$G_1$which is also a space group, of index$N$. The pairs$G\supset
G_1$and$G'\supset G_1'$are equivalent if there is an isomorphism between$G$and$G'$which maps$G_1$onto$G_1'$. This implies that there is actually an affine transformation$f$such that$G' = fGf^{-1}$and$G_1' =
fG_1f^{-1}$. If$N = 2$then$G\supset G_1$is called a \textit{black and white group}. These definitions go back to Heesch and Shubnikov; they can be found (in slightly different form) in the paper of van der Waerden \& Burckhardt (1961). There are two well known books by Shubnikov \& Belov (1964) and by Loeb (1971) which discuss the case$N = 2$in detail and which include coloured pictures iliustrating various coloured place groups. Both give complete descriptions of the 46 black and white groups but for$N = 2$the listings begun in these books are far from complete. In this paper we describe a method for obtaining a complete listing for any given value of$N$and give explicit results for all$N$up to 15. For further remarks on the significance of coloured groups in the enumeration of space, groups and in the study of twinning we refer to Schwarzenberger (1980). Other recent work on coloured space groups by Senechal (1975) and Harker (1976) has led to the development of arithmetic algorithms for the determination of coloured space groups. Senschal (1979) uses such an algorithm to count coloured plane groups for various values of$N$. When$N$is prime she shows that the number is 14, 15, 13, 16 when$N = 5$, 7, 11, 1 modulo 12 in agreement with our computations; when$N$is composite there were some discrepancies between our preliminary results even for$N = 4$. Meanwhile, Wieting (1980) has developed two quite different methods of computation: one using generators and relations for$N\le 5$, and the other using pairs of plane ornamental groups for$N\le 60$. Comparison of the preliminary results both of Senechal and of ourselves with the results of Wieting made us aware of several errors which have been corrected in the present version. We are grateful to Senechal and Wieting for their generous cooperation but accept sole responsibility for any errors which remain. \section{The Hermann decomposition} Let$G\supset G_1$be a coloured group and$T\supset T_1$the corresponding pair of lattices. These groups yield point groups$H = G/T$and$H_1 = G_1/T_1$with homomorphisms $\begin{matrix} 0 & \rightarrow & T & \rightarrow & G & \overset{\rightarrow}{p} &H&\rightarrow&1\\ & & \uparrow & & \uparrow & & \uparrow \\ 0 & \rightarrow &T_1& \rightarrow &G_1&\rightarrow&H_1&\rightarrow&1\\ \end{matrix}$ where vertical arrows denote embeddings of subgroups. Following Hermann (1929) we call the coloured group$G\supset G_1$\textit{lattice equivalent} if$T = T_1$and \textit{class equivalent} if$H = H_1$. The main result, which is due to Hermann and holds for arbitrary dimension$n$, is: \textit{Theorem.} Any coloured group$G\supset G_1$can be expressed uniquely as the composition of a lattice equivalent coloured group$G\supset G'$and a class equivalent coloured group$G'\supset G_1$. \textit{Proof.} A subgroup$G'$of$G$satisfies the required conditions if and only if it has lattice$T$and$p(G') = H_1$. There is one and only one subgroup with these properties, namely$G' = p^{-1}(H_1)$. The pair$G\supset G'$is then lattice equivalent of index$r$and the pair$G'\supset G_1$is class equivalent of index$s$where$r\times s = N$. \textit{Remark.} If two coloured groups are equivalent then so are their lattice equivalent and class equivalent parts. The converse is not true (see \S 4). Table 1 lists the numbers of distinct coloured plane groups corresponding to various factorizations$N = r \times
s$for$n = 2$. An indication of the method used to obtain these results is given in \S\S 3 and 4. \section{The lattice equivalent and class equivalent cases} The lattice equivalent coloured groups$G\supset G'$are finite in number for given dimension$n$. To obtain the list for$n = 2$it is sufficient to consider the possible pairs$H\supset
H_1$of point groups. For completeness, and for use in \S 4, we list the 52 lattice equivalent coloured plane groups explicitly in Table 2. The class equivalent coloured groups$G'\supset G_1$depend on the possible pairs$T\supset T_1$of lattices. We consider these according to the Bravais type of$T$; the number which occur is infinite but is finite for given$s$. The groups$G'\supset G_1$which occur depend on the choice of integers$p$,$q$; in the tabulations which follow, the symbol for$G_1$is placed below the symbol for$G'$to indicate existence of the corresponding coloured group$G'\supset G_1$. \begin{center} Table 1. \textit{Number of coloured plane groups with}\\ \textit{$N = r\times s$colours for$N = 2,\dots,15$} \end{center} $\begin{array}{rlccclr} & \text{Lattice} & & & & \text{Class} & \\ N&\text{equivalent}&&\text{Other}&&\text{equivalent}&\text{Total}\\ 2&(2\times1)\ \ 26&&&&(1\times2)\ \ 20&46 \\ 3&(3\times1)\ \ \ 5&&&&(1\times3)\ \ 18&23 \\ 4&(4\times1)\ \ 12&&(2\times2)\ 46&&(1\times4)\ \ 38&96 \\ 5&&&&&(1\times5)\ \ 14&14 \\ 6&(6\times1)\ \ \ 6&(3\times2)\ 11&&(2\times3)\ \ 44&(1\times6)\ \ 29&90 \\ 7&&&&&(1\times7)\ \ 15&15 \\ 8&(8\times1)\ \ \ 2&(4\times2)\ 26&&(2\times4)\ \ 98&(1\times8)\ \ 44&170 \\ 9&&&(3\times3)\ 10&&(1\times9)\ \ 30&40 \\ 10&&&&(2\times5)\ 45&(1\times10)\ 30&75 \\ 11&&&&&(1\times11)\ 13&13 \\ 12&(12\times1)\ \ 1&(6\times2)\ \ 9&&(2\times6)\ 98&(1\times12)\ 58&221 \\ & &(4\times3)\ 25&&(3\times4)\ 30&& \\ 13&&&&&(1\times13)\ 16&16 \\ 14&&&&(2\times7)\ 53&(1\times14)\ 29&82 \\ 15&&&&(3\times5)\ 10&(1\times15)\ 24&34 \\ \ \\ \text{Total}&\multicolumn{1}{c}{52}&&\multicolumn{1}{c}{505}&& \multicolumn{1}{c}{378}&935 \end{array}$ \begin{center} Table 2. \textit{The 52 lattice equivalent groups$G\supset G'$} \end{center} {\tiny $\begin{array}{lllllllllll} r=2 &G &P2 &Pm &Pg &Cm &Pmm&Pmg &Pgg &Cmm &P4 \\ &G'&P1 &P1 &P1 &P1 &P2 &P2 &P2 &P2 &P2 \\ &G &Pmm &Pmg &Pmg &Pgg &P4mm&P4gm&Cmm &P4mm&P4gm\\ &G'&Pm &Pm &Pg &Pg &Pmm &Pgg &Cm &Cmm &Cmm \\ &G &P4mm&P4gm&P31m&P3m1&P6 &P6mm&P6mm&P6mm& \\ &G'&P4 &P4 &P3 &P3 &P3 &P31m&P3m1&P6 & \\ r=3 &G &P3 &P6 &P31m&P3m1&P6mm& & & & \\ &G'&P1 &P2 &Cm &Cm &Cmm & & & & \\ r=4 &G &Pmm &Pmg &Pgg &Cmm &P4 &P4mm&P4gm& & \\ &G'&P1 &P1 &P1 &P1 &P1 &P2 &P2 & & \\ &G &P4mm&P4gm&P4mm&P4gm&P6mm& & & & \\ &G'&Pm &Pg &Cm &Cm &P3 & & & \\ r=6 &G &P31m&P3m1&P6 &P6mm&P6mm&P6mm& & & \\ &G'&P1 &P1 &P1 &P2 &Cm1 &C1m & & & \\ r=8,\ 12&G &P4mm&P4gm&P6mm& & & & & & \\ &G'&P1 &P1 &P1 & & & & & & \end{array}$ } \renewcommand{\thesubsection}{(\roman{subsection})} \subsection{\textit{$T = P$parallelogram}} Each sublattice$P_1$is determined by the highest common factor$d = \text{h.c.f.}(p,q)$of a factorization$s =
pq$. For$s\le 15$the possible values of$d$are 1 (for all$s$), 2 (for$s = 4$, 8, 12) and 3 (for$s = 9). Each value of a gives two coloured groups: \begin{align*} G' &= P1\ \ \ \ P2 \\ G_1 & =P_11\ \ \ P_12 \end{align*} \subsection{\textit{T = P$rectangle (primitive orthogonal)}} Each primitive sublattice$P_1$compatible with reflections is determined by a factorization$s = pq$and generators$(p,0)$,$(0q)$with respect to orthogonal coordinates. If$p \ne q$there are eight coloured groups whereas if$p = q$there are five because of the equivalences marked$\sim$: {\scriptsize $\hspace{-1mm}\begin{array}{rllllllllllll} &p,q\ \text{odd}&Pm1&&P1m&Pg1&&P1g&Pgm&&Pgm&Pmm&Pmg \\ G'=&p,q\ \text{even}&Pm1&&P1m&Pm1&&P1m&Pmm&&Pmm&Pmm&Pmm\\ &p-q\ \text{odd}&Pm1&&P1m&Pm1&&P1g&Pmm&&Pmg&Pmm&Pmg\\ G_1=&&P_1m&\sim&P_1m&P_1g&\sim&P_1g&P_1gm&\sim&P_1mg&P_1mm&P_1gg \end{array}$ } Similarly each centred sublattice$C_1$compatible with reflections is determined by a factorization$s = 2pq$and generators$(2p,0)$,$(0,2q)$,$(p,q)$. If$p \ne q$there are three coloured groups reducing to two if$p = q: \begin{align*} G' &= Pm1\ \ \ \,P1m\ \ Pmm \\ G_1 &= C_1m \sim C_1m\ \ C_1mm \end{align*} \subsection{\textit{T = C$diamond (centred orthogonal)}} Each primitive sublattice$P_1$compatible with relections is determined by a factorization$s = 2pq$and generators$(p,0)$,$(0,q)$. If$p \ne q$there are eight coloured groups reducing to five if$p = q: \begin{align*} G' &= Cm1\ \ \ C1m\ \ Cm1\ \ C1m\ \ Cmm\ \ \ Cmm\ Cmm\ Cmm \\ G_1 &= P_1m \sim P_1m\ \ P_1g \sim P_1g\ \ \ P_1mg \sim P_1gm\ P_1mm\ P_1gg. \end{align*} Similarly each centred subiatticeC_1$compatible with reflections is determined by a factorization$s = pq$where$p$,$q$have the same parity and generators$(p,0)$.$(0,q)$,$(\frac{1}{2}p,\frac{1}{2}q)$. If$p \ne q$there are three coloured groups reducing to two if$p = q: \begin{align*} G' &= Cm1\ \ \ \,C1m\ \ Cmm \\ G_1 &= C_1m \sim C_1m\ \ C_1mm \end{align*} \subsection{\textit{T = P$square}} The possible sublattices invariant under rotations of order 4 are \medskip \noindent$P_1$with generators$(p,0)$,$(0,p)$and$s = p^2$\medskip \noindent$P_1$with generators$(2p,0)$,$(0,2p)$,$(p,p)$and$s = 2p^2$\medskip \noindent$P_1$with generators$(p,q)$,$(-q,p)$and$s=p^2+q^2$,$p\ne q$\medskip \noindent of which only the first two are invariant also under reflections. In the range$2 \le s \le 15$the coloured groups which arise are: $\begin{array}{rcllll} G' &=&P4 & P4mm & P4mm & P4gm \\ G_1&=&P_14 & P_14mm & P_14gm & P_14gm \\ s &=&2,4,5,8, & 2,4,8,9& 2,4,8 & 9 \\ & &9,10,13 \end{array}$ \subsection{\textit{$T = P$hexagonal}} The possible sublattices invariant under rotations of order 3 or 6 are (with generators now expressed relative to inclined axes). Finally, counting up the various ways of expressing$s = 2$,\dots, 15 in terms of$p$,$q$we obtain the 378 class equivalent coloured plane groups enumerated in Table 3. \medskip \noindent$P_1$with generators$(p,0)$,$(0,p)$and$s = p^2$\medskip \noindent$P_1$with generators$(3p,0)$,$(0,3p)$,$(p,p)$and$s = 3p^2$\medskip \noindent$P_1$with generators$(p,q)$,$(-q,p+q)$and$s=p^2+pq+q^2$,$p\ne q$\medskip \noindent of which only the first two are invariant under reflections. In the range$2 \le s \le 15$the coloured groups which arise are: $\begin{array}{rclllllll} G'&=&P3 &P31m &P31m &P3m1 &P3m1 &P6 &P6mm \\ G &=&P_13 &P_131m&P_13m1&P_13m1&P_131m&P_16 &P_16mm \\ s &=&3,4,7,9,&4,9 &3,12 &4,9 &3,12 &3,4,7,9,&3,4,9,12 \\ & &12,13 & & & & &12,13 & \end{array}$ \begin{center} Table 3. \textit{Number of class equivalent groups$G'\supset G_1$with$s$colours arranged according to Bravais type} \end{center} {\tiny $\begin{array}{llcrrrrrrrrrrrrrr} &G &T_1&s=2& 3& 4& 5& 6& 7& 8& 9&10&11&12&13&14&15\\ \ \\ (i)&P1^* & & 1& 1& 2& 1& 1& 1& 2& 2& 1& 1& 2& 1& 1& 1\\ &P2^* & & 1& 1& 2& 1& 1& 1& 2& 2& 1& 1& 2& 1& 1& 1\\ (ii)&Pm &P_1& 3& 2& 5& 2& 6& 2& 7& 3& 6& 2&10& 2& 6& 4\\ & &C_1& 1& -& 2& -& 2& -& 3& -& 2& -& 4& -& 2& -\\ &Pmm^* &P_1& 2& 1& 5& 1& 4& 1& 6& 2& 4& 1& 8& 1& 4& 2\\ & &C_1& 1& -& 1& -& 1& -& 2& -& 1& -& 2& -& 1& -\\ &Pg & & 1& 2& 1& 2& 2& 2& 1& 3& 2& 2& 2& 2& 2& 4\\ &Pmg & & 2& 2& 2& 2& 4& 2& 2& 3& 4& 2& 4& 2& 4& 4\\ &Pgg & & -& 1& -& 1& -& 1& -& 2& -& 1& -& 1& -& 2\\ (iii)&Cm &P_1& 2& -& 4& -& 4& -& 6& -& 4& -& 8& -& 4& -\\ & &C_1& -& 2& 1& 2& -& 2& 2& 3& -& 2& 2& 2& -& 4\\ &Cmm^* &P_1& 3& -& 4& -& 4& -& 7& -& 4& -& 8& -& 4& -\\ & &C_1& -& 1& 1& 1& -& 1& 1& 2& -& 1& 1& 1& -& 2\\ (iv)&P4 & & 1& -& 1& 1& -& -& 1& 1& 1& -& -& 1& -& -\\ &P4mm & & 2& -& 2& -& -& -& 2& 1& -& -& -& -& -& -\\ &P4gm & & -& -& -& -& -& -& -& 1& -& -& -& -& -& -\\ (v)&P3^* & & -& 1& 1& -& -& 1& -& 1& -& -& 1& 1& -& -\\ &P6 & & -& 1& 1& -& -& 1& -& 1& -& -& 1& 1& -& -\\ &P31m & & -& 1& 1& -& -& -& -& 1& -& -& 1& -& -& -\\ &P3m1^*& & -& 1& 1& -& -& -& -& 1& -& -& 1& -& -& -\\ &P6mm & & -& 1& 1& -& -& -& -& 1& -& -& 1& -& -& -\\ \text{Total} &&& 20&18&38&14&29&15&44&30&30&13&58&16&29&24 \end{array}$ \begin{center}$^*$Consult Table 4 before using this information in conjunction with Table 2. \end{center} } \section{The mixed case} Previous sections have dealt with all coloured plane groups arising from factorizations$N = r \times s$with$r = 1$(class equivalent,$s \le 15$) or$s = 1$(lattice equivalent). Unless$N$is prime there are further coloured groups to be obtained from a careful comparison of Table 2 and Table 3. Thus, for each of the 52 lattice equivalent coloured groups$G\supset G'$listed in Table 2, we determine from Table 3 the number of possible class equivalent groups$G'\supset G_1$so as to obtain the total number of compositions$G \supset G' \supset G_1$). There is one difficulty: it may be necessary to distinguish between subgroups$G_1$of$G'$which, although equivalent in$G'$, are not equivalent in$G$(that is, the coloured groups$G'\supset G_1$are equivalent although the compositions$G\supset G_1$are not). An example may clarify this phenomenon. Consider the lattice equivalent coloured group$P6\supset P2$of index 3 from Table 2 and the unique class equivalent coloured group$P2\supset P_12$of index 2 from Table 3. The group P6 defines centres of rotation of order six (6 centres) and, between these, centres of rotation of order two (2 centres). Which of these occur as 2 centres for the group$P_12$? Either a mixture of 6 centres and 2 centres of$P6$or else only 2 centres of P6, giving two distinct coloured groups$P6\supset P_12$. Note that it is not possible for the 2 centres of$P_12$to consist only of 6 centres of$P6$(although this can happen when$P2\supset P_12$is of index 4 instead of index 2). In this way we obtain the entry 2 in the column of Table 4 for$s = 2$. Other entries of Table 4 are obtained by a similar argument. By using Tables 2, 3 and 4 together we obtain the figures in the middle column of Table 1. For any of the seventeen plane groups G we may, using Tables 2, 3 and 4, find the total number of coloured groups$G\supset G_1$of index$N = 2$,\dots, 15. The results are listed in Table 5. \begin{center} Table 4. \textit{Coloured groups$G'\supset G_1$giving inequivalent compositions$G\supset G'\supset G_1$for$s=2$,\dots, 7} \end{center} {\small $\begin{array}{lcrclrrrrrr} \multicolumn{1}{c}{G}&r&G'&\supset&G_1&s=2&3&4&5&6&7\\ \ \\ Pm,Pg,Pmg&2,4&&&&3&3&6&4&9&5\\ Pmm,Pgg,Cmm,&4,8&&&&2&2&4&3&5&3\\ P4,P4mm,P4gm&&&&&&&&&&\\ Cm&2&P1&\supset&P_11&2&3&5&4&7&5\\ P3,P31m,P3m1,&3,6,12&&&&1&2&3&2&3&2\\ P6,P6mm&&&&&\\ \ \\ Pmm,Pmg &2 & & & &2&2&4&3&5&3\\ Pmg &2 & & & &3&3&6&4&9&5\\ Cmm,P4,P4mm,P4gm &2,4 &P2 &\supset&P_12 &3&2&6&3&7&3\\ P6,P6mm &3,6 & & & &2&2&7&2&6&2\\ & &Pmm &\supset&P_1mm &1&1&3&1&2&1\\ P4mm &2 &Pmm &\supset&P_1mg &1&0&3&0&2&0\\ & &Pmm &\supset&P_1gg &0&0&2&0&0&0\\ & &Pmm &\supset&C_1mm &2&0&3&0&3&0\\ \ \\ & &Cmm &\supset&P_1mm &1&0&2&0&2&0\\ P6mm &3 &Cmm &\supset&P_1mg &2&0&4&0&4&0\\ & &Cmm &\supset&P_1gg &1&0&2&0&2&0\\ & &Cmm &\supset&C_1mm &0&2&2&2&0&2\\ P6,P31m,P6mm &2,4 &P3 &\supset&P_13 &0&2&2&0&0&2\\ P6mm &2 &P3m1&\supset&P_131m &0&2&0&0&0&0\\ & &P3m1&\supset&P_13m1 &0&0&2&0&0&0 \end{array}$ } \begin{center} Table 5. \textit{Number of coloured groups$G\supset G_1$with$N$colours corresponding to each plane group$G$for$N=2$,\dots, 15} \end{center} $\begin{array}{lrrrrrrrrrrrrrr} \ \ \ G &N=2& 3& 4& 5& 6& 7& 8& 9&10&11&12&13&14&15\\ \ \\ P1 & 1& 1& 2& 1& 1& 1& 2& 2& 1& 1& 2& 1& 1& 1\\ P2 & 2& 1& 3& 1& 2& 1& 4& 2& 2& 1& 3& 1& 2& 1\\ Pm & 5& 2&10& 2&11& 2&16& 3&12& 2&23& 2&13& 4\\ Pmm & 5& 1&13& 1& 9& 1&21& 2&10& 1&25& 1&10& 2\\ Pg & 2& 2& 4& 2& 5& 2& 7& 3& 6& 2&11& 2& 7& 4\\ Pmg & 5& 2&11& 2&11& 2&19& 3&12& 2&26& 2&13& 4\\ Pgg & 2& 1& 4& 1& 4& 1& 7& 2& 5& 1& 9& 1& 5& 2\\ Cm & 3& 2& 7& 2& 7& 2&13& 3& 8& 2&17& 2& 9& 4\\ Cmm & 5& 1&11& 1& 8& 1&21& 2& 9& 1&22& 1& 9& 2\\ P4 & 2& 0& 5& 1& 2& 0& 9& 1& 4& 0& 9& 1& 3& 0\\ P4mm & 5& 0&13& 0& 2& 0&29& 1& 3& 0&17& 0& 2& 0\\ P4gm & 3& 0& 7& 0& 2& 0&13& 1& 3& 0&10& 0& 2& 0\\ P3 & 0& 2& 1& 0& 1& 1& 0& 3& 0& 0& 4& 1& 0& 2\\ P6 & 1& 2& 1& 0& 5& 1& 2& 3& 0& 0& 9& 1& 2& 2\\ P31m & 1& 2& 1& 0& 5& 0& 2& 3& 0& 0& 7& 0& 2& 2\\ P3m1 & 1& 2& 1& 0& 4& 0& 1& 3& 0& 0& 7& 0& 1& 2\\ P6mm & 3& 2& 2& 0&11& 0& 4& 3& 0& 0&20& 0& 1& 2\\ \ \\ \text{Total}&46&23&96&14&90&15&170&40&75&13&221&16&82&34 \end{array}$ \section*{References} % Set up for reference list \newcommand{\hi}{\par\noindent\hangindent=1em} \hi \textsc{Barker, D.} (1976). \textit{Acta Cryst.}\ \textbf{A32}, 133--139. \hi \textsc{Hermann, C.} (1929). \textit{Z.\ Kristallogr.}\ \textbf{69}, 533. \hi \textsc{Loeb, A. 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\textsc{Senechal, M.} (1975). \textit{Z.\ Kristallogr.}\ \textbf{142},
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\textsc{Senechal, M.} (1979). \textit{Discrete Appl.\ Math.}\ \
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\textsc{Shubnikov, A. V. \& Belov}, N. V. (1964), \textit{Colored
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\textsc{Waerden, B. L. van der \& Burckhardt, J. J.} (1965).
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\textsc{Wieting, T.} (1980). To be published.

\bigskip\bigskip

\noindent From: \textit{Acta Cryst.} (1980) \textbf{A36}, 884--888
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