Polytope of Type {3,2,6,12}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {3,2,6,12}*1728b
if this polytope has a name.
Group : SmallGroup(1728,47874)
Rank : 5
Schlafli Type : {3,2,6,12}
Number of vertices, edges, etc : 3, 3, 12, 72, 24
Order of s₀s₁s₂s₃s₄ : 6
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 : {3,2,3,12}*864
   3-fold quotients : {3,2,6,4}*576
   4-fold quotients : {3,2,6,6}*432c
   6-fold quotients : {3,2,3,4}*288, {3,2,6,4}*288b, {3,2,6,4}*288c
   8-fold quotients : {3,2,3,6}*216
   12-fold quotients : {3,2,3,4}*144, {3,2,6,2}*144
   24-fold quotients : {3,2,3,2}*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 := ( 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);;
s3 := ( 4,56)( 5,57)( 6,59)( 7,58)( 8,52)( 9,53)(10,55)(11,54)(12,60)(13,61)
(14,63)(15,62)(16,44)(17,45)(18,47)(19,46)(20,40)(21,41)(22,43)(23,42)(24,48)
(25,49)(26,51)(27,50)(28,68)(29,69)(30,71)(31,70)(32,64)(33,65)(34,67)(35,66)
(36,72)(37,73)(38,75)(39,74);;
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);;
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, s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
s2*s3*s4*s3*s2*s3*s2*s3*s4*s3*s2*s3, 
s4*s2*s3*s4*s2*s3*s4*s2*s3*s4*s2*s3*s4*s2*s3*s4*s2*s3, 
s4*s2*s3*s4*s3*s4*s3*s4*s3*s4*s2*s3*s4*s3*s4*s3*s4*s3 ];;
poly := F / rels;;

Permutation Representation (Magma) :

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

to this polytope