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

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {12,12,2,3}*1728c
if this polytope has a name.
Group : SmallGroup(1728,37586)
Rank : 5
Schlafli Type : {12,12,2,3}
Number of vertices, edges, etc : 12, 72, 12, 3, 3
Order of s₀s₁s₂s₃s₄ : 12
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 : {12,6,2,3}*864b, {6,12,2,3}*864c
   3-fold quotients : {12,4,2,3}*576a
   4-fold quotients : {6,6,2,3}*432c
   6-fold quotients : {12,2,2,3}*288, {6,4,2,3}*288a
   8-fold quotients : {3,6,2,3}*216
   9-fold quotients : {4,4,2,3}*192
   12-fold quotients : {6,2,2,3}*144
   18-fold quotients : {2,4,2,3}*96, {4,2,2,3}*96
   24-fold quotients : {3,2,2,3}*72
   36-fold quotients : {2,2,2,3}*48
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

to this polytope