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}*1728c
if this polytope has a name.
Group : SmallGroup(1728,37593)
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}*864c
   3-fold quotients : {3,2,6,8}*576
   4-fold quotients : {3,2,6,6}*432c
   6-fold quotients : {3,2,6,4}*288a
   8-fold quotients : {3,2,3,6}*216
   9-fold quotients : {3,2,2,8}*192
   12-fold quotients : {3,2,6,2}*144
   18-fold quotients : {3,2,2,4}*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)(23,24)(25,28)
(26,30)(27,29)(32,33)(34,37)(35,39)(36,38)(41,42)(43,46)(44,48)(45,47)(50,51)
(52,55)(53,57)(54,56)(59,60)(61,64)(62,66)(63,65)(68,69)(70,73)(71,75)
(72,74);;
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 := ( 7,10)( 8,11)( 9,12)(16,19)(17,20)(18,21)(22,31)(23,32)(24,33)(25,37)
(26,38)(27,39)(28,34)(29,35)(30,36)(40,58)(41,59)(42,60)(43,64)(44,65)(45,66)
(46,61)(47,62)(48,63)(49,67)(50,68)(51,69)(52,73)(53,74)(54,75)(55,70)(56,71)
(57,72);;
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, s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
s4*s2*s3*s4*s3*s2*s3*s4*s2*s3*s4*s3*s2*s3, 
s4*s2*s3*s4*s3*s4*s3*s4*s3*s4*s2*s3*s4*s3*s4*s3*s4*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)(23,24)
(25,28)(26,30)(27,29)(32,33)(34,37)(35,39)(36,38)(41,42)(43,46)(44,48)(45,47)
(50,51)(52,55)(53,57)(54,56)(59,60)(61,64)(62,66)(63,65)(68,69)(70,73)(71,75)
(72,74);
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)!( 7,10)( 8,11)( 9,12)(16,19)(17,20)(18,21)(22,31)(23,32)(24,33)
(25,37)(26,38)(27,39)(28,34)(29,35)(30,36)(40,58)(41,59)(42,60)(43,64)(44,65)
(45,66)(46,61)(47,62)(48,63)(49,67)(50,68)(51,69)(52,73)(53,74)(54,75)(55,70)
(56,71)(57,72);
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, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
s4*s2*s3*s4*s3*s2*s3*s4*s2*s3*s4*s3*s2*s3, 
s4*s2*s3*s4*s3*s4*s3*s4*s3*s4*s2*s3*s4*s3*s4*s3*s4*s3 >;

to this polytope