Polytope of Type {6,18}

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

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

Permutation Representation (Magma) :

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

References : None.
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