Polytope of Type {9,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {9,6}*324d
if this polytope has a name.
Group : SmallGroup(324,40)
Rank : 3
Schlafli Type : {9,6}
Number of vertices, edges, etc : 27, 81, 18
Order of s₀s₁s₂ : 18
Order of s₀s₁s₂s₁ : 6
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
Facet Of :
   {9,6,2} of size 648
   {9,6,4} of size 1296
   {9,6,6} of size 1944
Vertex Figure Of :
   {2,9,6} of size 648
   {4,9,6} of size 1296
   {6,9,6} of size 1944
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {3,6}*108
   9-fold quotients : {3,6}*36
   27-fold quotients : {3,2}*12
Covers (Minimal Covers in Boldface) :
   2-fold covers : {18,6}*648e
   3-fold covers : {9,6}*972b, {9,18}*972d, {9,18}*972e, {9,18}*972g, {9,6}*972e
   4-fold covers : {36,6}*1296e, {18,12}*1296h, {9,6}*1296a, {9,12}*1296a
   5-fold covers : {45,6}*1620d
   6-fold covers : {18,6}*1944d, {18,18}*1944l, {18,18}*1944o, {18,18}*1944t, {18,6}*1944i, {18,6}*1944r
Permutation Representation (GAP) :

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

References : None.
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