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

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {3,2,8,18}*1728
if this polytope has a name.
Group : SmallGroup(1728,15957)
Rank : 5
Schlafli Type : {3,2,8,18}
Number of vertices, edges, etc : 3, 3, 8, 72, 18
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 : {3,2,4,18}*864a
   3-fold quotients : {3,2,8,6}*576
   4-fold quotients : {3,2,2,18}*432
   6-fold quotients : {3,2,4,6}*288a
   8-fold quotients : {3,2,2,9}*216
   9-fold quotients : {3,2,8,2}*192
   12-fold quotients : {3,2,2,6}*144
   18-fold quotients : {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 := (22,31)(23,32)(24,33)(25,34)(26,35)(27,36)(28,37)(29,38)(30,39)(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)(55,73)(56,74)(57,75);;
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)(22,67)(23,69)(24,68)
(25,74)(26,73)(27,75)(28,71)(29,70)(30,72)(31,58)(32,60)(33,59)(34,65)(35,64)
(36,66)(37,62)(38,61)(39,63);;
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)(40,43)(41,45)(42,44)(46,47)(49,52)
(50,54)(51,53)(55,56)(58,61)(59,63)(60,62)(64,65)(67,70)(68,72)(69,71)
(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, 
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, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*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 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(75)!(2,3);
s1 := Sym(75)!(1,2);
s2 := Sym(75)!(22,31)(23,32)(24,33)(25,34)(26,35)(27,36)(28,37)(29,38)(30,39)
(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)(55,73)(56,74)(57,75);
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)(22,67)(23,69)
(24,68)(25,74)(26,73)(27,75)(28,71)(29,70)(30,72)(31,58)(32,60)(33,59)(34,65)
(35,64)(36,66)(37,62)(38,61)(39,63);
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)(40,43)(41,45)(42,44)(46,47)
(49,52)(50,54)(51,53)(55,56)(58,61)(59,63)(60,62)(64,65)(67,70)(68,72)(69,71)
(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, 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, s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*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 >;

to this polytope