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Polytope of Type {6,6,3}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,6,3}*288
Also Known As : {{6,6|2},{6,3}4}. if this polytope has another name.
Group : SmallGroup(288,1028)
Rank : 4
Schlafli Type : {6,6,3}
Number of vertices, edges, etc : 6, 24, 12, 4
Order of s0s1s2s3 : 12
Order of s0s1s2s3s2s1 : 2
Special Properties :
   Universal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {6,6,3,2} of size 576
   {6,6,3,3} of size 1440
Vertex Figure Of :
   {2,6,6,3} of size 576
   {3,6,6,3} of size 864
   {4,6,6,3} of size 1152
   {3,6,6,3} of size 1152
   {4,6,6,3} of size 1152
   {6,6,6,3} of size 1728
   {6,6,6,3} of size 1728
   {6,6,6,3} of size 1728
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {2,6,3}*96
   6-fold quotients : {2,3,3}*48
Covers (Minimal Covers in Boldface) :
   2-fold covers : {12,6,3}*576, {6,12,3}*576, {6,6,6}*576a
   3-fold covers : {18,6,3}*864, {6,6,3}*864
   4-fold covers : {12,12,3}*1152, {12,6,3}*1152, {6,6,3}*1152, {24,6,3}*1152, {12,6,6}*1152a, {6,6,12}*1152b, {6,12,6}*1152a, {6,6,12}*1152c, {6,12,6}*1152c, {6,6,6}*1152a
   5-fold covers : {6,6,15}*1440, {30,6,3}*1440
   6-fold covers : {36,6,3}*1728, {18,12,3}*1728, {18,6,6}*1728, {6,12,3}*1728, {12,6,3}*1728, {6,6,6}*1728a, {6,6,6}*1728b
Permutation Representation (GAP) :
s0 := ( 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);;
s1 := ( 1, 5)( 2, 7)( 3, 6)( 4, 8)(10,11)(13,29)(14,31)(15,30)(16,32)(17,25)
(18,27)(19,26)(20,28)(21,33)(22,35)(23,34)(24,36)(37,41)(38,43)(39,42)(40,44)
(46,47)(49,65)(50,67)(51,66)(52,68)(53,61)(54,63)(55,62)(56,64)(57,69)(58,71)
(59,70)(60,72);;
s2 := ( 1,49)( 2,50)( 3,52)( 4,51)( 5,53)( 6,54)( 7,56)( 8,55)( 9,57)(10,58)
(11,60)(12,59)(13,37)(14,38)(15,40)(16,39)(17,41)(18,42)(19,44)(20,43)(21,45)
(22,46)(23,48)(24,47)(25,61)(26,62)(27,64)(28,63)(29,65)(30,66)(31,68)(32,67)
(33,69)(34,70)(35,72)(36,71);;
s3 := ( 1,40)( 2,38)( 3,39)( 4,37)( 5,44)( 6,42)( 7,43)( 8,41)( 9,48)(10,46)
(11,47)(12,45)(13,64)(14,62)(15,63)(16,61)(17,68)(18,66)(19,67)(20,65)(21,72)
(22,70)(23,71)(24,69)(25,52)(26,50)(27,51)(28,49)(29,56)(30,54)(31,55)(32,53)
(33,60)(34,58)(35,59)(36,57);;
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, s2*s3*s2*s3*s2*s3, 
s0*s1*s2*s1*s0*s1*s2*s1, s1*s3*s2*s1*s3*s2*s1*s3*s2*s1*s3*s2, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 ];;
poly := F / rels;;
 
Permutation Representation (Magma) :
s0 := 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);
s1 := Sym(72)!( 1, 5)( 2, 7)( 3, 6)( 4, 8)(10,11)(13,29)(14,31)(15,30)(16,32)
(17,25)(18,27)(19,26)(20,28)(21,33)(22,35)(23,34)(24,36)(37,41)(38,43)(39,42)
(40,44)(46,47)(49,65)(50,67)(51,66)(52,68)(53,61)(54,63)(55,62)(56,64)(57,69)
(58,71)(59,70)(60,72);
s2 := Sym(72)!( 1,49)( 2,50)( 3,52)( 4,51)( 5,53)( 6,54)( 7,56)( 8,55)( 9,57)
(10,58)(11,60)(12,59)(13,37)(14,38)(15,40)(16,39)(17,41)(18,42)(19,44)(20,43)
(21,45)(22,46)(23,48)(24,47)(25,61)(26,62)(27,64)(28,63)(29,65)(30,66)(31,68)
(32,67)(33,69)(34,70)(35,72)(36,71);
s3 := Sym(72)!( 1,40)( 2,38)( 3,39)( 4,37)( 5,44)( 6,42)( 7,43)( 8,41)( 9,48)
(10,46)(11,47)(12,45)(13,64)(14,62)(15,63)(16,61)(17,68)(18,66)(19,67)(20,65)
(21,72)(22,70)(23,71)(24,69)(25,52)(26,50)(27,51)(28,49)(29,56)(30,54)(31,55)
(32,53)(33,60)(34,58)(35,59)(36,57);
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, 
s2*s3*s2*s3*s2*s3, s0*s1*s2*s1*s0*s1*s2*s1, 
s1*s3*s2*s1*s3*s2*s1*s3*s2*s1*s3*s2, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 >; 
 
References : None.
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