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

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,12,2,6}*1728a
if this polytope has a name.
Group : SmallGroup(1728,47319)
Rank : 5
Schlafli Type : {6,12,2,6}
Number of vertices, edges, etc : 6, 36, 12, 6, 6
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 : {6,12,2,3}*864a, {6,6,2,6}*864a
   3-fold quotients : {2,12,2,6}*576, {6,12,2,2}*576a, {6,4,2,6}*576a
   4-fold quotients : {6,6,2,3}*432a
   6-fold quotients : {2,12,2,3}*288, {6,4,2,3}*288a, {2,6,2,6}*288, {6,2,2,6}*288, {6,6,2,2}*288a
   9-fold quotients : {2,12,2,2}*192, {2,4,2,6}*192, {6,4,2,2}*192a
   12-fold quotients : {2,3,2,6}*144, {2,6,2,3}*144, {3,2,2,6}*144, {6,2,2,3}*144
   18-fold quotients : {2,4,2,3}*96, {2,2,2,6}*96, {2,6,2,2}*96, {6,2,2,2}*96
   24-fold quotients : {2,3,2,3}*72, {3,2,2,3}*72
   27-fold quotients : {2,4,2,2}*64
   36-fold quotients : {2,2,2,3}*48, {2,3,2,2}*48, {3,2,2,2}*48
   54-fold quotients : {2,2,2,2}*32
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

s0 := ( 2, 3)( 5, 6)( 8, 9)(11,12)(14,15)(17,18)(20,21)(23,24)(26,27)(29,30)
(32,33)(35,36)(38,39)(41,42)(44,45)(47,48)(50,51)(53,54)(56,57)(59,60)(62,63)
(65,66)(68,69)(71,72);;
s1 := ( 1,38)( 2,37)( 3,39)( 4,44)( 5,43)( 6,45)( 7,41)( 8,40)( 9,42)(10,47)
(11,46)(12,48)(13,53)(14,52)(15,54)(16,50)(17,49)(18,51)(19,65)(20,64)(21,66)
(22,71)(23,70)(24,72)(25,68)(26,67)(27,69)(28,56)(29,55)(30,57)(31,62)(32,61)
(33,63)(34,59)(35,58)(36,60);;
s2 := ( 1,58)( 2,59)( 3,60)( 4,55)( 5,56)( 6,57)( 7,61)( 8,62)( 9,63)(10,67)
(11,68)(12,69)(13,64)(14,65)(15,66)(16,70)(17,71)(18,72)(19,40)(20,41)(21,42)
(22,37)(23,38)(24,39)(25,43)(26,44)(27,45)(28,49)(29,50)(30,51)(31,46)(32,47)
(33,48)(34,52)(35,53)(36,54);;
s3 := (75,76)(77,78);;
s4 := (73,77)(74,75)(76,78);;
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, 
s0*s1*s2*s1*s0*s1*s2*s1, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(78)!( 2, 3)( 5, 6)( 8, 9)(11,12)(14,15)(17,18)(20,21)(23,24)(26,27)
(29,30)(32,33)(35,36)(38,39)(41,42)(44,45)(47,48)(50,51)(53,54)(56,57)(59,60)
(62,63)(65,66)(68,69)(71,72);
s1 := Sym(78)!( 1,38)( 2,37)( 3,39)( 4,44)( 5,43)( 6,45)( 7,41)( 8,40)( 9,42)
(10,47)(11,46)(12,48)(13,53)(14,52)(15,54)(16,50)(17,49)(18,51)(19,65)(20,64)
(21,66)(22,71)(23,70)(24,72)(25,68)(26,67)(27,69)(28,56)(29,55)(30,57)(31,62)
(32,61)(33,63)(34,59)(35,58)(36,60);
s2 := Sym(78)!( 1,58)( 2,59)( 3,60)( 4,55)( 5,56)( 6,57)( 7,61)( 8,62)( 9,63)
(10,67)(11,68)(12,69)(13,64)(14,65)(15,66)(16,70)(17,71)(18,72)(19,40)(20,41)
(21,42)(22,37)(23,38)(24,39)(25,43)(26,44)(27,45)(28,49)(29,50)(30,51)(31,46)
(32,47)(33,48)(34,52)(35,53)(36,54);
s3 := Sym(78)!(75,76)(77,78);
s4 := Sym(78)!(73,77)(74,75)(76,78);
poly := sub<Sym(78)|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, s0*s1*s2*s1*s0*s1*s2*s1, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 >;

to this polytope