Polytope of Type {18,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {18,6}*648i
if this polytope has a name.
Group : SmallGroup(648,554)
Rank : 3
Schlafli Type : {18,6}
Number of vertices, edges, etc : 54, 162, 18
Order of s₀s₁s₂ : 18
Order of s₀s₁s₂s₁ : 6
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
   Self-Petrie
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
Facet Of :
   {18,6,2} of size 1296
   {18,6,3} of size 1944
Vertex Figure Of :
   {2,18,6} of size 1296
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {18,6}*216a, {18,6}*216b, {6,6}*216d
   6-fold quotients : {9,6}*108
   9-fold quotients : {18,2}*72, {6,6}*72a, {6,6}*72b, {6,6}*72c
   18-fold quotients : {9,2}*36, {3,6}*36, {6,3}*36
   27-fold quotients : {2,6}*24, {6,2}*24
   54-fold quotients : {2,3}*12, {3,2}*12
   81-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   2-fold covers : {36,6}*1296l, {18,12}*1296l
   3-fold covers : {18,18}*1944ad, {18,18}*1944af, {18,6}*1944m, {18,6}*1944n, {18,6}*1944o, {54,6}*1944g
Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

References : None.
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