Polytope of Type {3,2,3,4,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {3,2,3,4,6}*1728
if this polytope has a name.
Group : SmallGroup(1728,47874)
Rank : 6
Schlafli Type : {3,2,3,4,6}
Number of vertices, edges, etc : 3, 3, 6, 12, 24, 6
Order of s₀s₁s₂s₃s₄s₅ : 6
Order of s₀s₁s₂s₃s₄s₅s₄s₃s₂s₁ : 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) :
   3-fold quotients : {3,2,3,4,2}*576
   4-fold quotients : {3,2,3,2,6}*432
   6-fold quotients : {3,2,3,4,2}*288
   8-fold quotients : {3,2,3,2,3}*216
   12-fold quotients : {3,2,3,2,2}*144
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

s0 := (2,3);;
s1 := (1,2);;
s2 := ( 5, 6)( 9,10)(13,14)(16,28)(17,30)(18,29)(19,31)(20,32)(21,34)(22,33)
(23,35)(24,36)(25,38)(26,37)(27,39)(41,42)(45,46)(49,50)(52,64)(53,66)(54,65)
(55,67)(56,68)(57,70)(58,69)(59,71)(60,72)(61,74)(62,73)(63,75);;
s3 := ( 4,16)( 5,17)( 6,19)( 7,18)( 8,20)( 9,21)(10,23)(11,22)(12,24)(13,25)
(14,27)(15,26)(30,31)(34,35)(38,39)(40,52)(41,53)(42,55)(43,54)(44,56)(45,57)
(46,59)(47,58)(48,60)(49,61)(50,63)(51,62)(66,67)(70,71)(74,75);;
s4 := ( 4, 7)( 5, 6)( 8,15)( 9,14)(10,13)(11,12)(16,19)(17,18)(20,27)(21,26)
(22,25)(23,24)(28,31)(29,30)(32,39)(33,38)(34,37)(35,36)(40,43)(41,42)(44,51)
(45,50)(46,49)(47,48)(52,55)(53,54)(56,63)(57,62)(58,61)(59,60)(64,67)(65,66)
(68,75)(69,74)(70,73)(71,72);;
s5 := ( 4,44)( 5,45)( 6,46)( 7,47)( 8,40)( 9,41)(10,42)(11,43)(12,48)(13,49)
(14,50)(15,51)(16,56)(17,57)(18,58)(19,59)(20,52)(21,53)(22,54)(23,55)(24,60)
(25,61)(26,62)(27,63)(28,68)(29,69)(30,70)(31,71)(32,64)(33,65)(34,66)(35,67)
(36,72)(37,73)(38,74)(39,75);;
poly := Group([s0,s1,s2,s3,s4,s5]);;

Finitely Presented Group Representation (GAP) :

F := FreeGroup("s0","s1","s2","s3","s4","s5");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;  s4 := F.5;;  s5 := F.6;;  
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s4*s4, s5*s5, 
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*s5*s0*s5, s1*s5*s1*s5, 
s2*s5*s2*s5, s3*s5*s3*s5, s0*s1*s0*s1*s0*s1, 
s2*s3*s2*s3*s2*s3, s3*s4*s3*s4*s3*s4*s3*s4, 
s3*s4*s5*s4*s3*s4*s5*s4, s4*s5*s4*s5*s4*s5*s4*s5*s4*s5*s4*s5 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(75)!(2,3);
s1 := Sym(75)!(1,2);
s2 := Sym(75)!( 5, 6)( 9,10)(13,14)(16,28)(17,30)(18,29)(19,31)(20,32)(21,34)
(22,33)(23,35)(24,36)(25,38)(26,37)(27,39)(41,42)(45,46)(49,50)(52,64)(53,66)
(54,65)(55,67)(56,68)(57,70)(58,69)(59,71)(60,72)(61,74)(62,73)(63,75);
s3 := Sym(75)!( 4,16)( 5,17)( 6,19)( 7,18)( 8,20)( 9,21)(10,23)(11,22)(12,24)
(13,25)(14,27)(15,26)(30,31)(34,35)(38,39)(40,52)(41,53)(42,55)(43,54)(44,56)
(45,57)(46,59)(47,58)(48,60)(49,61)(50,63)(51,62)(66,67)(70,71)(74,75);
s4 := Sym(75)!( 4, 7)( 5, 6)( 8,15)( 9,14)(10,13)(11,12)(16,19)(17,18)(20,27)
(21,26)(22,25)(23,24)(28,31)(29,30)(32,39)(33,38)(34,37)(35,36)(40,43)(41,42)
(44,51)(45,50)(46,49)(47,48)(52,55)(53,54)(56,63)(57,62)(58,61)(59,60)(64,67)
(65,66)(68,75)(69,74)(70,73)(71,72);
s5 := Sym(75)!( 4,44)( 5,45)( 6,46)( 7,47)( 8,40)( 9,41)(10,42)(11,43)(12,48)
(13,49)(14,50)(15,51)(16,56)(17,57)(18,58)(19,59)(20,52)(21,53)(22,54)(23,55)
(24,60)(25,61)(26,62)(27,63)(28,68)(29,69)(30,70)(31,71)(32,64)(33,65)(34,66)
(35,67)(36,72)(37,73)(38,74)(39,75);
poly := sub<Sym(75)|s0,s1,s2,s3,s4,s5>;

Finitely Presented Group Representation (Magma) :

poly<s0,s1,s2,s3,s4,s5> := Group< s0,s1,s2,s3,s4,s5 | s0*s0, s1*s1, s2*s2, 
s3*s3, s4*s4, s5*s5, 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*s5*s0*s5, 
s1*s5*s1*s5, s2*s5*s2*s5, s3*s5*s3*s5, 
s0*s1*s0*s1*s0*s1, s2*s3*s2*s3*s2*s3, 
s3*s4*s3*s4*s3*s4*s3*s4, s3*s4*s5*s4*s3*s4*s5*s4, 
s4*s5*s4*s5*s4*s5*s4*s5*s4*s5*s4*s5 >;

to this polytope