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Polytope of Type {4,4}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,4}*144
Also Known As : {4,4}(3,3), {4,4}6if this polytope has another name.
Group : SmallGroup(144,186)
Rank : 3
Schlafli Type : {4,4}
Number of vertices, edges, etc : 18, 36, 18
Order of s0s1s2 : 6
Order of s0s1s2s1 : 6
Special Properties :
   Toroidal
   Locally Spherical
   Orientable
   Self-Dual
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
   Halving Operation
   Skewing Operation
Facet Of :
   {4,4,2} of size 288
   {4,4,4} of size 576
   {4,4,6} of size 864
   {4,4,8} of size 1152
   {4,4,3} of size 1440
   {4,4,3} of size 1440
   {4,4,4} of size 1440
   {4,4,4} of size 1440
   {4,4,10} of size 1440
   {4,4,12} of size 1728
Vertex Figure Of :
   {2,4,4} of size 288
   {4,4,4} of size 576
   {6,4,4} of size 864
   {8,4,4} of size 1152
   {3,4,4} of size 1440
   {3,4,4} of size 1440
   {4,4,4} of size 1440
   {4,4,4} of size 1440
   {10,4,4} of size 1440
   {12,4,4} of size 1728
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {4,4}*72
   18-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   2-fold covers : {4,4}*288
   3-fold covers : {4,12}*432a, {12,4}*432a, {4,12}*432b, {12,4}*432b
   4-fold covers : {4,4}*576, {4,8}*576a, {8,4}*576a, {4,8}*576b, {8,4}*576b
   5-fold covers : {4,20}*720, {20,4}*720
   6-fold covers : {4,12}*864a, {12,4}*864a, {4,12}*864d, {12,4}*864c
   7-fold covers : {4,28}*1008, {28,4}*1008
   8-fold covers : {4,8}*1152a, {8,4}*1152a, {8,8}*1152a, {8,8}*1152b, {8,8}*1152c, {8,8}*1152d, {4,16}*1152a, {16,4}*1152a, {4,16}*1152b, {16,4}*1152b, {4,8}*1152b, {8,4}*1152b, {4,4}*1152
   9-fold covers : {4,4}*1296, {4,36}*1296, {36,4}*1296, {4,12}*1296, {12,4}*1296, {12,12}*1296a, {12,12}*1296b, {12,12}*1296c, {12,12}*1296d, {12,12}*1296e, {12,12}*1296f, {12,12}*1296g, {12,12}*1296h
   10-fold covers : {4,20}*1440, {20,4}*1440
   11-fold covers : {4,44}*1584, {44,4}*1584
   12-fold covers : {4,12}*1728b, {12,4}*1728a, {4,24}*1728b, {8,12}*1728b, {12,8}*1728b, {24,4}*1728b, {4,24}*1728d, {8,12}*1728c, {12,8}*1728c, {24,4}*1728d, {4,24}*1728f, {24,4}*1728e, {8,12}*1728e, {12,8}*1728e, {4,24}*1728h, {24,4}*1728h, {8,12}*1728f, {12,8}*1728f, {4,12}*1728d, {12,4}*1728d, {4,12}*1728e, {12,4}*1728e, {12,12}*1728z
   13-fold covers : {4,52}*1872, {52,4}*1872
Permutation Representation (GAP) :
s0 := ( 4, 7)( 5, 8)( 6, 9)(13,16)(14,17)(15,18)(22,25)(23,26)(24,27)(31,34)
(32,35)(33,36)(37,46)(38,47)(39,48)(40,52)(41,53)(42,54)(43,49)(44,50)(45,51)
(55,64)(56,65)(57,66)(58,70)(59,71)(60,72)(61,67)(62,68)(63,69);;
s1 := ( 1,37)( 2,40)( 3,43)( 4,38)( 5,41)( 6,44)( 7,39)( 8,42)( 9,45)(10,46)
(11,49)(12,52)(13,47)(14,50)(15,53)(16,48)(17,51)(18,54)(19,55)(20,58)(21,61)
(22,56)(23,59)(24,62)(25,57)(26,60)(27,63)(28,64)(29,67)(30,70)(31,65)(32,68)
(33,71)(34,66)(35,69)(36,72);;
s2 := ( 1,29)( 2,28)( 3,30)( 4,32)( 5,31)( 6,33)( 7,35)( 8,34)( 9,36)(10,20)
(11,19)(12,21)(13,23)(14,22)(15,24)(16,26)(17,25)(18,27)(37,56)(38,55)(39,57)
(40,59)(41,58)(42,60)(43,62)(44,61)(45,63)(46,65)(47,64)(48,66)(49,68)(50,67)
(51,69)(52,71)(53,70)(54,72);;
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*s0*s1*s0*s1*s0*s1, 
s1*s2*s1*s2*s1*s2*s1*s2, s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1 ];;
poly := F / rels;;
 
Permutation Representation (Magma) :
s0 := Sym(72)!( 4, 7)( 5, 8)( 6, 9)(13,16)(14,17)(15,18)(22,25)(23,26)(24,27)
(31,34)(32,35)(33,36)(37,46)(38,47)(39,48)(40,52)(41,53)(42,54)(43,49)(44,50)
(45,51)(55,64)(56,65)(57,66)(58,70)(59,71)(60,72)(61,67)(62,68)(63,69);
s1 := Sym(72)!( 1,37)( 2,40)( 3,43)( 4,38)( 5,41)( 6,44)( 7,39)( 8,42)( 9,45)
(10,46)(11,49)(12,52)(13,47)(14,50)(15,53)(16,48)(17,51)(18,54)(19,55)(20,58)
(21,61)(22,56)(23,59)(24,62)(25,57)(26,60)(27,63)(28,64)(29,67)(30,70)(31,65)
(32,68)(33,71)(34,66)(35,69)(36,72);
s2 := Sym(72)!( 1,29)( 2,28)( 3,30)( 4,32)( 5,31)( 6,33)( 7,35)( 8,34)( 9,36)
(10,20)(11,19)(12,21)(13,23)(14,22)(15,24)(16,26)(17,25)(18,27)(37,56)(38,55)
(39,57)(40,59)(41,58)(42,60)(43,62)(44,61)(45,63)(46,65)(47,64)(48,66)(49,68)
(50,67)(51,69)(52,71)(53,70)(54,72);
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*s0*s1*s0*s1*s0*s1, 
s1*s2*s1*s2*s1*s2*s1*s2, s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1 >; 
 
References : None.
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