Polytope of Type {2,72}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,72}*288
if this polytope has a name.
Group : SmallGroup(288,114)
Rank : 3
Schlafli Type : {2,72}
Number of vertices, edges, etc : 2, 72, 72
Order of s₀s₁s₂ : 72
Order of s₀s₁s₂s₁ : 2
Special Properties :
   Degenerate
   Universal
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {2,72,2} of size 576
   {2,72,4} of size 1152
   {2,72,4} of size 1152
   {2,72,4} of size 1152
   {2,72,4} of size 1152
   {2,72,6} of size 1728
   {2,72,6} of size 1728
Vertex Figure Of :
   {2,2,72} of size 576
   {3,2,72} of size 864
   {4,2,72} of size 1152
   {5,2,72} of size 1440
   {6,2,72} of size 1728
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {2,36}*144
   3-fold quotients : {2,24}*96
   4-fold quotients : {2,18}*72
   6-fold quotients : {2,12}*48
   8-fold quotients : {2,9}*36
   9-fold quotients : {2,8}*32
   12-fold quotients : {2,6}*24
   18-fold quotients : {2,4}*16
   24-fold quotients : {2,3}*12
   36-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   2-fold covers : {4,72}*576a, {2,144}*576
   3-fold covers : {2,216}*864, {6,72}*864a, {6,72}*864b
   4-fold covers : {4,72}*1152a, {8,72}*1152b, {8,72}*1152c, {4,144}*1152a, {4,144}*1152b, {2,288}*1152, {4,72}*1152c
   5-fold covers : {10,72}*1440, {2,360}*1440
   6-fold covers : {4,216}*1728a, {2,432}*1728, {6,144}*1728a, {6,144}*1728b, {12,72}*1728a, {12,72}*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);;
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*s1*s0*s1, s0*s2*s0*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*s1*s2 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(74)!(1,2);
s1 := Sym(74)!( 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(74)!( 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);
poly := sub<Sym(74)|s0,s1,s2>;

Finitely Presented Group Representation (Magma) :

poly<s0,s1,s2> := Group< s0,s1,s2 | s0*s0, s1*s1, s2*s2, 
s0*s1*s0*s1, s0*s2*s0*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*s1*s2 >;

to this polytope