Polytope of Type {12,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {12,6}*648
if this polytope has a name.
Group : SmallGroup(648,547)
Rank : 3
Schlafli Type : {12,6}
Number of vertices, edges, etc : 54, 162, 27
Order of s₀s₁s₂ : 12
Order of s₀s₁s₂s₁ : 6
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Non-Orientable
   Self-Petrie
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
Facet Of :
   {12,6,2} of size 1296
Vertex Figure Of :
   {2,12,6} of size 1296
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {4,6}*216, {12,6}*216a, {12,6}*216b, {12,6}*216c
   9-fold quotients : {4,6}*72
Covers (Minimal Covers in Boldface) :
   2-fold covers : {12,6}*1296o
   3-fold covers : {36,6}*1944
Permutation Representation (GAP) :

s0 := ( 2, 3)( 4, 7)( 5, 9)( 6, 8)(10,66)(11,65)(12,64)(13,72)(14,71)(15,70)
(16,69)(17,68)(18,67)(19,46)(20,48)(21,47)(22,52)(23,54)(24,53)(25,49)(26,51)
(27,50)(28,55)(29,57)(30,56)(31,61)(32,63)(33,62)(34,58)(35,60)(36,59)(37,39)
(40,45)(41,44)(42,43)(74,75)(76,79)(77,81)(78,80);;
s1 := ( 1, 4)( 2, 6)( 3, 5)( 8, 9)(10,22)(11,24)(12,23)(13,19)(14,21)(15,20)
(16,25)(17,27)(18,26)(28,41)(29,40)(30,42)(31,38)(32,37)(33,39)(34,44)(35,43)
(36,45)(46,50)(47,49)(48,51)(52,53)(55,76)(56,78)(57,77)(58,73)(59,75)(60,74)
(61,79)(62,81)(63,80)(64,67)(65,69)(66,68)(71,72);;
s2 := ( 1,38)( 2,39)( 3,37)( 4,41)( 5,42)( 6,40)( 7,44)( 8,45)( 9,43)(10,30)
(11,28)(12,29)(13,33)(14,31)(15,32)(16,36)(17,34)(18,35)(19,46)(20,47)(21,48)
(22,49)(23,50)(24,51)(25,52)(26,53)(27,54)(55,65)(56,66)(57,64)(58,68)(59,69)
(60,67)(61,71)(62,72)(63,70);;
poly := Group([s0,s1,s2]);;

Finitely Presented Group Representation (GAP) :

F := FreeGroup("s0","s1","s2");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  
rels := [ s0*s0, s1*s1, s2*s2, s0*s2*s0*s2, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2, 
s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(81)!( 2, 3)( 4, 7)( 5, 9)( 6, 8)(10,66)(11,65)(12,64)(13,72)(14,71)
(15,70)(16,69)(17,68)(18,67)(19,46)(20,48)(21,47)(22,52)(23,54)(24,53)(25,49)
(26,51)(27,50)(28,55)(29,57)(30,56)(31,61)(32,63)(33,62)(34,58)(35,60)(36,59)
(37,39)(40,45)(41,44)(42,43)(74,75)(76,79)(77,81)(78,80);
s1 := Sym(81)!( 1, 4)( 2, 6)( 3, 5)( 8, 9)(10,22)(11,24)(12,23)(13,19)(14,21)
(15,20)(16,25)(17,27)(18,26)(28,41)(29,40)(30,42)(31,38)(32,37)(33,39)(34,44)
(35,43)(36,45)(46,50)(47,49)(48,51)(52,53)(55,76)(56,78)(57,77)(58,73)(59,75)
(60,74)(61,79)(62,81)(63,80)(64,67)(65,69)(66,68)(71,72);
s2 := Sym(81)!( 1,38)( 2,39)( 3,37)( 4,41)( 5,42)( 6,40)( 7,44)( 8,45)( 9,43)
(10,30)(11,28)(12,29)(13,33)(14,31)(15,32)(16,36)(17,34)(18,35)(19,46)(20,47)
(21,48)(22,49)(23,50)(24,51)(25,52)(26,53)(27,54)(55,65)(56,66)(57,64)(58,68)
(59,69)(60,67)(61,71)(62,72)(63,70);
poly := sub<Sym(81)|s0,s1,s2>;

Finitely Presented Group Representation (Magma) :

poly<s0,s1,s2> := Group< s0,s1,s2 | s0*s0, s1*s1, s2*s2, 
s0*s2*s0*s2, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2, 
s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1 >;

References : None.
to this polytope