Polytope of Type {4,6,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,6,6}*864k
if this polytope has a name.
Group : SmallGroup(864,4686)
Rank : 4
Schlafli Type : {4,6,6}
Number of vertices, edges, etc : 12, 36, 54, 6
Order of s₀s₁s₂s₃ : 12
Order of s₀s₁s₂s₃s₂s₁ : 6
Special Properties :
   Universal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {4,6,6,2} of size 1728
Vertex Figure Of :
   {2,4,6,6} of size 1728
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {4,6,3}*432b
   9-fold quotients : {4,2,6}*96
   18-fold quotients : {4,2,3}*48, {2,2,6}*48
   27-fold quotients : {4,2,2}*32
   36-fold quotients : {2,2,3}*24
   54-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
   2-fold covers : {4,6,12}*1728l, {8,6,6}*1728g, {4,12,6}*1728o
Permutation Representation (GAP) :

s0 := ( 4,10)( 5,11)( 6,12)( 7,19)( 8,20)( 9,21)(16,22)(17,23)(18,24)(31,37)
(32,38)(33,39)(34,46)(35,47)(36,48)(43,49)(44,50)(45,51);;
s1 := (10,19)(11,20)(12,21)(13,22)(14,23)(15,24)(16,25)(17,26)(18,27)(37,46)
(38,47)(39,48)(40,49)(41,50)(42,51)(43,52)(44,53)(45,54);;
s2 := ( 1,13)( 2,15)( 3,14)( 4,10)( 5,12)( 6,11)( 7,16)( 8,18)( 9,17)(19,22)
(20,24)(21,23)(26,27)(28,40)(29,42)(30,41)(31,37)(32,39)(33,38)(34,43)(35,45)
(36,44)(46,49)(47,51)(48,50)(53,54);;
s3 := ( 1,29)( 2,28)( 3,30)( 4,35)( 5,34)( 6,36)( 7,32)( 8,31)( 9,33)(10,47)
(11,46)(12,48)(13,53)(14,52)(15,54)(16,50)(17,49)(18,51)(19,38)(20,37)(21,39)
(22,44)(23,43)(24,45)(25,41)(26,40)(27,42);;
poly := Group([s0,s1,s2,s3]);;

Finitely Presented Group Representation (GAP) :

F := FreeGroup("s0","s1","s2","s3");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;  
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s0*s2*s0*s2, 
s0*s3*s0*s3, s1*s3*s1*s3, s0*s1*s0*s1*s0*s1*s0*s1, 
s3*s1*s2*s1*s2*s3*s1*s2*s1*s2, s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(54)!( 4,10)( 5,11)( 6,12)( 7,19)( 8,20)( 9,21)(16,22)(17,23)(18,24)
(31,37)(32,38)(33,39)(34,46)(35,47)(36,48)(43,49)(44,50)(45,51);
s1 := Sym(54)!(10,19)(11,20)(12,21)(13,22)(14,23)(15,24)(16,25)(17,26)(18,27)
(37,46)(38,47)(39,48)(40,49)(41,50)(42,51)(43,52)(44,53)(45,54);
s2 := Sym(54)!( 1,13)( 2,15)( 3,14)( 4,10)( 5,12)( 6,11)( 7,16)( 8,18)( 9,17)
(19,22)(20,24)(21,23)(26,27)(28,40)(29,42)(30,41)(31,37)(32,39)(33,38)(34,43)
(35,45)(36,44)(46,49)(47,51)(48,50)(53,54);
s3 := Sym(54)!( 1,29)( 2,28)( 3,30)( 4,35)( 5,34)( 6,36)( 7,32)( 8,31)( 9,33)
(10,47)(11,46)(12,48)(13,53)(14,52)(15,54)(16,50)(17,49)(18,51)(19,38)(20,37)
(21,39)(22,44)(23,43)(24,45)(25,41)(26,40)(27,42);
poly := sub<Sym(54)|s0,s1,s2,s3>;

Finitely Presented Group Representation (Magma) :

poly<s0,s1,s2,s3> := Group< s0,s1,s2,s3 | s0*s0, s1*s1, s2*s2, 
s3*s3, s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s0*s1*s0*s1*s0*s1*s0*s1, s3*s1*s2*s1*s2*s3*s1*s2*s1*s2, 
s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 >;

References : None.
to this polytope