Polytope of Type {8,18}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {8,18}*288
Also Known As : {8,18|2}. if this polytope has another name.
Group : SmallGroup(288,120)
Rank : 3
Schlafli Type : {8,18}
Number of vertices, edges, etc : 8, 72, 18
Order of s₀s₁s₂ : 72
Order of s₀s₁s₂s₁ : 2
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {8,18,2} of size 576
   {8,18,4} of size 1152
   {8,18,4} of size 1152
   {8,18,6} of size 1728
   {8,18,6} of size 1728
Vertex Figure Of :
   {2,8,18} of size 576
   {4,8,18} of size 1152
   {4,8,18} of size 1152
   {6,8,18} of size 1728
   {3,8,18} of size 1728
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {4,18}*144a
   3-fold quotients : {8,6}*96
   4-fold quotients : {2,18}*72
   6-fold quotients : {4,6}*48a
   8-fold quotients : {2,9}*36
   9-fold quotients : {8,2}*32
   12-fold quotients : {2,6}*24
   18-fold quotients : {4,2}*16
   24-fold quotients : {2,3}*12
   36-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   2-fold covers : {8,36}*576a, {16,18}*576
   3-fold covers : {8,54}*864, {24,18}*864a, {24,18}*864b
   4-fold covers : {8,36}*1152a, {8,72}*1152a, {8,72}*1152c, {16,36}*1152a, {16,36}*1152b, {32,18}*1152, {8,18}*1152g
   5-fold covers : {40,18}*1440, {8,90}*1440
   6-fold covers : {8,108}*1728a, {16,54}*1728, {48,18}*1728a, {24,36}*1728b, {24,36}*1728c, {48,18}*1728b
Permutation Representation (GAP) :

s0 := (19,28)(20,29)(21,30)(22,31)(23,32)(24,33)(25,34)(26,35)(27,36)(37,55)
(38,56)(39,57)(40,58)(41,59)(42,60)(43,61)(44,62)(45,63)(46,64)(47,65)(48,66)
(49,67)(50,68)(51,69)(52,70)(53,71)(54,72);;
s1 := ( 1,37)( 2,39)( 3,38)( 4,44)( 5,43)( 6,45)( 7,41)( 8,40)( 9,42)(10,46)
(11,48)(12,47)(13,53)(14,52)(15,54)(16,50)(17,49)(18,51)(19,64)(20,66)(21,65)
(22,71)(23,70)(24,72)(25,68)(26,67)(27,69)(28,55)(29,57)(30,56)(31,62)(32,61)
(33,63)(34,59)(35,58)(36,60);;
s2 := ( 1, 4)( 2, 6)( 3, 5)( 7, 8)(10,13)(11,15)(12,14)(16,17)(19,22)(20,24)
(21,23)(25,26)(28,31)(29,33)(30,32)(34,35)(37,40)(38,42)(39,41)(43,44)(46,49)
(47,51)(48,50)(52,53)(55,58)(56,60)(57,59)(61,62)(64,67)(65,69)(66,68)
(70,71);;
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, s0*s1*s2*s1*s0*s1*s2*s1, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
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(72)!(19,28)(20,29)(21,30)(22,31)(23,32)(24,33)(25,34)(26,35)(27,36)
(37,55)(38,56)(39,57)(40,58)(41,59)(42,60)(43,61)(44,62)(45,63)(46,64)(47,65)
(48,66)(49,67)(50,68)(51,69)(52,70)(53,71)(54,72);
s1 := Sym(72)!( 1,37)( 2,39)( 3,38)( 4,44)( 5,43)( 6,45)( 7,41)( 8,40)( 9,42)
(10,46)(11,48)(12,47)(13,53)(14,52)(15,54)(16,50)(17,49)(18,51)(19,64)(20,66)
(21,65)(22,71)(23,70)(24,72)(25,68)(26,67)(27,69)(28,55)(29,57)(30,56)(31,62)
(32,61)(33,63)(34,59)(35,58)(36,60);
s2 := Sym(72)!( 1, 4)( 2, 6)( 3, 5)( 7, 8)(10,13)(11,15)(12,14)(16,17)(19,22)
(20,24)(21,23)(25,26)(28,31)(29,33)(30,32)(34,35)(37,40)(38,42)(39,41)(43,44)
(46,49)(47,51)(48,50)(52,53)(55,58)(56,60)(57,59)(61,62)(64,67)(65,69)(66,68)
(70,71);
poly := sub<Sym(72)|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, s0*s1*s2*s1*s0*s1*s2*s1, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
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 >;

References : None.
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