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

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {3,2,4,36}*1728a
if this polytope has a name.
Group : SmallGroup(1728,14461)
Rank : 5
Schlafli Type : {3,2,4,36}
Number of vertices, edges, etc : 3, 3, 4, 72, 36
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 : {3,2,2,36}*864, {3,2,4,18}*864a
   3-fold quotients : {3,2,4,12}*576a
   4-fold quotients : {3,2,2,18}*432
   6-fold quotients : {3,2,2,12}*288, {3,2,4,6}*288a
   8-fold quotients : {3,2,2,9}*216
   9-fold quotients : {3,2,4,4}*192
   12-fold quotients : {3,2,2,6}*144
   18-fold quotients : {3,2,2,4}*96, {3,2,4,2}*96
   24-fold quotients : {3,2,2,3}*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 := (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);;
s3 := ( 4,40)( 5,42)( 6,41)( 7,47)( 8,46)( 9,48)(10,44)(11,43)(12,45)(13,49)
(14,51)(15,50)(16,56)(17,55)(18,57)(19,53)(20,52)(21,54)(22,58)(23,60)(24,59)
(25,65)(26,64)(27,66)(28,62)(29,61)(30,63)(31,67)(32,69)(33,68)(34,74)(35,73)
(36,75)(37,71)(38,70)(39,72);;
s4 := ( 4, 7)( 5, 9)( 6, 8)(10,11)(13,16)(14,18)(15,17)(19,20)(22,25)(23,27)
(24,26)(28,29)(31,34)(32,36)(33,35)(37,38)(40,61)(41,63)(42,62)(43,58)(44,60)
(45,59)(46,65)(47,64)(48,66)(49,70)(50,72)(51,71)(52,67)(53,69)(54,68)(55,74)
(56,73)(57,75);;
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*s2*s3*s2*s3*s2*s3, 
s2*s3*s4*s3*s2*s3*s4*s3, s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4 ];;
poly := F / rels;;

Permutation Representation (Magma) :

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

to this polytope