Polytope of Type {18,4}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {18,4}*648
if this polytope has a name.
Group : SmallGroup(648,252)
Rank : 3
Schlafli Type : {18,4}
Number of vertices, edges, etc : 81, 162, 18
Order of s₀s₁s₂ : 4
Order of s₀s₁s₂s₁ : 18
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Non-Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
   Skewing Operation
Facet Of :
   {18,4,2} of size 1296
Vertex Figure Of :
   {2,18,4} of size 1296
Quotients (Maximal Quotients in Boldface) :
   9-fold quotients : {6,4}*72
Covers (Minimal Covers in Boldface) :
   2-fold covers : {18,4}*1296a
   3-fold covers : {18,4}*1944a, {18,12}*1944a, {18,12}*1944b, {18,12}*1944g
Permutation Representation (GAP) :

s0 := ( 2, 3)( 4, 7)( 5, 9)( 6, 8)(10,26)(11,25)(12,27)(13,23)(14,22)(15,24)
(16,20)(17,19)(18,21)(28,61)(29,63)(30,62)(31,58)(32,60)(33,59)(34,55)(35,57)
(36,56)(37,77)(38,76)(39,78)(40,74)(41,73)(42,75)(43,80)(44,79)(45,81)(46,68)
(47,67)(48,69)(49,65)(50,64)(51,66)(52,71)(53,70)(54,72);;
s1 := ( 1,28)( 2,32)( 3,36)( 4,34)( 5,29)( 6,33)( 7,31)( 8,35)( 9,30)(10,40)
(11,44)(12,39)(13,37)(14,41)(15,45)(16,43)(17,38)(18,42)(19,52)(20,47)(21,51)
(22,49)(23,53)(24,48)(25,46)(26,50)(27,54)(55,61)(57,60)(59,62)(65,68)(66,72)
(67,70)(73,76)(74,80)(78,81);;
s2 := ( 2, 3)( 4, 6)( 7, 8)(10,30)(11,29)(12,28)(13,32)(14,31)(15,33)(16,34)
(17,36)(18,35)(19,56)(20,55)(21,57)(22,58)(23,60)(24,59)(25,63)(26,62)(27,61)
(37,38)(41,42)(43,45)(46,64)(47,66)(48,65)(49,69)(50,68)(51,67)(52,71)(53,70)
(54,72)(73,75)(76,77)(80,81);;
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*s1*s2*s1*s2, 
s2*s0*s1*s2*s0*s1*s2*s0*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*s0*s1, 
s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s2*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(81)!( 2, 3)( 4, 7)( 5, 9)( 6, 8)(10,26)(11,25)(12,27)(13,23)(14,22)
(15,24)(16,20)(17,19)(18,21)(28,61)(29,63)(30,62)(31,58)(32,60)(33,59)(34,55)
(35,57)(36,56)(37,77)(38,76)(39,78)(40,74)(41,73)(42,75)(43,80)(44,79)(45,81)
(46,68)(47,67)(48,69)(49,65)(50,64)(51,66)(52,71)(53,70)(54,72);
s1 := Sym(81)!( 1,28)( 2,32)( 3,36)( 4,34)( 5,29)( 6,33)( 7,31)( 8,35)( 9,30)
(10,40)(11,44)(12,39)(13,37)(14,41)(15,45)(16,43)(17,38)(18,42)(19,52)(20,47)
(21,51)(22,49)(23,53)(24,48)(25,46)(26,50)(27,54)(55,61)(57,60)(59,62)(65,68)
(66,72)(67,70)(73,76)(74,80)(78,81);
s2 := Sym(81)!( 2, 3)( 4, 6)( 7, 8)(10,30)(11,29)(12,28)(13,32)(14,31)(15,33)
(16,34)(17,36)(18,35)(19,56)(20,55)(21,57)(22,58)(23,60)(24,59)(25,63)(26,62)
(27,61)(37,38)(41,42)(43,45)(46,64)(47,66)(48,65)(49,69)(50,68)(51,67)(52,71)
(53,70)(54,72)(73,75)(76,77)(80,81);
poly := sub<Sym(81)|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*s1*s2*s1*s2, 
s2*s0*s1*s2*s0*s1*s2*s0*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*s0*s1, 
s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s2*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 >;

References : None.
to this polytope