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

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

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

Permutation Representation (Magma) :

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

to this polytope