Polytope of Type {18}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {18}*36
Also Known As : 18-gon, {18}. if this polytope has another name.
Group : SmallGroup(36,4)
Rank : 2
Schlafli Type : {18}
Number of vertices, edges, etc : 18, 18
Order of s0s1 : 18
Special Properties :
   Universal
   Spherical
   Locally Spherical
   Orientable
   Self-Dual
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {18,2} of size 72
   {18,4} of size 144
   {18,4} of size 144
   {18,4} of size 144
   {18,6} of size 216
   {18,6} of size 216
   {18,8} of size 288
   {18,4} of size 288
   {18,9} of size 324
   {18,6} of size 324
   {18,6} of size 324
   {18,3} of size 324
   {18,6} of size 324
   {18,10} of size 360
   {18,12} of size 432
   {18,12} of size 432
   {18,12} of size 432
   {18,14} of size 504
   {18,16} of size 576
   {18,4} of size 576
   {18,8} of size 576
   {18,4} of size 576
   {18,8} of size 576
   {18,8} of size 576
   {18,4} of size 648
   {18,18} of size 648
   {18,18} of size 648
   {18,18} of size 648
   {18,6} of size 648
   {18,6} of size 648
   {18,6} of size 648
   {18,6} of size 648
   {18,6} of size 648
   {18,6} of size 648
   {18,6} of size 648
   {18,6} of size 648
   {18,6} of size 648
   {18,20} of size 720
   {18,20} of size 720
   {18,22} of size 792
   {18,24} of size 864
   {18,24} of size 864
   {18,6} of size 864
   {18,12} of size 864
   {18,12} of size 864
   {18,10} of size 900
   {18,26} of size 936
   {18,9} of size 972
   {18,18} of size 972
   {18,3} of size 972
   {18,18} of size 972
   {18,6} of size 972
   {18,9} of size 972
   {18,6} of size 972
   {18,9} of size 972
   {18,9} of size 972
   {18,18} of size 972
   {18,18} of size 972
   {18,9} of size 972
   {18,18} of size 972
   {18,27} of size 972
   {18,6} of size 972
   {18,9} of size 972
   {18,9} of size 972
   {18,9} of size 972
   {18,18} of size 972
   {18,18} of size 972
   {18,9} of size 972
   {18,18} of size 972
   {18,18} of size 972
   {18,6} of size 972
   {18,9} of size 972
   {18,3} of size 972
   {18,6} of size 972
   {18,28} of size 1008
   {18,28} of size 1008
   {18,3} of size 1008
   {18,3} of size 1008
   {18,7} of size 1008
   {18,7} of size 1008
   {18,7} of size 1008
   {18,7} of size 1008
   {18,7} of size 1008
   {18,7} of size 1008
   {18,9} of size 1008
   {18,9} of size 1008
   {18,9} of size 1008
   {18,9} of size 1008
   {18,30} of size 1080
   {18,30} of size 1080
   {18,32} of size 1152
   {18,8} of size 1152
   {18,8} of size 1152
   {18,8} of size 1152
   {18,4} of size 1152
   {18,8} of size 1152
   {18,8} of size 1152
   {18,8} of size 1152
   {18,8} of size 1152
   {18,4} of size 1152
   {18,34} of size 1224
   {18,36} of size 1296
   {18,36} of size 1296
   {18,12} of size 1296
   {18,12} of size 1296
   {18,12} of size 1296
   {18,12} of size 1296
   {18,36} of size 1296
   {18,12} of size 1296
   {18,12} of size 1296
   {18,12} of size 1296
   {18,12} of size 1296
   {18,9} of size 1296
   {18,36} of size 1296
   {18,3} of size 1296
   {18,12} of size 1296
   {18,12} of size 1296
   {18,12} of size 1296
   {18,4} of size 1296
   {18,12} of size 1296
   {18,4} of size 1296
   {18,3} of size 1296
   {18,4} of size 1296
   {18,4} of size 1296
   {18,9} of size 1296
   {18,9} of size 1296
   {18,12} of size 1296
   {18,12} of size 1296
   {18,12} of size 1296
   {18,12} of size 1296
   {18,38} of size 1368
   {18,40} of size 1440
   {18,20} of size 1440
   {18,42} of size 1512
   {18,42} of size 1512
   {18,44} of size 1584
   {18,44} of size 1584
   {18,45} of size 1620
   {18,30} of size 1620
   {18,30} of size 1620
   {18,15} of size 1620
   {18,30} of size 1620
   {18,46} of size 1656
   {18,48} of size 1728
   {18,48} of size 1728
   {18,24} of size 1728
   {18,12} of size 1728
   {18,6} of size 1728
   {18,12} of size 1728
   {18,12} of size 1728
   {18,24} of size 1728
   {18,24} of size 1728
   {18,24} of size 1728
   {18,24} of size 1728
   {18,12} of size 1728
   {18,6} of size 1728
   {18,6} of size 1728
   {18,14} of size 1764
   {18,50} of size 1800
   {18,10} of size 1800
   {18,10} of size 1800
   {18,52} of size 1872
   {18,52} of size 1872
   {18,4} of size 1944
   {18,12} of size 1944
   {18,12} of size 1944
   {18,4} of size 1944
   {18,12} of size 1944
   {18,12} of size 1944
   {18,4} of size 1944
   {18,12} of size 1944
   {18,12} of size 1944
   {18,18} of size 1944
   {18,18} of size 1944
   {18,18} of size 1944
   {18,6} of size 1944
   {18,6} of size 1944
   {18,18} of size 1944
   {18,6} of size 1944
   {18,6} of size 1944
   {18,18} of size 1944
   {18,18} of size 1944
   {18,6} of size 1944
   {18,6} of size 1944
   {18,18} of size 1944
   {18,18} of size 1944
   {18,18} of size 1944
   {18,18} of size 1944
   {18,18} of size 1944
   {18,18} of size 1944
   {18,18} of size 1944
   {18,18} of size 1944
   {18,18} of size 1944
   {18,54} of size 1944
   {18,54} of size 1944
   {18,6} of size 1944
   {18,6} of size 1944
   {18,18} of size 1944
   {18,18} of size 1944
   {18,18} of size 1944
   {18,18} of size 1944
   {18,18} of size 1944
   {18,18} of size 1944
   {18,18} of size 1944
   {18,18} of size 1944
   {18,18} of size 1944
   {18,18} of size 1944
   {18,18} of size 1944
   {18,18} of size 1944
   {18,6} of size 1944
   {18,6} of size 1944
   {18,18} of size 1944
   {18,18} of size 1944
   {18,6} of size 1944
   {18,6} of size 1944
   {18,12} of size 1944
   {18,12} of size 1944
   {18,12} of size 1944
   {18,18} of size 1944
   {18,18} of size 1944
   {18,18} of size 1944
   {18,6} of size 1944
   {18,6} of size 1944
   {18,6} of size 1944
   {18,6} of size 1944
   {18,6} of size 1944
   {18,6} of size 1944
   {18,6} of size 1944
   {18,6} of size 1944
   {18,6} of size 1944
Vertex Figure Of :
   {2,18} of size 72
   {4,18} of size 144
   {4,18} of size 144
   {4,18} of size 144
   {6,18} of size 216
   {6,18} of size 216
   {8,18} of size 288
   {4,18} of size 288
   {9,18} of size 324
   {6,18} of size 324
   {6,18} of size 324
   {3,18} of size 324
   {6,18} of size 324
   {10,18} of size 360
   {12,18} of size 432
   {12,18} of size 432
   {12,18} of size 432
   {14,18} of size 504
   {16,18} of size 576
   {4,18} of size 576
   {8,18} of size 576
   {4,18} of size 576
   {8,18} of size 576
   {8,18} of size 576
   {4,18} of size 648
   {18,18} of size 648
   {18,18} of size 648
   {18,18} of size 648
   {6,18} of size 648
   {6,18} of size 648
   {6,18} of size 648
   {6,18} of size 648
   {6,18} of size 648
   {6,18} of size 648
   {6,18} of size 648
   {6,18} of size 648
   {6,18} of size 648
   {20,18} of size 720
   {20,18} of size 720
   {22,18} of size 792
   {24,18} of size 864
   {24,18} of size 864
   {6,18} of size 864
   {12,18} of size 864
   {12,18} of size 864
   {10,18} of size 900
   {26,18} of size 936
   {9,18} of size 972
   {18,18} of size 972
   {3,18} of size 972
   {18,18} of size 972
   {6,18} of size 972
   {9,18} of size 972
   {6,18} of size 972
   {9,18} of size 972
   {9,18} of size 972
   {18,18} of size 972
   {18,18} of size 972
   {9,18} of size 972
   {18,18} of size 972
   {27,18} of size 972
   {6,18} of size 972
   {9,18} of size 972
   {9,18} of size 972
   {9,18} of size 972
   {18,18} of size 972
   {18,18} of size 972
   {9,18} of size 972
   {18,18} of size 972
   {18,18} of size 972
   {6,18} of size 972
   {9,18} of size 972
   {3,18} of size 972
   {6,18} of size 972
   {28,18} of size 1008
   {28,18} of size 1008
   {3,18} of size 1008
   {3,18} of size 1008
   {7,18} of size 1008
   {7,18} of size 1008
   {7,18} of size 1008
   {7,18} of size 1008
   {7,18} of size 1008
   {7,18} of size 1008
   {9,18} of size 1008
   {9,18} of size 1008
   {9,18} of size 1008
   {9,18} of size 1008
   {30,18} of size 1080
   {30,18} of size 1080
   {32,18} of size 1152
   {8,18} of size 1152
   {8,18} of size 1152
   {8,18} of size 1152
   {4,18} of size 1152
   {8,18} of size 1152
   {8,18} of size 1152
   {8,18} of size 1152
   {8,18} of size 1152
   {4,18} of size 1152
   {34,18} of size 1224
   {36,18} of size 1296
   {36,18} of size 1296
   {12,18} of size 1296
   {12,18} of size 1296
   {12,18} of size 1296
   {12,18} of size 1296
   {36,18} of size 1296
   {12,18} of size 1296
   {12,18} of size 1296
   {12,18} of size 1296
   {12,18} of size 1296
   {9,18} of size 1296
   {36,18} of size 1296
   {3,18} of size 1296
   {12,18} of size 1296
   {12,18} of size 1296
   {12,18} of size 1296
   {4,18} of size 1296
   {12,18} of size 1296
   {4,18} of size 1296
   {3,18} of size 1296
   {4,18} of size 1296
   {4,18} of size 1296
   {9,18} of size 1296
   {9,18} of size 1296
   {12,18} of size 1296
   {12,18} of size 1296
   {12,18} of size 1296
   {12,18} of size 1296
   {38,18} of size 1368
   {40,18} of size 1440
   {20,18} of size 1440
   {42,18} of size 1512
   {42,18} of size 1512
   {44,18} of size 1584
   {44,18} of size 1584
   {45,18} of size 1620
   {30,18} of size 1620
   {30,18} of size 1620
   {15,18} of size 1620
   {30,18} of size 1620
   {46,18} of size 1656
   {48,18} of size 1728
   {48,18} of size 1728
   {24,18} of size 1728
   {12,18} of size 1728
   {6,18} of size 1728
   {12,18} of size 1728
   {12,18} of size 1728
   {24,18} of size 1728
   {24,18} of size 1728
   {24,18} of size 1728
   {24,18} of size 1728
   {12,18} of size 1728
   {6,18} of size 1728
   {6,18} of size 1728
   {14,18} of size 1764
   {50,18} of size 1800
   {10,18} of size 1800
   {10,18} of size 1800
   {52,18} of size 1872
   {52,18} of size 1872
   {4,18} of size 1944
   {12,18} of size 1944
   {12,18} of size 1944
   {4,18} of size 1944
   {12,18} of size 1944
   {12,18} of size 1944
   {4,18} of size 1944
   {12,18} of size 1944
   {12,18} of size 1944
   {18,18} of size 1944
   {18,18} of size 1944
   {18,18} of size 1944
   {6,18} of size 1944
   {6,18} of size 1944
   {18,18} of size 1944
   {6,18} of size 1944
   {6,18} of size 1944
   {18,18} of size 1944
   {18,18} of size 1944
   {6,18} of size 1944
   {6,18} of size 1944
   {18,18} of size 1944
   {18,18} of size 1944
   {18,18} of size 1944
   {18,18} of size 1944
   {18,18} of size 1944
   {18,18} of size 1944
   {18,18} of size 1944
   {18,18} of size 1944
   {18,18} of size 1944
   {54,18} of size 1944
   {54,18} of size 1944
   {6,18} of size 1944
   {6,18} of size 1944
   {18,18} of size 1944
   {18,18} of size 1944
   {18,18} of size 1944
   {18,18} of size 1944
   {18,18} of size 1944
   {18,18} of size 1944
   {18,18} of size 1944
   {18,18} of size 1944
   {18,18} of size 1944
   {18,18} of size 1944
   {18,18} of size 1944
   {18,18} of size 1944
   {6,18} of size 1944
   {6,18} of size 1944
   {18,18} of size 1944
   {18,18} of size 1944
   {6,18} of size 1944
   {6,18} of size 1944
   {12,18} of size 1944
   {12,18} of size 1944
   {12,18} of size 1944
   {18,18} of size 1944
   {18,18} of size 1944
   {18,18} of size 1944
   {6,18} of size 1944
   {6,18} of size 1944
   {6,18} of size 1944
   {6,18} of size 1944
   {6,18} of size 1944
   {6,18} of size 1944
   {6,18} of size 1944
   {6,18} of size 1944
   {6,18} of size 1944
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {9}*18
   3-fold quotients : {6}*12
   6-fold quotients : {3}*6
   9-fold quotients : {2}*4
Covers (Minimal Covers in Boldface) :
   2-fold covers : {36}*72
   3-fold covers : {54}*108
   4-fold covers : {72}*144
   5-fold covers : {90}*180
   6-fold covers : {108}*216
   7-fold covers : {126}*252
   8-fold covers : {144}*288
   9-fold covers : {162}*324
   10-fold covers : {180}*360
   11-fold covers : {198}*396
   12-fold covers : {216}*432
   13-fold covers : {234}*468
   14-fold covers : {252}*504
   15-fold covers : {270}*540
   16-fold covers : {288}*576
   17-fold covers : {306}*612
   18-fold covers : {324}*648
   19-fold covers : {342}*684
   20-fold covers : {360}*720
   21-fold covers : {378}*756
   22-fold covers : {396}*792
   23-fold covers : {414}*828
   24-fold covers : {432}*864
   25-fold covers : {450}*900
   26-fold covers : {468}*936
   27-fold covers : {486}*972
   28-fold covers : {504}*1008
   29-fold covers : {522}*1044
   30-fold covers : {540}*1080
   31-fold covers : {558}*1116
   32-fold covers : {576}*1152
   33-fold covers : {594}*1188
   34-fold covers : {612}*1224
   35-fold covers : {630}*1260
   36-fold covers : {648}*1296
   37-fold covers : {666}*1332
   38-fold covers : {684}*1368
   39-fold covers : {702}*1404
   40-fold covers : {720}*1440
   41-fold covers : {738}*1476
   42-fold covers : {756}*1512
   43-fold covers : {774}*1548
   44-fold covers : {792}*1584
   45-fold covers : {810}*1620
   46-fold covers : {828}*1656
   47-fold covers : {846}*1692
   48-fold covers : {864}*1728
   49-fold covers : {882}*1764
   50-fold covers : {900}*1800
   51-fold covers : {918}*1836
   52-fold covers : {936}*1872
   53-fold covers : {954}*1908
   54-fold covers : {972}*1944
   55-fold covers : {990}*1980
Permutation Representation (GAP) :
s0 := ( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18);;
s1 := ( 1, 5)( 2, 3)( 4, 9)( 6, 7)( 8,13)(10,11)(12,17)(14,15)(16,18);;
poly := Group([s0,s1]);;
 
Finitely Presented Group Representation (GAP) :
F := FreeGroup("s0","s1");;
s0 := F.1;;  s1 := F.2;;  
rels := [ s0*s0, s1*s1, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 ];;
poly := F / rels;;
 
Permutation Representation (Magma) :
s0 := Sym(18)!( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18);
s1 := Sym(18)!( 1, 5)( 2, 3)( 4, 9)( 6, 7)( 8,13)(10,11)(12,17)(14,15)(16,18);
poly := sub<Sym(18)|s0,s1>;
 
Finitely Presented Group Representation (Magma) :
poly<s0,s1> := Group< s0,s1 | s0*s0, s1*s1, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 >; 
 
References : None.
to this polytope