{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Load Francy Package"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "true"
      ]
     },
     "execution_count": 1,
     "metadata": {
      "text/plain": ""
     },
     "output_type": "execute_result"
    },
    {
     "data": {
      "text/plain": [
       "true"
      ]
     },
     "execution_count": 2,
     "metadata": {
      "text/plain": ""
     },
     "output_type": "execute_result"
    },
    {
     "data": {
      "text/plain": [
       "true"
      ]
     },
     "execution_count": 3,
     "metadata": {
      "text/plain": ""
     },
     "output_type": "execute_result"
    }
   ],
   "source": [
    "LoadPackage(\"francy\");\n",
    "LoadPackage(\"digraph\");\n",
    "LoadPackage(\"num\");"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### An attempt to LatticeSubgroups By Markus"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
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     },
     "execution_count": 8,
     "metadata": {
      "application/vnd.francy+json": {}
     },
     "output_type": "execute_result"
    }
   ],
   "source": [
    "DisplayLatticeSubgroups := function(L)\n",
    "    local cls, len, sz, max, rep, z, t, i, j, k, graphHasse, nodes, last;\n",
    "    cls:=ConjugacyClassesSubgroups(L);\n",
    "    len:=[];\n",
    "    sz:=[];\n",
    "    for i in cls do\n",
    "        Add(len,Size(i));\n",
    "        AddSet(sz,Size(Representative(i)));\n",
    "    od;\n",
    "\n",
    "    graphHasse := Graph(GraphType.UNDIRECTED);\n",
    "#SetShowNeighbours(graphHasse,true);\n",
    "\n",
    "    nodes := [];\n",
    "    sz:=Reversed(sz);\n",
    "    # subgroup nodes, also acccording to size\n",
    "    for i in [1..Length(cls)] do\n",
    "        nodes[i] := [];\n",
    "        for j in [1..len[i]] do\n",
    "            if len[i]=1 then\n",
    "                nodes[i][j] := Shape(ShapeType.DIAMOND, String(i));\n",
    "                SetLayer(nodes[i][j], -Size(Representative(cls[i])));\n",
    "                # AppendTo(file,\"\\\"\",i,\"x\",j,\"\\\" [label=\\\"\",i,\"\\\", shape=box];\\n\");\n",
    "            else\n",
    "                nodes[i][j] := Shape(ShapeType.CIRCLE, String(i));\n",
    "                SetLayer(nodes[i][j], -Size(Representative(cls[i])));\n",
    "                # AppendTo(file,\"\\\"\",i,\"x\",j,\"\\\" [label=\\\"\",i,\"-\",j,\"\\\", shape=circle];\\n\");\n",
    "            fi;\n",
    "            Add(graphHasse, nodes[i][j]);\n",
    "        od;\n",
    "    od;\n",
    "\n",
    "    last:=rec(o:=0,n:=0);\n",
    "    for i in [1..Length(cls)] do\n",
    "        for j in [1..len[i]] do\n",
    "            if Layer(nodes[i][j]) <> last.o then\n",
    "                last.o := Layer(nodes[i][j]);\n",
    "                last.n := last.n - 2;\n",
    "            fi;\n",
    "            SetLayer(nodes[i][j], last.n);\n",
    "        od;\n",
    "    od;\n",
    "\n",
    "    max:=MaximalSubgroupsLattice(L);\n",
    "    for i in [1..Length(cls)] do\n",
    "        for j in max[i] do\n",
    "            rep:=ClassElementLattice(cls[i],1);\n",
    "            for k in [1..len[i]] do\n",
    "                if k=1 then\n",
    "                    z:=j[2];\n",
    "                else\n",
    "                    t:=cls[i]!.normalizerTransversal[k];\n",
    "                    z:=ClassElementLattice(cls[j[1]],1); # force computation of transv.\n",
    "                    z:=cls[j[1]]!.normalizerTransversal[j[2]]*t;\n",
    "                    z:=PositionCanonical(cls[j[1]]!.normalizerTransversal,z);\n",
    "                fi;\n",
    "                Add(graphHasse, Link(nodes[i][k],nodes[j[1]][z]));\n",
    "          od;\n",
    "        od;\n",
    "      od;\n",
    "    return graphHasse;\n",
    "end;;\n",
    "L:=LatticeSubgroups(DihedralGroup(44));;\n",
    "canvas := Canvas(\"Lattice Subgroups\");;\n",
    "Add(canvas, DisplayLatticeSubgroups(L));;\n",
    "Draw(canvas);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Digraphs"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {},
   "outputs": [
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\"directed\"},\"height\" : 600,\"id\" : \"FE5415C43900B437F6460443206EE3115\",\"menus\" : {},\"messages\" : {},\"renderer\" : \"GraphViz-Renderer.dot\",\"texTypesetting\" : true,\"title\" : \"Subgroups Digraph of Group( [ f1, f2, f3 ] )\",\"width\" : 800,\"zoomToFit\" : true},\"mime\" : \"application\\/vnd.francy+json\",\"version\" : \"2.0.0\"}"
     },
     "execution_count": 38,
     "metadata": {
      "application/vnd.francy+json": {}
     },
     "output_type": "execute_result"
    }
   ],
   "source": [
    "G := DihedralGroup(8);; #SymmetricGroup(5);; #FreeGroup( \"a\", \"b\" );;\n",
    "as := AllSubgroups(G);;\n",
    "d := Digraph(as, {H, K} -> IsSubgroup(H, K));;\n",
    "\n",
    "vertices := DigraphVertices(d);;\n",
    "edges := DigraphEdges(d);;\n",
    "\n",
    "graph := Graph(GraphType.DIRECTED);;\n",
    "#SetDrag(graph, true);;\n",
    "nodes := [];;\n",
    "\n",
    "simpleGroupMessage := FrancyMessage(FrancyMessageType.INFO, \"Simple Groups\", \"A group is simple if it is nontrivial and has no nontrivial normal subgroups.\");;\n",
    "\n",
    "IsGroupSimple := function(i)\n",
    "    Add(canvas, simpleGroupMessage);;\n",
    "    if IsSimpleGroup(as[i]) then\n",
    "        Add(canvas, FrancyMessage(\"Simple\", Concatenation(\"The vertex \", String(i), \", representing the subgroup \", String(as[i]), \", is simple.\")));;\n",
    "    else\n",
    "        Add(canvas, FrancyMessage(\"Not Simple\", Concatenation(\"The vertex \", String(i), \", representing the subgroup \", String(as[i]), \", is not simple.\")));;\n",
    "    fi;;\n",
    "    return Draw(canvas);\n",
    "end;;\n",
    "    \n",
    "for i in vertices do;\n",
    "    nodes[i] := Shape(ShapeType.CIRCLE, String(i));;\n",
    "#SetColor(nodes[i], \"#0c9460\");;\n",
    "    Add(nodes[i], Menu(\"Is this subgroup simple?\", Callback(IsGroupSimple, [i])));;\n",
    "    Add(graph, nodes[i]);;\n",
    "od;;\n",
    "\n",
    "for i in edges do\n",
    "    Add(graph, Link(nodes[i[1]], nodes[i[2]]));;\n",
    "od;;\n",
    "\n",
    "canvas := Canvas(Concatenation(\"Subgroups Digraph of \", String(G)));;\n",
    "Add(canvas, graph);;\n",
    "\n",
    "graphviz := FrancyRenderer(FrancyRendererType.GRAPHVIZ_DOT);;\n",
    "Add(canvas, graphviz);;\n",
    "\n",
    "Draw(canvas);"
   ]
  },
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   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
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