Polytope of Type {2,18}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,18}*72
if this polytope has a name.
Group : SmallGroup(72,17)
Rank : 3
Schlafli Type : {2,18}
Number of vertices, edges, etc : 2, 18, 18
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 :
{2,18,2} of size 144
{2,18,4} of size 288
{2,18,4} of size 288
{2,18,4} of size 288
{2,18,6} of size 432
{2,18,6} of size 432
{2,18,8} of size 576
{2,18,4} of size 576
{2,18,9} of size 648
{2,18,6} of size 648
{2,18,6} of size 648
{2,18,3} of size 648
{2,18,6} of size 648
{2,18,10} of size 720
{2,18,12} of size 864
{2,18,12} of size 864
{2,18,12} of size 864
{2,18,14} of size 1008
{2,18,16} of size 1152
{2,18,4} of size 1152
{2,18,8} of size 1152
{2,18,4} of size 1152
{2,18,8} of size 1152
{2,18,8} of size 1152
{2,18,4} of size 1296
{2,18,18} of size 1296
{2,18,18} of size 1296
{2,18,18} of size 1296
{2,18,6} of size 1296
{2,18,6} of size 1296
{2,18,6} of size 1296
{2,18,6} of size 1296
{2,18,6} of size 1296
{2,18,6} of size 1296
{2,18,6} of size 1296
{2,18,6} of size 1296
{2,18,6} of size 1296
{2,18,20} of size 1440
{2,18,20} of size 1440
{2,18,22} of size 1584
{2,18,24} of size 1728
{2,18,24} of size 1728
{2,18,6} of size 1728
{2,18,12} of size 1728
{2,18,12} of size 1728
{2,18,10} of size 1800
{2,18,26} of size 1872
{2,18,9} of size 1944
{2,18,18} of size 1944
{2,18,3} of size 1944
{2,18,18} of size 1944
{2,18,6} of size 1944
{2,18,9} of size 1944
{2,18,6} of size 1944
{2,18,9} of size 1944
{2,18,9} of size 1944
{2,18,18} of size 1944
{2,18,18} of size 1944
{2,18,9} of size 1944
{2,18,18} of size 1944
{2,18,27} of size 1944
{2,18,6} of size 1944
{2,18,9} of size 1944
{2,18,9} of size 1944
{2,18,9} of size 1944
{2,18,18} of size 1944
{2,18,18} of size 1944
{2,18,9} of size 1944
{2,18,18} of size 1944
{2,18,18} of size 1944
{2,18,6} of size 1944
{2,18,9} of size 1944
{2,18,3} of size 1944
{2,18,6} of size 1944
Vertex Figure Of :
{2,2,18} of size 144
{3,2,18} of size 216
{4,2,18} of size 288
{5,2,18} of size 360
{6,2,18} of size 432
{7,2,18} of size 504
{8,2,18} of size 576
{9,2,18} of size 648
{10,2,18} of size 720
{11,2,18} of size 792
{12,2,18} of size 864
{13,2,18} of size 936
{14,2,18} of size 1008
{15,2,18} of size 1080
{16,2,18} of size 1152
{17,2,18} of size 1224
{18,2,18} of size 1296
{19,2,18} of size 1368
{20,2,18} of size 1440
{21,2,18} of size 1512
{22,2,18} of size 1584
{23,2,18} of size 1656
{24,2,18} of size 1728
{25,2,18} of size 1800
{26,2,18} of size 1872
{27,2,18} of size 1944
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {2,9}*36
3-fold quotients : {2,6}*24
6-fold quotients : {2,3}*12
9-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
2-fold covers : {2,36}*144, {4,18}*144a
3-fold covers : {2,54}*216, {6,18}*216a, {6,18}*216b
4-fold covers : {4,36}*288a, {2,72}*288, {8,18}*288, {4,18}*288
5-fold covers : {10,18}*360, {2,90}*360
6-fold covers : {2,108}*432, {4,54}*432a, {6,36}*432a, {6,36}*432b, {12,18}*432a, {12,18}*432b
7-fold covers : {14,18}*504, {2,126}*504
8-fold covers : {4,72}*576a, {4,36}*576a, {4,72}*576b, {8,36}*576a, {8,36}*576b, {2,144}*576, {16,18}*576, {4,36}*576b, {4,18}*576b, {4,36}*576c, {8,18}*576b, {8,18}*576c
9-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
10-fold covers : {10,36}*720, {20,18}*720a, {2,180}*720, {4,90}*720a
11-fold covers : {22,18}*792, {2,198}*792
12-fold covers : {4,108}*864a, {2,216}*864, {8,54}*864, {6,72}*864a, {6,72}*864b, {24,18}*864a, {12,36}*864a, {12,36}*864b, {24,18}*864b, {4,54}*864, {6,18}*864, {6,36}*864, {12,18}*864a, {12,18}*864b
13-fold covers : {26,18}*936, {2,234}*936
14-fold covers : {14,36}*1008, {28,18}*1008a, {2,252}*1008, {4,126}*1008a
15-fold covers : {10,54}*1080, {2,270}*1080, {30,18}*1080a, {6,90}*1080a, {6,90}*1080b, {30,18}*1080b
16-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, {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
17-fold covers : {34,18}*1224, {2,306}*1224
18-fold covers : {2,324}*1296, {4,162}*1296a, {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, {6,36}*1296l, {12,18}*1296l, {4,18}*1296b, {4,36}*1296, {6,36}*1296m
19-fold covers : {38,18}*1368, {2,342}*1368
20-fold covers : {10,72}*1440, {40,18}*1440, {20,36}*1440, {4,180}*1440a, {2,360}*1440, {8,90}*1440, {20,18}*1440, {4,90}*1440
21-fold covers : {14,54}*1512, {2,378}*1512, {42,18}*1512a, {6,126}*1512a, {6,126}*1512b, {42,18}*1512b
22-fold covers : {22,36}*1584, {44,18}*1584a, {2,396}*1584, {4,198}*1584a
23-fold covers : {46,18}*1656, {2,414}*1656
24-fold covers : {4,216}*1728a, {4,108}*1728a, {4,216}*1728b, {8,108}*1728a, {8,108}*1728b, {2,432}*1728, {16,54}*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, {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
25-fold covers : {50,18}*1800, {2,450}*1800, {10,18}*1800a, {10,18}*1800b, {10,90}*1800a, {10,90}*1800b, {10,90}*1800c
26-fold covers : {26,36}*1872, {52,18}*1872a, {2,468}*1872, {4,234}*1872a
27-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
Permutation Representation (GAP) :
s0 := (1,2);;
s1 := ( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20);;
s2 := ( 3, 7)( 4, 5)( 6,11)( 8, 9)(10,15)(12,13)(14,19)(16,17)(18,20);;
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*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(20)!(1,2);
s1 := Sym(20)!( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20);
s2 := Sym(20)!( 3, 7)( 4, 5)( 6,11)( 8, 9)(10,15)(12,13)(14,19)(16,17)(18,20);
poly := sub<Sym(20)|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*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 >;
to this polytope