Polytope of Type {36,4}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {36,4}*288a
Also Known As : {36,4|2}. if this polytope has another name.
Group : SmallGroup(288,92)
Rank : 3
Schlafli Type : {36,4}
Number of vertices, edges, etc : 36, 72, 4
Order of s₀s₁s₂ : 36
Order of s₀s₁s₂s₁ : 2
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
   Flat
   Self-Petrie
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
   Skewing Operation
Facet Of :
   {36,4,2} of size 576
   {36,4,4} of size 1152
   {36,4,6} of size 1728
   {36,4,3} of size 1728
Vertex Figure Of :
   {2,36,4} of size 576
   {4,36,4} of size 1152
   {4,36,4} of size 1152
   {4,36,4} of size 1152
   {6,36,4} of size 1728
   {6,36,4} of size 1728
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {36,2}*144, {18,4}*144a
   3-fold quotients : {12,4}*96a
   4-fold quotients : {18,2}*72
   6-fold quotients : {12,2}*48, {6,4}*48a
   8-fold quotients : {9,2}*36
   9-fold quotients : {4,4}*32
   12-fold quotients : {6,2}*24
   18-fold quotients : {2,4}*16, {4,2}*16
   24-fold quotients : {3,2}*12
   36-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   2-fold covers : {72,4}*576a, {36,4}*576a, {72,4}*576b, {36,8}*576a, {36,8}*576b
   3-fold covers : {108,4}*864a, {36,12}*864a, {36,12}*864b
   4-fold covers : {36,8}*1152a, {72,4}*1152a, {72,8}*1152a, {72,8}*1152b, {72,8}*1152c, {72,8}*1152d, {36,16}*1152a, {144,4}*1152a, {36,16}*1152b, {144,4}*1152b, {36,4}*1152a, {72,4}*1152b, {36,8}*1152b, {36,4}*1152d
   5-fold covers : {36,20}*1440, {180,4}*1440a
   6-fold covers : {216,4}*1728a, {108,4}*1728a, {216,4}*1728b, {108,8}*1728a, {108,8}*1728b, {36,24}*1728a, {36,12}*1728a, {36,12}*1728b, {36,24}*1728b, {72,12}*1728a, {72,12}*1728b, {36,24}*1728c, {72,12}*1728c, {72,12}*1728d, {36,24}*1728d
Permutation Representation (GAP) :

s0 := ( 2, 3)( 4, 8)( 5, 7)( 6, 9)(11,12)(13,17)(14,16)(15,18)(20,21)(22,26)
(23,25)(24,27)(29,30)(31,35)(32,34)(33,36)(37,55)(38,57)(39,56)(40,62)(41,61)
(42,63)(43,59)(44,58)(45,60)(46,64)(47,66)(48,65)(49,71)(50,70)(51,72)(52,68)
(53,67)(54,69);;
s1 := ( 1,40)( 2,42)( 3,41)( 4,37)( 5,39)( 6,38)( 7,44)( 8,43)( 9,45)(10,49)
(11,51)(12,50)(13,46)(14,48)(15,47)(16,53)(17,52)(18,54)(19,58)(20,60)(21,59)
(22,55)(23,57)(24,56)(25,62)(26,61)(27,63)(28,67)(29,69)(30,68)(31,64)(32,66)
(33,65)(34,71)(35,70)(36,72);;
s2 := (37,46)(38,47)(39,48)(40,49)(41,50)(42,51)(43,52)(44,53)(45,54)(55,64)
(56,65)(57,66)(58,67)(59,68)(60,69)(61,70)(62,71)(63,72);;
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*s2*s1*s0*s1*s2*s1, 
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*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*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)!( 2, 3)( 4, 8)( 5, 7)( 6, 9)(11,12)(13,17)(14,16)(15,18)(20,21)
(22,26)(23,25)(24,27)(29,30)(31,35)(32,34)(33,36)(37,55)(38,57)(39,56)(40,62)
(41,61)(42,63)(43,59)(44,58)(45,60)(46,64)(47,66)(48,65)(49,71)(50,70)(51,72)
(52,68)(53,67)(54,69);
s1 := Sym(72)!( 1,40)( 2,42)( 3,41)( 4,37)( 5,39)( 6,38)( 7,44)( 8,43)( 9,45)
(10,49)(11,51)(12,50)(13,46)(14,48)(15,47)(16,53)(17,52)(18,54)(19,58)(20,60)
(21,59)(22,55)(23,57)(24,56)(25,62)(26,61)(27,63)(28,67)(29,69)(30,68)(31,64)
(32,66)(33,65)(34,71)(35,70)(36,72);
s2 := Sym(72)!(37,46)(38,47)(39,48)(40,49)(41,50)(42,51)(43,52)(44,53)(45,54)
(55,64)(56,65)(57,66)(58,67)(59,68)(60,69)(61,70)(62,71)(63,72);
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*s2*s1*s0*s1*s2*s1, 
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*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*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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