Polytope of Type {2,6,27,2}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,6,27,2}*1296
if this polytope has a name.
Group : SmallGroup(1296,1859)
Rank : 5
Schlafli Type : {2,6,27,2}
Number of vertices, edges, etc : 2, 6, 81, 27, 2
Order of s₀s₁s₂s₃s₄ : 54
Order of s₀s₁s₂s₃s₄s₃s₂s₁ : 2
Special Properties :
   Degenerate
   Universal
   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 : {2,2,27,2}*432, {2,6,9,2}*432
   9-fold quotients : {2,2,9,2}*144, {2,6,3,2}*144
   27-fold quotients : {2,2,3,2}*48
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

s0 := (1,2);;
s1 := (12,21)(13,22)(14,23)(15,24)(16,25)(17,26)(18,27)(19,28)(20,29)(39,48)
(40,49)(41,50)(42,51)(43,52)(44,53)(45,54)(46,55)(47,56)(66,75)(67,76)(68,77)
(69,78)(70,79)(71,80)(72,81)(73,82)(74,83);;
s2 := ( 3,12)( 4,14)( 5,13)( 6,19)( 7,18)( 8,20)( 9,16)(10,15)(11,17)(22,23)
(24,28)(25,27)(26,29)(30,69)(31,71)(32,70)(33,66)(34,68)(35,67)(36,73)(37,72)
(38,74)(39,60)(40,62)(41,61)(42,57)(43,59)(44,58)(45,64)(46,63)(47,65)(48,78)
(49,80)(50,79)(51,75)(52,77)(53,76)(54,82)(55,81)(56,83);;
s3 := ( 3,30)( 4,32)( 5,31)( 6,37)( 7,36)( 8,38)( 9,34)(10,33)(11,35)(12,48)
(13,50)(14,49)(15,55)(16,54)(17,56)(18,52)(19,51)(20,53)(21,39)(22,41)(23,40)
(24,46)(25,45)(26,47)(27,43)(28,42)(29,44)(57,60)(58,62)(59,61)(63,64)(66,78)
(67,80)(68,79)(69,75)(70,77)(71,76)(72,82)(73,81)(74,83);;
s4 := (84,85);;
poly := Group([s0,s1,s2,s3,s4]);;

Finitely Presented Group Representation (GAP) :

F := FreeGroup("s0","s1","s2","s3","s4");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;  s4 := F.5;;  
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s4*s4, s0*s1*s0*s1, 
s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s0*s4*s0*s4, s1*s4*s1*s4, s2*s4*s2*s4, 
s3*s4*s3*s4, s3*s1*s2*s1*s2*s3*s1*s2*s1*s2, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(85)!(1,2);
s1 := Sym(85)!(12,21)(13,22)(14,23)(15,24)(16,25)(17,26)(18,27)(19,28)(20,29)
(39,48)(40,49)(41,50)(42,51)(43,52)(44,53)(45,54)(46,55)(47,56)(66,75)(67,76)
(68,77)(69,78)(70,79)(71,80)(72,81)(73,82)(74,83);
s2 := Sym(85)!( 3,12)( 4,14)( 5,13)( 6,19)( 7,18)( 8,20)( 9,16)(10,15)(11,17)
(22,23)(24,28)(25,27)(26,29)(30,69)(31,71)(32,70)(33,66)(34,68)(35,67)(36,73)
(37,72)(38,74)(39,60)(40,62)(41,61)(42,57)(43,59)(44,58)(45,64)(46,63)(47,65)
(48,78)(49,80)(50,79)(51,75)(52,77)(53,76)(54,82)(55,81)(56,83);
s3 := Sym(85)!( 3,30)( 4,32)( 5,31)( 6,37)( 7,36)( 8,38)( 9,34)(10,33)(11,35)
(12,48)(13,50)(14,49)(15,55)(16,54)(17,56)(18,52)(19,51)(20,53)(21,39)(22,41)
(23,40)(24,46)(25,45)(26,47)(27,43)(28,42)(29,44)(57,60)(58,62)(59,61)(63,64)
(66,78)(67,80)(68,79)(69,75)(70,77)(71,76)(72,82)(73,81)(74,83);
s4 := Sym(85)!(84,85);
poly := sub<Sym(85)|s0,s1,s2,s3,s4>;

Finitely Presented Group Representation (Magma) :

poly<s0,s1,s2,s3,s4> := Group< s0,s1,s2,s3,s4 | s0*s0, s1*s1, s2*s2, 
s3*s3, s4*s4, s0*s1*s0*s1, s0*s2*s0*s2, 
s0*s3*s0*s3, s1*s3*s1*s3, s0*s4*s0*s4, 
s1*s4*s1*s4, s2*s4*s2*s4, s3*s4*s3*s4, 
s3*s1*s2*s1*s2*s3*s1*s2*s1*s2, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 >;

to this polytope