Polytope of Type {2,9,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,9,6}*648a
if this polytope has a name.
Group : SmallGroup(648,297)
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
   {2,9,6,3} of size 1944
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 : {2,9,6}*216, {2,3,6}*216
   9-fold quotients : {2,9,2}*72, {2,3,6}*72
   27-fold quotients : {2,3,2}*24
Covers (Minimal Covers in Boldface) :
   2-fold covers : {2,18,6}*1296a
   3-fold covers : {2,9,18}*1944a, {2,27,6}*1944a, {2,9,6}*1944d, {2,9,18}*1944h, {2,9,18}*1944i, {2,9,6}*1944e, {2,27,6}*1944b, {2,27,6}*1944c, {6,9,6}*1944a
Permutation Representation (GAP) :

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

to this polytope