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# Polytope of Type {2,18,4}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,18,4}*288a
if this polytope has a name.
Group : SmallGroup(288,356)
Rank : 4
Schlafli Type : {2,18,4}
Number of vertices, edges, etc : 2, 18, 36, 4
Order of s0s1s2s3 : 36
Order of s0s1s2s3s2s1 : 2
Special Properties :
Degenerate
Universal
Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{2,18,4,2} of size 576
{2,18,4,4} of size 1152
{2,18,4,6} of size 1728
{2,18,4,3} of size 1728
Vertex Figure Of :
{2,2,18,4} of size 576
{3,2,18,4} of size 864
{4,2,18,4} of size 1152
{5,2,18,4} of size 1440
{6,2,18,4} of size 1728
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {2,18,2}*144
3-fold quotients : {2,6,4}*96a
4-fold quotients : {2,9,2}*72
6-fold quotients : {2,6,2}*48
9-fold quotients : {2,2,4}*32
12-fold quotients : {2,3,2}*24
18-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
2-fold covers : {2,36,4}*576a, {4,18,4}*576a, {2,18,8}*576
3-fold covers : {2,54,4}*864a, {2,18,12}*864a, {6,18,4}*864a, {6,18,4}*864b, {2,18,12}*864b
4-fold covers : {4,36,4}*1152a, {2,36,8}*1152a, {2,72,4}*1152a, {2,36,8}*1152b, {2,72,4}*1152b, {2,36,4}*1152a, {4,18,8}*1152a, {8,18,4}*1152a, {2,18,16}*1152, {4,18,4}*1152b, {2,18,4}*1152b
5-fold covers : {2,18,20}*1440a, {10,18,4}*1440a, {2,90,4}*1440a
6-fold covers : {2,108,4}*1728a, {4,54,4}*1728a, {2,54,8}*1728, {4,18,12}*1728a, {12,18,4}*1728a, {6,36,4}*1728a, {6,36,4}*1728b, {2,18,24}*1728a, {6,18,8}*1728a, {6,18,8}*1728b, {2,36,12}*1728a, {2,36,12}*1728b, {2,18,24}*1728b, {4,18,12}*1728b, {12,18,4}*1728b
Permutation Representation (GAP) :
s0 := (1,2);;
s1 := ( 4, 5)( 6,10)( 7, 9)( 8,11)(13,14)(15,19)(16,18)(17,20)(22,23)(24,28)
(25,27)(26,29)(31,32)(33,37)(34,36)(35,38);;
s2 := ( 3, 6)( 4, 8)( 5, 7)( 9,10)(12,15)(13,17)(14,16)(18,19)(21,33)(22,35)
(23,34)(24,30)(25,32)(26,31)(27,37)(28,36)(29,38);;
s3 := ( 3,21)( 4,22)( 5,23)( 6,24)( 7,25)( 8,26)( 9,27)(10,28)(11,29)(12,30)
(13,31)(14,32)(15,33)(16,34)(17,35)(18,36)(19,37)(20,38);;
poly := Group([s0,s1,s2,s3]);;

Finitely Presented Group Representation (GAP) :
F := FreeGroup("s0","s1","s2","s3");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s0*s1*s0*s1,
s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3,
s1*s2*s3*s2*s1*s2*s3*s2, s2*s3*s2*s3*s2*s3*s2*s3,
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(38)!(1,2);
s1 := Sym(38)!( 4, 5)( 6,10)( 7, 9)( 8,11)(13,14)(15,19)(16,18)(17,20)(22,23)
(24,28)(25,27)(26,29)(31,32)(33,37)(34,36)(35,38);
s2 := Sym(38)!( 3, 6)( 4, 8)( 5, 7)( 9,10)(12,15)(13,17)(14,16)(18,19)(21,33)
(22,35)(23,34)(24,30)(25,32)(26,31)(27,37)(28,36)(29,38);
s3 := Sym(38)!( 3,21)( 4,22)( 5,23)( 6,24)( 7,25)( 8,26)( 9,27)(10,28)(11,29)
(12,30)(13,31)(14,32)(15,33)(16,34)(17,35)(18,36)(19,37)(20,38);
poly := sub<Sym(38)|s0,s1,s2,s3>;

Finitely Presented Group Representation (Magma) :
poly<s0,s1,s2,s3> := Group< s0,s1,s2,s3 | s0*s0, s1*s1, s2*s2,
s3*s3, s0*s1*s0*s1, s0*s2*s0*s2, s0*s3*s0*s3,
s1*s3*s1*s3, s1*s2*s3*s2*s1*s2*s3*s2,
s2*s3*s2*s3*s2*s3*s2*s3, 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