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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}*144a
Also Known As : {12,6|2}. if this polytope has another name.
Group : SmallGroup(144,144)
Rank : 3
Schlafli Type : {12,6}
Number of vertices, edges, etc : 12, 36, 6
Order of s0s1s2 : 12
Order of s0s1s2s1 : 2
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
Facet Of :
   {12,6,2} of size 288
   {12,6,3} of size 432
   {12,6,4} of size 576
   {12,6,3} of size 576
   {12,6,4} of size 576
   {12,6,6} of size 864
   {12,6,6} of size 864
   {12,6,6} of size 864
   {12,6,8} of size 1152
   {12,6,4} of size 1152
   {12,6,6} of size 1152
   {12,6,9} of size 1296
   {12,6,3} of size 1296
   {12,6,5} of size 1440
   {12,6,5} of size 1440
   {12,6,10} of size 1440
   {12,6,12} of size 1728
   {12,6,12} of size 1728
   {12,6,12} of size 1728
   {12,6,3} of size 1728
   {12,6,4} of size 1728
Vertex Figure Of :
   {2,12,6} of size 288
   {4,12,6} of size 576
   {4,12,6} of size 576
   {4,12,6} of size 576
   {3,12,6} of size 576
   {6,12,6} of size 864
   {6,12,6} of size 864
   {6,12,6} of size 864
   {3,12,6} of size 864
   {8,12,6} of size 1152
   {8,12,6} of size 1152
   {4,12,6} of size 1152
   {4,12,6} of size 1152
   {4,12,6} of size 1152
   {6,12,6} of size 1152
   {6,12,6} of size 1152
   {6,12,6} of size 1296
   {6,12,6} of size 1296
   {6,12,6} of size 1296
   {10,12,6} of size 1440
   {12,12,6} of size 1728
   {12,12,6} of size 1728
   {12,12,6} of size 1728
   {3,12,6} of size 1728
   {4,12,6} of size 1728
   {6,12,6} of size 1728
   {6,12,6} of size 1728
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {6,6}*72a
   3-fold quotients : {12,2}*48, {4,6}*48a
   6-fold quotients : {2,6}*24, {6,2}*24
   9-fold quotients : {4,2}*16
   12-fold quotients : {2,3}*12, {3,2}*12
   18-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   2-fold covers : {24,6}*288a, {12,12}*288a
   3-fold covers : {36,6}*432a, {12,18}*432a, {12,6}*432b, {12,6}*432g
   4-fold covers : {48,6}*576a, {12,12}*576a, {12,24}*576c, {24,12}*576c, {12,24}*576e, {24,12}*576e, {12,12}*576d, {12,6}*576b
   5-fold covers : {12,30}*720b, {60,6}*720b
   6-fold covers : {72,6}*864a, {24,18}*864a, {24,6}*864b, {12,36}*864a, {36,12}*864a, {12,12}*864c, {24,6}*864f, {12,12}*864h
   7-fold covers : {12,42}*1008b, {84,6}*1008b
   8-fold covers : {12,24}*1152b, {24,12}*1152b, {24,24}*1152b, {24,24}*1152g, {24,24}*1152i, {24,24}*1152k, {12,48}*1152b, {48,12}*1152b, {12,48}*1152e, {48,12}*1152e, {12,24}*1152e, {24,12}*1152e, {12,12}*1152c, {96,6}*1152c, {12,24}*1152i, {12,24}*1152k, {24,6}*1152d, {24,12}*1152o, {24,12}*1152q, {24,6}*1152h, {12,6}*1152d, {12,12}*1152i, {12,12}*1152k, {12,12}*1152n
   9-fold covers : {36,18}*1296a, {12,18}*1296a, {36,6}*1296b, {12,54}*1296a, {108,6}*1296a, {12,6}*1296a, {12,6}*1296b, {12,18}*1296b, {36,6}*1296f, {12,18}*1296c, {36,6}*1296g, {36,6}*1296l, {12,18}*1296l, {12,6}*1296g, {12,6}*1296h, {12,6}*1296i, {12,6}*1296t
   10-fold covers : {24,30}*1440b, {120,6}*1440b, {12,60}*1440b, {60,12}*1440b
   11-fold covers : {12,66}*1584b, {132,6}*1584b
   12-fold covers : {144,6}*1728a, {48,18}*1728a, {48,6}*1728b, {12,36}*1728a, {36,12}*1728a, {12,12}*1728c, {12,72}*1728a, {72,12}*1728a, {24,36}*1728c, {36,24}*1728c, {12,24}*1728d, {24,12}*1728d, {12,72}*1728c, {72,12}*1728c, {24,36}*1728d, {36,24}*1728d, {12,24}*1728f, {24,12}*1728f, {48,6}*1728f, {12,24}*1728o, {24,12}*1728o, {12,24}*1728p, {24,12}*1728p, {12,12}*1728h, {12,36}*1728c, {36,6}*1728b, {36,12}*1728e, {12,18}*1728c, {12,12}*1728j, {12,6}*1728b, {12,12}*1728v, {12,6}*1728h, {12,6}*1728i
   13-fold covers : {12,78}*1872b, {156,6}*1872b
Permutation Representation (GAP) :
s0 := ( 1,37)( 2,38)( 3,39)( 4,43)( 5,44)( 6,45)( 7,40)( 8,41)( 9,42)(10,46)
(11,47)(12,48)(13,52)(14,53)(15,54)(16,49)(17,50)(18,51)(19,64)(20,65)(21,66)
(22,70)(23,71)(24,72)(25,67)(26,68)(27,69)(28,55)(29,56)(30,57)(31,61)(32,62)
(33,63)(34,58)(35,59)(36,60);;
s1 := ( 1,58)( 2,60)( 3,59)( 4,55)( 5,57)( 6,56)( 7,61)( 8,63)( 9,62)(10,67)
(11,69)(12,68)(13,64)(14,66)(15,65)(16,70)(17,72)(18,71)(19,40)(20,42)(21,41)
(22,37)(23,39)(24,38)(25,43)(26,45)(27,44)(28,49)(29,51)(30,50)(31,46)(32,48)
(33,47)(34,52)(35,54)(36,53);;
s2 := ( 1, 2)( 4, 5)( 7, 8)(10,11)(13,14)(16,17)(19,20)(22,23)(25,26)(28,29)
(31,32)(34,35)(37,38)(40,41)(43,44)(46,47)(49,50)(52,53)(55,56)(58,59)(61,62)
(64,65)(67,68)(70,71);;
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, s0*s1*s2*s1*s0*s1*s2*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 ];;
poly := F / rels;;
 
Permutation Representation (Magma) :
s0 := Sym(72)!( 1,37)( 2,38)( 3,39)( 4,43)( 5,44)( 6,45)( 7,40)( 8,41)( 9,42)
(10,46)(11,47)(12,48)(13,52)(14,53)(15,54)(16,49)(17,50)(18,51)(19,64)(20,65)
(21,66)(22,70)(23,71)(24,72)(25,67)(26,68)(27,69)(28,55)(29,56)(30,57)(31,61)
(32,62)(33,63)(34,58)(35,59)(36,60);
s1 := Sym(72)!( 1,58)( 2,60)( 3,59)( 4,55)( 5,57)( 6,56)( 7,61)( 8,63)( 9,62)
(10,67)(11,69)(12,68)(13,64)(14,66)(15,65)(16,70)(17,72)(18,71)(19,40)(20,42)
(21,41)(22,37)(23,39)(24,38)(25,43)(26,45)(27,44)(28,49)(29,51)(30,50)(31,46)
(32,48)(33,47)(34,52)(35,54)(36,53);
s2 := Sym(72)!( 1, 2)( 4, 5)( 7, 8)(10,11)(13,14)(16,17)(19,20)(22,23)(25,26)
(28,29)(31,32)(34,35)(37,38)(40,41)(43,44)(46,47)(49,50)(52,53)(55,56)(58,59)
(61,62)(64,65)(67,68)(70,71);
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, s0*s1*s2*s1*s0*s1*s2*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 >; 
 
References : None.
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