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Polytope of Type {12,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {12,6}*288a
if this polytope has a name.
Group : SmallGroup(288,1028)
Rank : 3
Schlafli Type : {12,6}
Number of vertices, edges, etc : 24, 72, 12
Order of s0s1s2 : 6
Order of s0s1s2s1 : 4
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
Facet Of :
   {12,6,2} of size 576
   {12,6,3} of size 1152
   {12,6,4} of size 1152
   {12,6,4} of size 1152
   {12,6,6} of size 1728
   {12,6,6} of size 1728
Vertex Figure Of :
   {2,12,6} of size 576
   {4,12,6} of size 1152
   {4,12,6} of size 1152
   {6,12,6} of size 1728
   {6,12,6} of size 1728
   {6,12,6} of size 1728
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {12,6}*144d
   3-fold quotients : {4,6}*96
   4-fold quotients : {6,6}*72a
   6-fold quotients : {4,3}*48, {4,6}*48b, {4,6}*48c
   12-fold quotients : {4,3}*24, {2,6}*24, {6,2}*24
   24-fold quotients : {2,3}*12, {3,2}*12
   36-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   2-fold covers : {12,12}*576f, {12,6}*576b, {24,6}*576c, {24,6}*576e, {12,12}*576k
   3-fold covers : {36,6}*864, {12,18}*864a, {12,6}*864b, {12,6}*864c
   4-fold covers : {24,6}*1152b, {24,12}*1152i, {24,12}*1152k, {24,6}*1152d, {12,6}*1152b, {24,6}*1152e, {12,24}*1152o, {12,24}*1152q, {24,6}*1152h, {12,6}*1152d, {12,12}*1152i, {12,12}*1152n, {24,12}*1152u, {24,12}*1152v, {12,24}*1152w, {12,24}*1152x, {12,12}*1152t
   5-fold covers : {12,30}*1440a, {60,6}*1440d
   6-fold covers : {36,12}*1728c, {36,6}*1728b, {72,6}*1728b, {72,6}*1728c, {36,12}*1728d, {12,36}*1728e, {12,18}*1728c, {12,12}*1728l, {12,6}*1728b, {24,18}*1728c, {24,6}*1728c, {24,18}*1728e, {24,6}*1728e, {12,36}*1728h, {12,12}*1728p, {24,6}*1728f, {24,6}*1728g, {12,12}*1728w, {12,6}*1728i, {12,12}*1728y
Permutation Representation (GAP) :
s0 := ( 1, 2)( 3, 4)( 5,10)( 6, 9)( 7,12)( 8,11)(13,14)(15,16)(17,22)(18,21)
(19,24)(20,23)(25,26)(27,28)(29,34)(30,33)(31,36)(32,35)(37,38)(39,40)(41,46)
(42,45)(43,48)(44,47)(49,50)(51,52)(53,58)(54,57)(55,60)(56,59)(61,62)(63,64)
(65,70)(66,69)(67,72)(68,71);;
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);;
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, 
s2*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s2*s0*s1*s0*s1, 
s2*s0*s1*s2*s1*s2*s0*s1*s2*s0*s1*s2*s1*s2*s0*s1, 
s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1 ];;
poly := F / rels;;
 
Permutation Representation (Magma) :
s0 := Sym(72)!( 1, 2)( 3, 4)( 5,10)( 6, 9)( 7,12)( 8,11)(13,14)(15,16)(17,22)
(18,21)(19,24)(20,23)(25,26)(27,28)(29,34)(30,33)(31,36)(32,35)(37,38)(39,40)
(41,46)(42,45)(43,48)(44,47)(49,50)(51,52)(53,58)(54,57)(55,60)(56,59)(61,62)
(63,64)(65,70)(66,69)(67,72)(68,71);
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);
poly := sub<Sym(72)|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, 
s2*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s2*s0*s1*s0*s1, 
s2*s0*s1*s2*s1*s2*s0*s1*s2*s0*s1*s2*s1*s2*s0*s1, 
s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1 >; 
 
References : None.
to this polytope