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

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

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

to this polytope