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

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {3,2,6,3,4}*1728
if this polytope has a name.
Group : SmallGroup(1728,47874)
Rank : 6
Schlafli Type : {3,2,6,3,4}
Number of vertices, edges, etc : 3, 3, 6, 18, 12, 8
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 : {3,2,6,3,4}*864
   3-fold quotients : {3,2,2,3,4}*576
   4-fold quotients : {3,2,6,3,2}*432
   6-fold quotients : {3,2,2,3,4}*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 := ( 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);;
s3 := ( 4, 8)( 5,10)( 6, 9)( 7,11)(13,14)(16,32)(17,34)(18,33)(19,35)(20,28)
(21,30)(22,29)(23,31)(24,36)(25,38)(26,37)(27,39)(40,44)(41,46)(42,45)(43,47)
(49,50)(52,68)(53,70)(54,69)(55,71)(56,64)(57,66)(58,65)(59,67)(60,72)(61,74)
(62,73)(63,75);;
s4 := ( 4,16)( 5,17)( 6,19)( 7,18)( 8,24)( 9,25)(10,27)(11,26)(12,20)(13,21)
(14,23)(15,22)(30,31)(32,36)(33,37)(34,39)(35,38)(40,52)(41,53)(42,55)(43,54)
(44,60)(45,61)(46,63)(47,62)(48,56)(49,57)(50,59)(51,58)(66,67)(68,72)(69,73)
(70,75)(71,74);;
s5 := ( 4,43)( 5,42)( 6,41)( 7,40)( 8,47)( 9,46)(10,45)(11,44)(12,51)(13,50)
(14,49)(15,48)(16,55)(17,54)(18,53)(19,52)(20,59)(21,58)(22,57)(23,56)(24,63)
(25,62)(26,61)(27,60)(28,67)(29,66)(30,65)(31,64)(32,71)(33,70)(34,69)(35,68)
(36,75)(37,74)(38,73)(39,72);;
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, s4*s5*s4*s5*s4*s5*s4*s5, 
s4*s2*s3*s2*s3*s4*s2*s3*s2*s3 ];;
poly := F / rels;;

Permutation Representation (Magma) :

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

to this polytope