Polytope of Type {18,2}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {18,2}*72
if this polytope has a name.
Group : SmallGroup(72,17)
Rank : 3
Schlafli Type : {18,2}
Number of vertices, edges, etc : 18, 18, 2
Order of s0s1s2 : 18
Order of s0s1s2s1 : 2
Special Properties :
Degenerate
Universal
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Flat
Self-Petrie
Related Polytopes :
Facet
Vertex Figure
Dual
Petrial
Facet Of :
{18,2,2} of size 144
{18,2,3} of size 216
{18,2,4} of size 288
{18,2,5} of size 360
{18,2,6} of size 432
{18,2,7} of size 504
{18,2,8} of size 576
{18,2,9} of size 648
{18,2,10} of size 720
{18,2,11} of size 792
{18,2,12} of size 864
{18,2,13} of size 936
{18,2,14} of size 1008
{18,2,15} of size 1080
{18,2,16} of size 1152
{18,2,17} of size 1224
{18,2,18} of size 1296
{18,2,19} of size 1368
{18,2,20} of size 1440
{18,2,21} of size 1512
{18,2,22} of size 1584
{18,2,23} of size 1656
{18,2,24} of size 1728
{18,2,25} of size 1800
{18,2,26} of size 1872
{18,2,27} of size 1944
Vertex Figure Of :
{2,18,2} of size 144
{4,18,2} of size 288
{4,18,2} of size 288
{4,18,2} of size 288
{6,18,2} of size 432
{6,18,2} of size 432
{8,18,2} of size 576
{4,18,2} of size 576
{9,18,2} of size 648
{6,18,2} of size 648
{6,18,2} of size 648
{3,18,2} of size 648
{6,18,2} of size 648
{10,18,2} of size 720
{12,18,2} of size 864
{12,18,2} of size 864
{12,18,2} of size 864
{14,18,2} of size 1008
{16,18,2} of size 1152
{4,18,2} of size 1152
{8,18,2} of size 1152
{4,18,2} of size 1152
{8,18,2} of size 1152
{8,18,2} of size 1152
{4,18,2} of size 1296
{18,18,2} of size 1296
{18,18,2} of size 1296
{18,18,2} of size 1296
{6,18,2} of size 1296
{6,18,2} of size 1296
{6,18,2} of size 1296
{6,18,2} of size 1296
{6,18,2} of size 1296
{6,18,2} of size 1296
{6,18,2} of size 1296
{6,18,2} of size 1296
{6,18,2} of size 1296
{20,18,2} of size 1440
{20,18,2} of size 1440
{22,18,2} of size 1584
{24,18,2} of size 1728
{24,18,2} of size 1728
{6,18,2} of size 1728
{12,18,2} of size 1728
{12,18,2} of size 1728
{10,18,2} of size 1800
{26,18,2} of size 1872
{9,18,2} of size 1944
{18,18,2} of size 1944
{3,18,2} of size 1944
{18,18,2} of size 1944
{6,18,2} of size 1944
{9,18,2} of size 1944
{6,18,2} of size 1944
{9,18,2} of size 1944
{9,18,2} of size 1944
{18,18,2} of size 1944
{18,18,2} of size 1944
{9,18,2} of size 1944
{18,18,2} of size 1944
{27,18,2} of size 1944
{6,18,2} of size 1944
{9,18,2} of size 1944
{9,18,2} of size 1944
{9,18,2} of size 1944
{18,18,2} of size 1944
{18,18,2} of size 1944
{9,18,2} of size 1944
{18,18,2} of size 1944
{18,18,2} of size 1944
{6,18,2} of size 1944
{9,18,2} of size 1944
{3,18,2} of size 1944
{6,18,2} of size 1944
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {9,2}*36
3-fold quotients : {6,2}*24
6-fold quotients : {3,2}*12
9-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
2-fold covers : {36,2}*144, {18,4}*144a
3-fold covers : {54,2}*216, {18,6}*216a, {18,6}*216b
4-fold covers : {36,4}*288a, {72,2}*288, {18,8}*288, {18,4}*288
5-fold covers : {18,10}*360, {90,2}*360
6-fold covers : {108,2}*432, {54,4}*432a, {36,6}*432a, {36,6}*432b, {18,12}*432a, {18,12}*432b
7-fold covers : {18,14}*504, {126,2}*504
8-fold covers : {72,4}*576a, {36,4}*576a, {72,4}*576b, {36,8}*576a, {36,8}*576b, {144,2}*576, {18,16}*576, {36,4}*576b, {18,4}*576b, {36,4}*576c, {18,8}*576b, {18,8}*576c
9-fold covers : {162,2}*648, {18,18}*648a, {18,18}*648c, {18,6}*648a, {18,6}*648b, {54,6}*648a, {54,6}*648b, {18,6}*648i
10-fold covers : {36,10}*720, {18,20}*720a, {180,2}*720, {90,4}*720a
11-fold covers : {18,22}*792, {198,2}*792
12-fold covers : {108,4}*864a, {216,2}*864, {54,8}*864, {72,6}*864a, {72,6}*864b, {18,24}*864a, {36,12}*864a, {36,12}*864b, {18,24}*864b, {54,4}*864, {18,6}*864, {36,6}*864, {18,12}*864a, {18,12}*864b
13-fold covers : {18,26}*936, {234,2}*936
14-fold covers : {36,14}*1008, {18,28}*1008a, {252,2}*1008, {126,4}*1008a
15-fold covers : {54,10}*1080, {270,2}*1080, {18,30}*1080a, {90,6}*1080a, {90,6}*1080b, {18,30}*1080b
16-fold covers : {36,8}*1152a, {72,4}*1152a, {72,8}*1152a, {72,8}*1152b, {72,8}*1152c, {72,8}*1152d, {36,16}*1152a, {144,4}*1152a, {36,16}*1152b, {144,4}*1152b, {36,4}*1152a, {72,4}*1152b, {36,8}*1152b, {18,32}*1152, {288,2}*1152, {36,4}*1152d, {36,8}*1152e, {36,8}*1152f, {18,4}*1152a, {18,8}*1152d, {18,8}*1152e, {18,8}*1152f, {36,8}*1152g, {36,8}*1152h, {72,4}*1152c, {72,4}*1152d, {18,8}*1152g, {36,4}*1152e, {72,4}*1152e, {18,4}*1152b, {72,4}*1152f
17-fold covers : {18,34}*1224, {306,2}*1224
18-fold covers : {324,2}*1296, {162,4}*1296a, {18,36}*1296a, {36,18}*1296a, {36,18}*1296b, {18,12}*1296a, {36,6}*1296a, {36,6}*1296b, {54,12}*1296a, {108,6}*1296a, {108,6}*1296b, {18,36}*1296c, {18,12}*1296e, {54,12}*1296b, {36,6}*1296l, {18,12}*1296l, {18,4}*1296b, {36,4}*1296, {36,6}*1296m
19-fold covers : {18,38}*1368, {342,2}*1368
20-fold covers : {72,10}*1440, {18,40}*1440, {36,20}*1440, {180,4}*1440a, {360,2}*1440, {90,8}*1440, {18,20}*1440, {90,4}*1440
21-fold covers : {54,14}*1512, {378,2}*1512, {18,42}*1512a, {126,6}*1512a, {126,6}*1512b, {18,42}*1512b
22-fold covers : {36,22}*1584, {18,44}*1584a, {396,2}*1584, {198,4}*1584a
23-fold covers : {18,46}*1656, {414,2}*1656
24-fold covers : {216,4}*1728a, {108,4}*1728a, {216,4}*1728b, {108,8}*1728a, {108,8}*1728b, {432,2}*1728, {54,16}*1728, {144,6}*1728a, {144,6}*1728b, {18,48}*1728a, {36,24}*1728a, {36,12}*1728a, {36,12}*1728b, {36,24}*1728b, {72,12}*1728a, {72,12}*1728b, {36,24}*1728c, {72,12}*1728c, {72,12}*1728d, {36,24}*1728d, {18,48}*1728b, {108,4}*1728b, {54,4}*1728b, {108,4}*1728c, {54,8}*1728b, {54,8}*1728c, {36,12}*1728c, {36,6}*1728a, {36,6}*1728b, {18,12}*1728a, {18,6}*1728a, {72,6}*1728b, {36,6}*1728c, {72,6}*1728c, {18,12}*1728b, {36,12}*1728d, {36,12}*1728e, {36,12}*1728f, {18,12}*1728c, {36,12}*1728g, {18,24}*1728b, {18,24}*1728c, {18,24}*1728d, {18,24}*1728e, {18,12}*1728d, {36,12}*1728h
25-fold covers : {18,50}*1800, {450,2}*1800, {18,10}*1800a, {18,10}*1800b, {90,10}*1800a, {90,10}*1800b, {90,10}*1800c
26-fold covers : {36,26}*1872, {18,52}*1872a, {468,2}*1872, {234,4}*1872a
27-fold covers : {486,2}*1944, {18,18}*1944a, {18,18}*1944c, {18,54}*1944a, {54,18}*1944a, {54,18}*1944b, {54,6}*1944a, {54,6}*1944b, {18,6}*1944g, {18,6}*1944h, {18,18}*1944s, {18,18}*1944u, {18,18}*1944x, {18,18}*1944y, {18,6}*1944i, {18,6}*1944j, {54,6}*1944c, {54,6}*1944d, {54,6}*1944e, {54,6}*1944f, {162,6}*1944a, {162,6}*1944b, {18,18}*1944ad, {18,18}*1944af, {18,6}*1944m, {18,6}*1944n, {18,6}*1944o, {54,6}*1944g
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);;
s2 := (19,20);;
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*s2*s0*s2, s1*s2*s1*s2,
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(20)!( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18);
s1 := Sym(20)!( 1, 5)( 2, 3)( 4, 9)( 6, 7)( 8,13)(10,11)(12,17)(14,15)(16,18);
s2 := Sym(20)!(19,20);
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*s2*s0*s2, s1*s2*s1*s2, 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 >;
to this polytope