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

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,24,2,3}*1728a
if this polytope has a name.
Group : SmallGroup(1728,33799)
Rank : 5
Schlafli Type : {6,24,2,3}
Number of vertices, edges, etc : 6, 72, 24, 3, 3
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 : {6,12,2,3}*864a
   3-fold quotients : {2,24,2,3}*576, {6,8,2,3}*576
   4-fold quotients : {6,6,2,3}*432a
   6-fold quotients : {2,12,2,3}*288, {6,4,2,3}*288a
   9-fold quotients : {2,8,2,3}*192
   12-fold quotients : {2,6,2,3}*144, {6,2,2,3}*144
   18-fold quotients : {2,4,2,3}*96
   24-fold quotients : {2,3,2,3}*72, {3,2,2,3}*72
   36-fold quotients : {2,2,2,3}*48
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

to this polytope