Polytope of Type {2,6,18}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,6,18}*648a
if this polytope has a name.
Group : SmallGroup(648,297)
Rank : 4
Schlafli Type : {2,6,18}
Number of vertices, edges, etc : 2, 9, 81, 27
Order of s₀s₁s₂s₃ : 18
Order of s₀s₁s₂s₃s₂s₁ : 2
Special Properties :
   Degenerate
   Universal
   Non-Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {2,6,18,2} of size 1296
Vertex Figure Of :
   {2,2,6,18} of size 1296
   {3,2,6,18} of size 1944
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {2,6,6}*216
Covers (Minimal Covers in Boldface) :
   2-fold covers : {2,6,18}*1296b
   3-fold covers : {2,18,18}*1944a, {2,6,54}*1944a, {2,6,18}*1944c, {2,18,18}*1944g, {2,18,18}*1944i, {2,6,18}*1944d, {2,6,54}*1944b, {2,6,54}*1944c, {6,6,18}*1944a
Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

to this polytope