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}*1728c
if this polytope has a name.
Group : SmallGroup(1728,37593)
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}*864c
   3-fold quotients : {3,2,8,6}*576
   4-fold quotients : {3,2,6,6}*432b
   6-fold quotients : {3,2,4,6}*288a
   8-fold quotients : {3,2,6,3}*216
   9-fold quotients : {3,2,8,2}*192
   12-fold quotients : {3,2,2,6}*144
   18-fold quotients : {3,2,4,2}*96
   24-fold quotients : {3,2,2,3}*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 := ( 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);;
s3 := ( 4,43)( 5,45)( 6,44)( 7,40)( 8,42)( 9,41)(10,46)(11,48)(12,47)(13,52)
(14,54)(15,53)(16,49)(17,51)(18,50)(19,55)(20,57)(21,56)(22,70)(23,72)(24,71)
(25,67)(26,69)(27,68)(28,73)(29,75)(30,74)(31,61)(32,63)(33,62)(34,58)(35,60)
(36,59)(37,64)(38,66)(39,65);;
s4 := ( 4, 5)( 7,11)( 8,10)( 9,12)(13,14)(16,20)(17,19)(18,21)(22,23)(25,29)
(26,28)(27,30)(31,32)(34,38)(35,37)(36,39)(40,41)(43,47)(44,46)(45,48)(49,50)
(52,56)(53,55)(54,57)(58,59)(61,65)(62,64)(63,66)(67,68)(70,74)(71,73)
(72,75);;
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*s3*s4*s3*s4, 
s4*s2*s3*s4*s3*s2*s3*s4*s2*s3*s4*s3*s2*s3, 
s4*s2*s3*s2*s3*s2*s3*s2*s3*s4*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)!( 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);
s3 := Sym(75)!( 4,43)( 5,45)( 6,44)( 7,40)( 8,42)( 9,41)(10,46)(11,48)(12,47)
(13,52)(14,54)(15,53)(16,49)(17,51)(18,50)(19,55)(20,57)(21,56)(22,70)(23,72)
(24,71)(25,67)(26,69)(27,68)(28,73)(29,75)(30,74)(31,61)(32,63)(33,62)(34,58)
(35,60)(36,59)(37,64)(38,66)(39,65);
s4 := Sym(75)!( 4, 5)( 7,11)( 8,10)( 9,12)(13,14)(16,20)(17,19)(18,21)(22,23)
(25,29)(26,28)(27,30)(31,32)(34,38)(35,37)(36,39)(40,41)(43,47)(44,46)(45,48)
(49,50)(52,56)(53,55)(54,57)(58,59)(61,65)(62,64)(63,66)(67,68)(70,74)(71,73)
(72,75);
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*s3*s4*s3*s4, 
s4*s2*s3*s4*s3*s2*s3*s4*s2*s3*s4*s3*s2*s3, 
s4*s2*s3*s2*s3*s2*s3*s2*s3*s4*s2*s3*s2*s3*s2*s3*s2*s3 >;

to this polytope