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

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {3,2,24,6}*1728b
if this polytope has a name.
Group : SmallGroup(1728,33799)
Rank : 5
Schlafli Type : {3,2,24,6}
Number of vertices, edges, etc : 3, 3, 24, 72, 6
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 : {3,2,12,6}*864b
   3-fold quotients : {3,2,24,2}*576
   4-fold quotients : {3,2,6,6}*432c
   6-fold quotients : {3,2,12,2}*288
   8-fold quotients : {3,2,3,6}*216
   9-fold quotients : {3,2,8,2}*192
   12-fold quotients : {3,2,6,2}*144
   18-fold quotients : {3,2,4,2}*96
   24-fold quotients : {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)( 7,10)( 8,12)( 9,11)(14,15)(16,19)(17,21)(18,20)(22,31)(23,33)
(24,32)(25,37)(26,39)(27,38)(28,34)(29,36)(30,35)(40,58)(41,60)(42,59)(43,64)
(44,66)(45,65)(46,61)(47,63)(48,62)(49,67)(50,69)(51,68)(52,73)(53,75)(54,74)
(55,70)(56,72)(57,71);;
s3 := ( 4,44)( 5,43)( 6,45)( 7,41)( 8,40)( 9,42)(10,47)(11,46)(12,48)(13,53)
(14,52)(15,54)(16,50)(17,49)(18,51)(19,56)(20,55)(21,57)(22,71)(23,70)(24,72)
(25,68)(26,67)(27,69)(28,74)(29,73)(30,75)(31,62)(32,61)(33,63)(34,59)(35,58)
(36,60)(37,65)(38,64)(39,66);;
s4 := ( 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);;
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, s4*s2*s3*s4*s3*s4*s2*s3*s4*s3, 
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*s2*s3*s2*s3*s2*s3*s2*s3*s2*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)( 7,10)( 8,12)( 9,11)(14,15)(16,19)(17,21)(18,20)(22,31)
(23,33)(24,32)(25,37)(26,39)(27,38)(28,34)(29,36)(30,35)(40,58)(41,60)(42,59)
(43,64)(44,66)(45,65)(46,61)(47,63)(48,62)(49,67)(50,69)(51,68)(52,73)(53,75)
(54,74)(55,70)(56,72)(57,71);
s3 := Sym(75)!( 4,44)( 5,43)( 6,45)( 7,41)( 8,40)( 9,42)(10,47)(11,46)(12,48)
(13,53)(14,52)(15,54)(16,50)(17,49)(18,51)(19,56)(20,55)(21,57)(22,71)(23,70)
(24,72)(25,68)(26,67)(27,69)(28,74)(29,73)(30,75)(31,62)(32,61)(33,63)(34,59)
(35,58)(36,60)(37,65)(38,64)(39,66);
s4 := 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);
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, 
s4*s2*s3*s4*s3*s4*s2*s3*s4*s3, 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*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 >;

to this polytope