Polytope of Type {3,2,4,9}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {3,2,4,9}*864
if this polytope has a name.
Group : SmallGroup(864,3999)
Rank : 5
Schlafli Type : {3,2,4,9}
Number of vertices, edges, etc : 3, 3, 8, 36, 18
Order of s₀s₁s₂s₃s₄ : 18
Order of s₀s₁s₂s₃s₄s₃s₂s₁ : 2
Special Properties :
   Degenerate
   Universal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {3,2,4,9,2} of size 1728
Vertex Figure Of :
   {2,3,2,4,9} of size 1728
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {3,2,4,9}*432
   3-fold quotients : {3,2,4,3}*288
   4-fold quotients : {3,2,2,9}*216
   6-fold quotients : {3,2,4,3}*144
   12-fold quotients : {3,2,2,3}*72
Covers (Minimal Covers in Boldface) :
   2-fold covers : {3,2,8,9}*1728, {3,2,4,18}*1728, {6,2,4,9}*1728
Permutation Representation (GAP) :

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

to this polytope