Polytope of Type {3,2,6,24}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {3,2,6,24}*1728a
if this polytope has a name.
Group : SmallGroup(1728,33799)
Rank : 5
Schlafli Type : {3,2,6,24}
Number of vertices, edges, etc : 3, 3, 6, 72, 24
Order of s₀s₁s₂s₃s₄ : 24
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 : {3,2,6,12}*864a
   3-fold quotients : {3,2,2,24}*576, {3,2,6,8}*576
   4-fold quotients : {3,2,6,6}*432a
   6-fold quotients : {3,2,2,12}*288, {3,2,6,4}*288a
   9-fold quotients : {3,2,2,8}*192
   12-fold quotients : {3,2,2,6}*144, {3,2,6,2}*144
   18-fold quotients : {3,2,2,4}*96
   24-fold quotients : {3,2,2,3}*72, {3,2,3,2}*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 := ( 5, 6)( 8, 9)(11,12)(14,15)(17,18)(20,21)(23,24)(26,27)(29,30)(32,33)
(35,36)(38,39)(41,42)(44,45)(47,48)(50,51)(53,54)(56,57)(59,60)(62,63)(65,66)
(68,69)(71,72)(74,75);;
s3 := ( 4, 5)( 7,11)( 8,10)( 9,12)(13,14)(16,20)(17,19)(18,21)(22,32)(23,31)
(24,33)(25,38)(26,37)(27,39)(28,35)(29,34)(30,36)(40,59)(41,58)(42,60)(43,65)
(44,64)(45,66)(46,62)(47,61)(48,63)(49,68)(50,67)(51,69)(52,74)(53,73)(54,75)
(55,71)(56,70)(57,72);;
s4 := ( 4,43)( 5,44)( 6,45)( 7,40)( 8,41)( 9,42)(10,46)(11,47)(12,48)(13,52)
(14,53)(15,54)(16,49)(17,50)(18,51)(19,55)(20,56)(21,57)(22,70)(23,71)(24,72)
(25,67)(26,68)(27,69)(28,73)(29,74)(30,75)(31,61)(32,62)(33,63)(34,58)(35,59)
(36,60)(37,64)(38,65)(39,66);;
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, 
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*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)!( 5, 6)( 8, 9)(11,12)(14,15)(17,18)(20,21)(23,24)(26,27)(29,30)
(32,33)(35,36)(38,39)(41,42)(44,45)(47,48)(50,51)(53,54)(56,57)(59,60)(62,63)
(65,66)(68,69)(71,72)(74,75);
s3 := Sym(75)!( 4, 5)( 7,11)( 8,10)( 9,12)(13,14)(16,20)(17,19)(18,21)(22,32)
(23,31)(24,33)(25,38)(26,37)(27,39)(28,35)(29,34)(30,36)(40,59)(41,58)(42,60)
(43,65)(44,64)(45,66)(46,62)(47,61)(48,63)(49,68)(50,67)(51,69)(52,74)(53,73)
(54,75)(55,71)(56,70)(57,72);
s4 := Sym(75)!( 4,43)( 5,44)( 6,45)( 7,40)( 8,41)( 9,42)(10,46)(11,47)(12,48)
(13,52)(14,53)(15,54)(16,49)(17,50)(18,51)(19,55)(20,56)(21,57)(22,70)(23,71)
(24,72)(25,67)(26,68)(27,69)(28,73)(29,74)(30,75)(31,61)(32,62)(33,63)(34,58)
(35,59)(36,60)(37,64)(38,65)(39,66);
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, 
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*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4 >;

to this polytope