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

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

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

to this polytope