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

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {3,2,4,3,6}*1728
if this polytope has a name.
Group : SmallGroup(1728,47874)
Rank : 6
Schlafli Type : {3,2,4,3,6}
Number of vertices, edges, etc : 3, 3, 8, 12, 18, 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) :
   2-fold quotients : {3,2,4,3,6}*864
   3-fold quotients : {3,2,4,3,2}*576
   4-fold quotients : {3,2,2,3,6}*432
   6-fold quotients : {3,2,4,3,2}*288
   12-fold quotients : {3,2,2,3,2}*144
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

to this polytope