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

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

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

Permutation Representation (Magma) :

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

to this polytope