Polytope of Type {2,9}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,9}*36
if this polytope has a name.
Group : SmallGroup(36,4)
Rank : 3
Schlafli Type : {2,9}
Number of vertices, edges, etc : 2, 9, 9
Order of s₀s₁s₂ : 18
Order of s₀s₁s₂s₁ : 2
Special Properties :
   Degenerate
   Universal
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {2,9,2} of size 72
   {2,9,4} of size 144
   {2,9,6} of size 216
   {2,9,4} of size 288
   {2,9,8} of size 576
   {2,9,18} of size 648
   {2,9,6} of size 648
   {2,9,6} of size 648
   {2,9,6} of size 648
   {2,9,6} of size 648
   {2,9,6} of size 864
   {2,9,12} of size 864
   {2,9,3} of size 1008
   {2,9,7} of size 1008
   {2,9,7} of size 1008
   {2,9,7} of size 1008
   {2,9,9} of size 1008
   {2,9,9} of size 1008
   {2,9,8} of size 1152
   {2,9,3} of size 1296
   {2,9,4} of size 1296
   {2,9,9} of size 1296
   {2,9,9} of size 1296
   {2,9,12} of size 1296
   {2,9,12} of size 1296
   {2,9,12} of size 1728
   {2,9,24} of size 1728
   {2,9,10} of size 1800
   {2,9,18} of size 1944
   {2,9,6} of size 1944
   {2,9,6} of size 1944
   {2,9,18} of size 1944
   {2,9,6} of size 1944
   {2,9,18} of size 1944
   {2,9,18} of size 1944
   {2,9,18} of size 1944
   {2,9,6} of size 1944
   {2,9,18} of size 1944
   {2,9,18} of size 1944
   {2,9,18} of size 1944
   {2,9,18} of size 1944
   {2,9,6} of size 1944
   {2,9,18} of size 1944
Vertex Figure Of :
   {2,2,9} of size 72
   {3,2,9} of size 108
   {4,2,9} of size 144
   {5,2,9} of size 180
   {6,2,9} of size 216
   {7,2,9} of size 252
   {8,2,9} of size 288
   {9,2,9} of size 324
   {10,2,9} of size 360
   {11,2,9} of size 396
   {12,2,9} of size 432
   {13,2,9} of size 468
   {14,2,9} of size 504
   {15,2,9} of size 540
   {16,2,9} of size 576
   {17,2,9} of size 612
   {18,2,9} of size 648
   {19,2,9} of size 684
   {20,2,9} of size 720
   {21,2,9} of size 756
   {22,2,9} of size 792
   {23,2,9} of size 828
   {24,2,9} of size 864
   {25,2,9} of size 900
   {26,2,9} of size 936
   {27,2,9} of size 972
   {28,2,9} of size 1008
   {29,2,9} of size 1044
   {30,2,9} of size 1080
   {31,2,9} of size 1116
   {32,2,9} of size 1152
   {33,2,9} of size 1188
   {34,2,9} of size 1224
   {35,2,9} of size 1260
   {36,2,9} of size 1296
   {37,2,9} of size 1332
   {38,2,9} of size 1368
   {39,2,9} of size 1404
   {40,2,9} of size 1440
   {41,2,9} of size 1476
   {42,2,9} of size 1512
   {43,2,9} of size 1548
   {44,2,9} of size 1584
   {45,2,9} of size 1620
   {46,2,9} of size 1656
   {47,2,9} of size 1692
   {48,2,9} of size 1728
   {49,2,9} of size 1764
   {50,2,9} of size 1800
   {51,2,9} of size 1836
   {52,2,9} of size 1872
   {53,2,9} of size 1908
   {54,2,9} of size 1944
   {55,2,9} of size 1980
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {2,3}*12
Covers (Minimal Covers in Boldface) :
   2-fold covers : {2,18}*72
   3-fold covers : {2,27}*108, {6,9}*108
   4-fold covers : {2,36}*144, {4,18}*144a, {4,9}*144
   5-fold covers : {2,45}*180
   6-fold covers : {2,54}*216, {6,18}*216a, {6,18}*216b
   7-fold covers : {2,63}*252
   8-fold covers : {4,36}*288a, {2,72}*288, {8,18}*288, {8,9}*288, {4,18}*288
   9-fold covers : {2,81}*324, {18,9}*324, {6,9}*324a, {6,27}*324
   10-fold covers : {10,18}*360, {2,90}*360
   11-fold covers : {2,99}*396
   12-fold covers : {2,108}*432, {4,54}*432a, {4,27}*432, {6,36}*432a, {6,36}*432b, {12,18}*432a, {12,18}*432b, {6,9}*432, {12,9}*432
   13-fold covers : {2,117}*468
   14-fold covers : {14,18}*504, {2,126}*504
   15-fold covers : {2,135}*540, {6,45}*540
   16-fold covers : {4,72}*576a, {4,36}*576a, {4,72}*576b, {8,36}*576a, {8,36}*576b, {2,144}*576, {16,18}*576, {8,9}*576, {4,36}*576b, {4,18}*576b, {4,36}*576c, {8,18}*576b, {8,18}*576c
   17-fold covers : {2,153}*612
   18-fold covers : {2,162}*648, {18,18}*648a, {18,18}*648b, {6,18}*648a, {6,18}*648b, {6,54}*648a, {6,54}*648b, {6,18}*648i
   19-fold covers : {2,171}*684
   20-fold covers : {10,36}*720, {20,18}*720a, {2,180}*720, {4,90}*720a, {4,45}*720
   21-fold covers : {2,189}*756, {6,63}*756
   22-fold covers : {22,18}*792, {2,198}*792
   23-fold covers : {2,207}*828
   24-fold covers : {4,108}*864a, {2,216}*864, {8,54}*864, {8,27}*864, {6,72}*864a, {6,72}*864b, {24,18}*864a, {12,36}*864a, {12,36}*864b, {24,18}*864b, {4,54}*864, {12,9}*864, {24,9}*864, {6,18}*864, {6,36}*864, {12,18}*864a, {12,18}*864b
   25-fold covers : {2,225}*900, {10,9}*900, {10,45}*900
   26-fold covers : {26,18}*936, {2,234}*936
   27-fold covers : {2,243}*972, {18,9}*972a, {18,27}*972, {6,27}*972a, {6,9}*972d, {18,9}*972h, {18,9}*972i, {6,9}*972e, {6,27}*972b, {6,27}*972c, {6,81}*972
   28-fold covers : {14,36}*1008, {28,18}*1008a, {2,252}*1008, {4,126}*1008a, {4,63}*1008
   29-fold covers : {2,261}*1044
   30-fold covers : {10,54}*1080, {2,270}*1080, {30,18}*1080a, {6,90}*1080a, {6,90}*1080b, {30,18}*1080b
   31-fold covers : {2,279}*1116
   32-fold covers : {8,36}*1152a, {4,72}*1152a, {8,72}*1152a, {8,72}*1152b, {8,72}*1152c, {8,72}*1152d, {16,36}*1152a, {4,144}*1152a, {16,36}*1152b, {4,144}*1152b, {4,36}*1152a, {4,72}*1152b, {8,36}*1152b, {32,18}*1152, {2,288}*1152, {8,9}*1152, {8,18}*1152a, {4,36}*1152d, {8,36}*1152e, {8,36}*1152f, {4,18}*1152a, {8,18}*1152d, {8,18}*1152e, {8,18}*1152f, {8,36}*1152g, {8,36}*1152h, {4,72}*1152c, {4,72}*1152d, {8,18}*1152g, {4,36}*1152e, {4,72}*1152e, {4,18}*1152b, {4,72}*1152f
   33-fold covers : {2,297}*1188, {6,99}*1188
   34-fold covers : {34,18}*1224, {2,306}*1224
   35-fold covers : {2,315}*1260
   36-fold covers : {2,324}*1296, {4,162}*1296a, {4,81}*1296, {18,36}*1296a, {18,36}*1296b, {36,18}*1296a, {12,18}*1296a, {6,36}*1296a, {6,36}*1296b, {12,54}*1296a, {6,108}*1296a, {6,108}*1296b, {36,18}*1296c, {12,18}*1296e, {12,54}*1296b, {6,27}*1296, {12,27}*1296, {18,9}*1296a, {36,9}*1296, {6,9}*1296b, {12,9}*1296c, {6,36}*1296l, {12,18}*1296l, {4,18}*1296b, {4,36}*1296, {6,36}*1296m
   37-fold covers : {2,333}*1332
   38-fold covers : {38,18}*1368, {2,342}*1368
   39-fold covers : {2,351}*1404, {6,117}*1404
   40-fold covers : {10,72}*1440, {40,18}*1440, {20,36}*1440, {4,180}*1440a, {2,360}*1440, {8,90}*1440, {8,45}*1440, {20,18}*1440, {4,90}*1440
   41-fold covers : {2,369}*1476
   42-fold covers : {14,54}*1512, {2,378}*1512, {42,18}*1512a, {6,126}*1512a, {6,126}*1512b, {42,18}*1512b
   43-fold covers : {2,387}*1548
   44-fold covers : {22,36}*1584, {44,18}*1584a, {2,396}*1584, {4,198}*1584a, {4,99}*1584
   45-fold covers : {2,405}*1620, {18,45}*1620, {6,45}*1620a, {6,135}*1620
   46-fold covers : {46,18}*1656, {2,414}*1656
   47-fold covers : {2,423}*1692
   48-fold covers : {4,216}*1728a, {4,108}*1728a, {4,216}*1728b, {8,108}*1728a, {8,108}*1728b, {2,432}*1728, {16,54}*1728, {8,27}*1728, {6,144}*1728a, {6,144}*1728b, {48,18}*1728a, {24,36}*1728a, {12,36}*1728a, {12,36}*1728b, {24,36}*1728b, {12,72}*1728a, {12,72}*1728b, {24,36}*1728c, {12,72}*1728c, {12,72}*1728d, {24,36}*1728d, {48,18}*1728b, {4,108}*1728b, {4,54}*1728b, {4,108}*1728c, {8,54}*1728b, {8,54}*1728c, {6,9}*1728, {24,9}*1728, {12,36}*1728c, {6,36}*1728a, {6,36}*1728b, {12,18}*1728a, {6,18}*1728a, {6,72}*1728b, {6,36}*1728c, {6,72}*1728c, {12,18}*1728b, {12,36}*1728d, {12,36}*1728e, {12,36}*1728f, {12,18}*1728c, {12,36}*1728g, {24,18}*1728b, {24,18}*1728c, {24,18}*1728d, {24,18}*1728e, {12,18}*1728d, {12,36}*1728h, {12,9}*1728, {6,18}*1728c
   49-fold covers : {2,441}*1764, {14,9}*1764, {14,63}*1764
   50-fold covers : {50,18}*1800, {2,450}*1800, {10,18}*1800a, {10,18}*1800b, {10,90}*1800a, {10,90}*1800b, {10,90}*1800c
   51-fold covers : {2,459}*1836, {6,153}*1836
   52-fold covers : {26,36}*1872, {52,18}*1872a, {2,468}*1872, {4,234}*1872a, {4,117}*1872
   53-fold covers : {2,477}*1908
   54-fold covers : {2,486}*1944, {18,18}*1944b, {18,18}*1944c, {18,54}*1944a, {18,54}*1944b, {54,18}*1944a, {6,54}*1944a, {6,54}*1944b, {6,18}*1944g, {6,18}*1944h, {18,18}*1944v, {18,18}*1944w, {18,18}*1944z, {18,18}*1944aa, {6,18}*1944i, {6,18}*1944j, {6,54}*1944c, {6,54}*1944d, {6,54}*1944e, {6,54}*1944f, {6,162}*1944a, {6,162}*1944b, {18,18}*1944ad, {18,18}*1944ae, {6,18}*1944m, {6,18}*1944n, {6,18}*1944o, {6,54}*1944g
   55-fold covers : {2,495}*1980
Permutation Representation (GAP) :

s0 := (1,2);;
s1 := ( 4, 5)( 6, 7)( 8, 9)(10,11);;
s2 := ( 3, 4)( 5, 6)( 7, 8)( 9,10);;
poly := Group([s0,s1,s2]);;

Finitely Presented Group Representation (GAP) :

F := FreeGroup("s0","s1","s2");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  
rels := [ s0*s0, s1*s1, s2*s2, s0*s1*s0*s1, s0*s2*s0*s2, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(11)!(1,2);
s1 := Sym(11)!( 4, 5)( 6, 7)( 8, 9)(10,11);
s2 := Sym(11)!( 3, 4)( 5, 6)( 7, 8)( 9,10);
poly := sub<Sym(11)|s0,s1,s2>;

Finitely Presented Group Representation (Magma) :

poly<s0,s1,s2> := Group< s0,s1,s2 | s0*s0, s1*s1, s2*s2, 
s0*s1*s0*s1, s0*s2*s0*s2, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 >;

to this polytope