Polytope of Type {8,18,2,3}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {8,18,2,3}*1728
if this polytope has a name.
Group : SmallGroup(1728,15957)
Rank : 5
Schlafli Type : {8,18,2,3}
Number of vertices, edges, etc : 8, 72, 18, 3, 3
Order of s₀s₁s₂s₃s₄ : 72
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 : {4,18,2,3}*864a
   3-fold quotients : {8,6,2,3}*576
   4-fold quotients : {2,18,2,3}*432
   6-fold quotients : {4,6,2,3}*288a
   8-fold quotients : {2,9,2,3}*216
   9-fold quotients : {8,2,2,3}*192
   12-fold quotients : {2,6,2,3}*144
   18-fold quotients : {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 := (19,28)(20,29)(21,30)(22,31)(23,32)(24,33)(25,34)(26,35)(27,36)(37,55)
(38,56)(39,57)(40,58)(41,59)(42,60)(43,61)(44,62)(45,63)(46,64)(47,65)(48,66)
(49,67)(50,68)(51,69)(52,70)(53,71)(54,72);;
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)(19,64)(20,66)(21,65)
(22,71)(23,70)(24,72)(25,68)(26,67)(27,69)(28,55)(29,57)(30,56)(31,62)(32,61)
(33,63)(34,59)(35,58)(36,60);;
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)(37,40)(38,42)(39,41)(43,44)(46,49)
(47,51)(48,50)(52,53)(55,58)(56,60)(57,59)(61,62)(64,67)(65,69)(66,68)
(70,71);;
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*s2*s1*s0*s1*s2*s1, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*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 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(75)!(19,28)(20,29)(21,30)(22,31)(23,32)(24,33)(25,34)(26,35)(27,36)
(37,55)(38,56)(39,57)(40,58)(41,59)(42,60)(43,61)(44,62)(45,63)(46,64)(47,65)
(48,66)(49,67)(50,68)(51,69)(52,70)(53,71)(54,72);
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)(19,64)(20,66)
(21,65)(22,71)(23,70)(24,72)(25,68)(26,67)(27,69)(28,55)(29,57)(30,56)(31,62)
(32,61)(33,63)(34,59)(35,58)(36,60);
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)(37,40)(38,42)(39,41)(43,44)
(46,49)(47,51)(48,50)(52,53)(55,58)(56,60)(57,59)(61,62)(64,67)(65,69)(66,68)
(70,71);
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*s2*s1*s0*s1*s2*s1, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*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 >;

to this polytope