Polytope of Type {6,18,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,18,6}*1944a
if this polytope has a name.
Group : SmallGroup(1944,2340)
Rank : 4
Schlafli Type : {6,18,6}
Number of vertices, edges, etc : 6, 81, 81, 9
Order of s₀s₁s₂s₃ : 18
Order of s₀s₁s₂s₃s₂s₁ : 2
Special Properties :
   Universal
   Non-Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {2,18,6}*648a, {6,6,6}*648a
   9-fold quotients : {2,6,6}*216
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

References : None.
to this polytope