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Polytope of Type {2,72,2}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,72,2}*576
if this polytope has a name.
Group : SmallGroup(576,1725)
Rank : 4
Schlafli Type : {2,72,2}
Number of vertices, edges, etc : 2, 72, 72, 2
Order of s0s1s2s3 : 72
Order of s0s1s2s3s2s1 : 2
Special Properties :
   Degenerate
   Universal
   Orientable
   Flat
   Self-Dual
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {2,72,2,2} of size 1152
   {2,72,2,3} of size 1728
Vertex Figure Of :
   {2,2,72,2} of size 1152
   {3,2,72,2} of size 1728
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {2,36,2}*288
   3-fold quotients : {2,24,2}*192
   4-fold quotients : {2,18,2}*144
   6-fold quotients : {2,12,2}*96
   8-fold quotients : {2,9,2}*72
   9-fold quotients : {2,8,2}*64
   12-fold quotients : {2,6,2}*48
   18-fold quotients : {2,4,2}*32
   24-fold quotients : {2,3,2}*24
   36-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
   2-fold covers : {2,72,4}*1152a, {4,72,2}*1152a, {2,144,2}*1152
   3-fold covers : {2,216,2}*1728, {2,72,6}*1728a, {2,72,6}*1728b, {6,72,2}*1728a, {6,72,2}*1728b
Permutation Representation (GAP) :
s0 := (1,2);;
s1 := ( 4, 5)( 6,10)( 7, 9)( 8,11)(13,14)(15,19)(16,18)(17,20)(21,30)(22,32)
(23,31)(24,37)(25,36)(26,38)(27,34)(28,33)(29,35)(39,57)(40,59)(41,58)(42,64)
(43,63)(44,65)(45,61)(46,60)(47,62)(48,66)(49,68)(50,67)(51,73)(52,72)(53,74)
(54,70)(55,69)(56,71);;
s2 := ( 3,42)( 4,44)( 5,43)( 6,39)( 7,41)( 8,40)( 9,46)(10,45)(11,47)(12,51)
(13,53)(14,52)(15,48)(16,50)(17,49)(18,55)(19,54)(20,56)(21,69)(22,71)(23,70)
(24,66)(25,68)(26,67)(27,73)(28,72)(29,74)(30,60)(31,62)(32,61)(33,57)(34,59)
(35,58)(36,64)(37,63)(38,65);;
s3 := (75,76);;
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*s1*s0*s1, 
s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s2*s3*s2*s3, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 ];;
poly := F / rels;;
 
Permutation Representation (Magma) :
s0 := Sym(76)!(1,2);
s1 := Sym(76)!( 4, 5)( 6,10)( 7, 9)( 8,11)(13,14)(15,19)(16,18)(17,20)(21,30)
(22,32)(23,31)(24,37)(25,36)(26,38)(27,34)(28,33)(29,35)(39,57)(40,59)(41,58)
(42,64)(43,63)(44,65)(45,61)(46,60)(47,62)(48,66)(49,68)(50,67)(51,73)(52,72)
(53,74)(54,70)(55,69)(56,71);
s2 := Sym(76)!( 3,42)( 4,44)( 5,43)( 6,39)( 7,41)( 8,40)( 9,46)(10,45)(11,47)
(12,51)(13,53)(14,52)(15,48)(16,50)(17,49)(18,55)(19,54)(20,56)(21,69)(22,71)
(23,70)(24,66)(25,68)(26,67)(27,73)(28,72)(29,74)(30,60)(31,62)(32,61)(33,57)
(34,59)(35,58)(36,64)(37,63)(38,65);
s3 := Sym(76)!(75,76);
poly := sub<Sym(76)|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*s1*s0*s1, s0*s2*s0*s2, s0*s3*s0*s3, 
s1*s3*s1*s3, s2*s3*s2*s3, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 >; 
 

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