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

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

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

Permutation Representation (Magma) :

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

to this polytope