Polytope of Type {3,2,36,4}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {3,2,36,4}*1728a
if this polytope has a name.
Group : SmallGroup(1728,14461)
Rank : 5
Schlafli Type : {3,2,36,4}
Number of vertices, edges, etc : 3, 3, 36, 72, 4
Order of s₀s₁s₂s₃s₄ : 36
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) :
   2-fold quotients : {3,2,36,2}*864, {3,2,18,4}*864a
   3-fold quotients : {3,2,12,4}*576a
   4-fold quotients : {3,2,18,2}*432
   6-fold quotients : {3,2,12,2}*288, {3,2,6,4}*288a
   8-fold quotients : {3,2,9,2}*216
   9-fold quotients : {3,2,4,4}*192
   12-fold quotients : {3,2,6,2}*144
   18-fold quotients : {3,2,2,4}*96, {3,2,4,2}*96
   24-fold quotients : {3,2,3,2}*72
   36-fold quotients : {3,2,2,2}*48
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

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

to this polytope