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Polytope of Type {3,6,6}

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