Polytope of Type {2,6,9}

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

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

Permutation Representation (Magma) :

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

to this polytope