Polytope of Type {4,6,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,6,6}*288e
if this polytope has a name.
Group : SmallGroup(288,1028)
Rank : 4
Schlafli Type : {4,6,6}
Number of vertices, edges, etc : 4, 12, 18, 6
Order of s₀s₁s₂s₃ : 6
Order of s₀s₁s₂s₃s₂s₁ : 2
Special Properties :
   Universal
   Non-Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {4,6,6,2} of size 576
   {4,6,6,3} of size 864
   {4,6,6,4} of size 1152
   {4,6,6,6} of size 1728
   {4,6,6,6} of size 1728
Vertex Figure Of :
   {2,4,6,6} of size 576
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {4,3,6}*144
   3-fold quotients : {4,6,2}*96c
   6-fold quotients : {4,3,2}*48
Covers (Minimal Covers in Boldface) :
   2-fold covers : {4,12,6}*576e, {4,12,6}*576g, {4,6,12}*576e, {4,6,6}*576b
   3-fold covers : {4,18,6}*864d, {4,6,6}*864d, {4,6,6}*864i
   4-fold covers : {4,6,6}*1152b, {4,24,6}*1152h, {4,24,6}*1152j, {4,12,12}*1152e, {4,12,12}*1152g, {4,6,24}*1152e, {4,12,6}*1152f, {4,6,6}*1152e, {4,12,6}*1152i, {8,6,6}*1152c, {8,6,6}*1152e, {4,6,12}*1152d, {4,6,6}*1152g, {4,6,12}*1152f
   5-fold covers : {4,6,30}*1440d, {4,30,6}*1440e
   6-fold covers : {4,36,6}*1728d, {4,36,6}*1728f, {4,12,6}*1728d, {4,12,6}*1728f, {4,18,12}*1728d, {4,6,12}*1728e, {4,18,6}*1728b, {4,6,6}*1728a, {4,12,6}*1728l, {4,12,6}*1728m, {4,6,12}*1728j, {4,6,6}*1728c, {12,6,6}*1728c, {12,6,6}*1728d
Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

References : None.
to this polytope