Polytope of Type {2,9,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,9,6}*648d
if this polytope has a name.
Group : SmallGroup(648,300)
Rank : 4
Schlafli Type : {2,9,6}
Number of vertices, edges, etc : 2, 27, 81, 18
Order of s₀s₁s₂s₃ : 18
Order of s₀s₁s₂s₃s₂s₁ : 2
Special Properties :
   Degenerate
   Universal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {2,9,6,2} of size 1296
Vertex Figure Of :
   {2,2,9,6} of size 1296
   {3,2,9,6} of size 1944
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {2,3,6}*216
   9-fold quotients : {2,3,6}*72
   27-fold quotients : {2,3,2}*24
Covers (Minimal Covers in Boldface) :
   2-fold covers : {2,18,6}*1296e
   3-fold covers : {2,9,6}*1944b, {2,9,18}*1944d, {2,9,18}*1944e, {2,9,18}*1944g, {2,9,6}*1944e, {6,9,6}*1944g
Permutation Representation (GAP) :

s0 := (1,2);;
s1 := ( 4, 5)( 6, 8)( 9,10)(12,21)(13,23)(14,22)(15,26)(16,25)(17,24)(18,28)
(19,27)(20,29)(30,58)(31,57)(32,59)(33,60)(34,62)(35,61)(36,65)(37,64)(38,63)
(39,76)(40,75)(41,77)(42,78)(43,80)(44,79)(45,83)(46,82)(47,81)(48,67)(49,66)
(50,68)(51,69)(52,71)(53,70)(54,74)(55,73)(56,72);;
s2 := ( 3,45)( 4,47)( 5,46)( 6,39)( 7,41)( 8,40)( 9,42)(10,44)(11,43)(12,33)
(13,35)(14,34)(15,36)(16,38)(17,37)(18,30)(19,32)(20,31)(21,49)(22,48)(23,50)
(24,52)(25,51)(26,53)(27,55)(28,54)(29,56)(57,73)(58,72)(59,74)(60,67)(61,66)
(62,68)(63,70)(64,69)(65,71)(75,77)(78,80)(81,83);;
s3 := ( 6,10)( 7,11)( 8, 9)(12,21)(13,22)(14,23)(15,28)(16,29)(17,27)(18,26)
(19,24)(20,25)(33,37)(34,38)(35,36)(39,48)(40,49)(41,50)(42,55)(43,56)(44,54)
(45,53)(46,51)(47,52)(60,64)(61,65)(62,63)(66,75)(67,76)(68,77)(69,82)(70,83)
(71,81)(72,80)(73,78)(74,79);;
poly := Group([s0,s1,s2,s3]);;

Finitely Presented Group Representation (GAP) :

F := FreeGroup("s0","s1","s2","s3");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;  
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s0*s1*s0*s1, 
s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s1*s2*s3*s2*s1*s2*s1*s2*s3*s2*s1*s2, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
s2*s3*s2*s1*s3*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s2*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(83)!(1,2);
s1 := Sym(83)!( 4, 5)( 6, 8)( 9,10)(12,21)(13,23)(14,22)(15,26)(16,25)(17,24)
(18,28)(19,27)(20,29)(30,58)(31,57)(32,59)(33,60)(34,62)(35,61)(36,65)(37,64)
(38,63)(39,76)(40,75)(41,77)(42,78)(43,80)(44,79)(45,83)(46,82)(47,81)(48,67)
(49,66)(50,68)(51,69)(52,71)(53,70)(54,74)(55,73)(56,72);
s2 := Sym(83)!( 3,45)( 4,47)( 5,46)( 6,39)( 7,41)( 8,40)( 9,42)(10,44)(11,43)
(12,33)(13,35)(14,34)(15,36)(16,38)(17,37)(18,30)(19,32)(20,31)(21,49)(22,48)
(23,50)(24,52)(25,51)(26,53)(27,55)(28,54)(29,56)(57,73)(58,72)(59,74)(60,67)
(61,66)(62,68)(63,70)(64,69)(65,71)(75,77)(78,80)(81,83);
s3 := Sym(83)!( 6,10)( 7,11)( 8, 9)(12,21)(13,22)(14,23)(15,28)(16,29)(17,27)
(18,26)(19,24)(20,25)(33,37)(34,38)(35,36)(39,48)(40,49)(41,50)(42,55)(43,56)
(44,54)(45,53)(46,51)(47,52)(60,64)(61,65)(62,63)(66,75)(67,76)(68,77)(69,82)
(70,83)(71,81)(72,80)(73,78)(74,79);
poly := sub<Sym(83)|s0,s1,s2,s3>;

Finitely Presented Group Representation (Magma) :

poly<s0,s1,s2,s3> := Group< s0,s1,s2,s3 | s0*s0, s1*s1, s2*s2, 
s3*s3, s0*s1*s0*s1, s0*s2*s0*s2, s0*s3*s0*s3, 
s1*s3*s1*s3, s1*s2*s3*s2*s1*s2*s1*s2*s3*s2*s1*s2, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
s2*s3*s2*s1*s3*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s2*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 >;

to this polytope