Polytope of Type {3,2,6,4}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {3,2,6,4}*864
if this polytope has a name.
Group : SmallGroup(864,4686)
Rank : 5
Schlafli Type : {3,2,6,4}
Number of vertices, edges, etc : 3, 3, 18, 36, 12
Order of s₀s₁s₂s₃s₄ : 12
Order of s₀s₁s₂s₃s₄s₃s₂s₁ : 2
Special Properties :
   Degenerate
   Universal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {3,2,6,4,2} of size 1728
Vertex Figure Of :
   {2,3,2,6,4} of size 1728
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {3,2,6,4}*432
   9-fold quotients : {3,2,2,4}*96
   18-fold quotients : {3,2,2,2}*48
Covers (Minimal Covers in Boldface) :
   2-fold covers : {3,2,6,8}*1728, {3,2,12,4}*1728, {6,2,6,4}*1728
Permutation Representation (GAP) :

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

to this polytope