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}*1728a
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 : {3,2,6,12}*864d
   3-fold quotients : {3,2,6,4}*576
   4-fold quotients : {3,2,6,6}*432a
   6-fold quotients : {3,2,3,4}*288, {3,2,6,4}*288b, {3,2,6,4}*288c
   12-fold quotients : {3,2,3,4}*144, {3,2,2,6}*144, {3,2,6,2}*144
   24-fold quotients : {3,2,2,3}*72, {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)( 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)( 8,24)( 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);;
s4 := ( 4,47)( 5,46)( 6,45)( 7,44)( 8,43)( 9,42)(10,41)(11,40)(12,51)(13,50)
(14,49)(15,48)(16,59)(17,58)(18,57)(19,56)(20,55)(21,54)(22,53)(23,52)(24,63)
(25,62)(26,61)(27,60)(28,71)(29,70)(30,69)(31,68)(32,67)(33,66)(34,65)(35,64)
(36,75)(37,74)(38,73)(39,72);;
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, 
s4*s2*s3*s4*s2*s3*s2*s3*s4*s2*s3*s4*s2*s3*s2*s3, 
s4*s2*s3*s4*s3*s4*s2*s3*s4*s2*s3*s4*s3*s4*s2*s3, 
s2*s3*s4*s3*s2*s3*s4*s3*s2*s3*s4*s3*s2*s3*s4*s3 ];;
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)( 8,24)( 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);
s4 := Sym(75)!( 4,47)( 5,46)( 6,45)( 7,44)( 8,43)( 9,42)(10,41)(11,40)(12,51)
(13,50)(14,49)(15,48)(16,59)(17,58)(18,57)(19,56)(20,55)(21,54)(22,53)(23,52)
(24,63)(25,62)(26,61)(27,60)(28,71)(29,70)(30,69)(31,68)(32,67)(33,66)(34,65)
(35,64)(36,75)(37,74)(38,73)(39,72);
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, 
s4*s2*s3*s4*s2*s3*s2*s3*s4*s2*s3*s4*s2*s3*s2*s3, 
s4*s2*s3*s4*s3*s4*s2*s3*s4*s2*s3*s4*s3*s4*s2*s3, 
s2*s3*s4*s3*s2*s3*s4*s3*s2*s3*s4*s3*s2*s3*s4*s3 >;

to this polytope