Polytope of Type {9,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {9,6}*324a
if this polytope has a name.
Group : SmallGroup(324,37)
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
   {9,6,3} of size 972
   {9,6,4} of size 1296
   {9,6,6} of size 1944
   {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 : {9,6}*108, {3,6}*108
   9-fold quotients : {9,2}*36, {3,6}*36
   27-fold quotients : {3,2}*12
Covers (Minimal Covers in Boldface) :
   2-fold covers : {18,6}*648a
   3-fold covers : {9,18}*972a, {27,6}*972a, {9,6}*972d, {9,18}*972h, {9,18}*972i, {9,6}*972e, {27,6}*972b, {27,6}*972c
   4-fold covers : {36,6}*1296a, {18,12}*1296e, {9,6}*1296b, {9,12}*1296c
   5-fold covers : {45,6}*1620a
   6-fold covers : {18,18}*1944a, {54,6}*1944a, {18,6}*1944h, {18,18}*1944u, {18,18}*1944y, {18,6}*1944i, {54,6}*1944c, {54,6}*1944e, {18,6}*1944m
Permutation Representation (GAP) :

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