Polytope of Type {18,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {18,6}*972a
if this polytope has a name.
Group : SmallGroup(972,102)
Rank : 3
Schlafli Type : {18,6}
Number of vertices, edges, etc : 81, 243, 27
Order of s₀s₁s₂ : 9
Order of s₀s₁s₂s₁ : 6
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Non-Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
Facet Of :
   {18,6,2} of size 1944
Vertex Figure Of :
   {2,18,6} of size 1944
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {6,6}*324b, {18,6}*324b
   9-fold quotients : {6,6}*108
Covers (Minimal Covers in Boldface) :
   2-fold covers : {18,6}*1944c
Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

References : None.
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