Polytope of Type {18,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {18,6}*1944q
if this polytope has a name.
Group : SmallGroup(1944,2344)
Rank : 3
Schlafli Type : {18,6}
Number of vertices, edges, etc : 162, 486, 54
Order of s₀s₁s₂ : 6
Order of s₀s₁s₂s₁ : 6
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {18,6}*648d, {6,6}*648e
   6-fold quotients : {9,6}*324b
   9-fold quotients : {6,6}*216c, {6,6}*216d
   18-fold quotients : {3,6}*108
   27-fold quotients : {6,6}*72a, {6,6}*72b, {6,6}*72c
   54-fold quotients : {3,6}*36, {6,3}*36
   81-fold quotients : {2,6}*24, {6,2}*24
   162-fold quotients : {2,3}*12, {3,2}*12
   243-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

References : None.
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