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Polytope of Type {2,4,36}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,4,36}*576a
if this polytope has a name.
Group : SmallGroup(576,1471)
Rank : 4
Schlafli Type : {2,4,36}
Number of vertices, edges, etc : 2, 4, 72, 36
Order of s0s1s2s3 : 36
Order of s0s1s2s3s2s1 : 2
Special Properties :
   Degenerate
   Universal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {2,4,36,2} of size 1152
Vertex Figure Of :
   {2,2,4,36} of size 1152
   {3,2,4,36} of size 1728
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {2,2,36}*288, {2,4,18}*288a
   3-fold quotients : {2,4,12}*192a
   4-fold quotients : {2,2,18}*144
   6-fold quotients : {2,2,12}*96, {2,4,6}*96a
   8-fold quotients : {2,2,9}*72
   9-fold quotients : {2,4,4}*64
   12-fold quotients : {2,2,6}*48
   18-fold quotients : {2,2,4}*32, {2,4,2}*32
   24-fold quotients : {2,2,3}*24
   36-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
   2-fold covers : {4,4,36}*1152, {2,8,36}*1152a, {2,4,72}*1152a, {2,8,36}*1152b, {2,4,72}*1152b, {2,4,36}*1152a
   3-fold covers : {2,4,108}*1728a, {6,4,36}*1728, {2,12,36}*1728a, {2,12,36}*1728b
Permutation Representation (GAP) :
s0 := (1,2);;
s1 := (39,48)(40,49)(41,50)(42,51)(43,52)(44,53)(45,54)(46,55)(47,56)(57,66)
(58,67)(59,68)(60,69)(61,70)(62,71)(63,72)(64,73)(65,74);;
s2 := ( 3,39)( 4,41)( 5,40)( 6,46)( 7,45)( 8,47)( 9,43)(10,42)(11,44)(12,48)
(13,50)(14,49)(15,55)(16,54)(17,56)(18,52)(19,51)(20,53)(21,57)(22,59)(23,58)
(24,64)(25,63)(26,65)(27,61)(28,60)(29,62)(30,66)(31,68)(32,67)(33,73)(34,72)
(35,74)(36,70)(37,69)(38,71);;
s3 := ( 3, 6)( 4, 8)( 5, 7)( 9,10)(12,15)(13,17)(14,16)(18,19)(21,24)(22,26)
(23,25)(27,28)(30,33)(31,35)(32,34)(36,37)(39,60)(40,62)(41,61)(42,57)(43,59)
(44,58)(45,64)(46,63)(47,65)(48,69)(49,71)(50,70)(51,66)(52,68)(53,67)(54,73)
(55,72)(56,74);;
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, 
s1*s2*s1*s2*s1*s2*s1*s2, s1*s2*s3*s2*s1*s2*s3*s2, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 ];;
poly := F / rels;;
 
Permutation Representation (Magma) :
s0 := Sym(74)!(1,2);
s1 := Sym(74)!(39,48)(40,49)(41,50)(42,51)(43,52)(44,53)(45,54)(46,55)(47,56)
(57,66)(58,67)(59,68)(60,69)(61,70)(62,71)(63,72)(64,73)(65,74);
s2 := Sym(74)!( 3,39)( 4,41)( 5,40)( 6,46)( 7,45)( 8,47)( 9,43)(10,42)(11,44)
(12,48)(13,50)(14,49)(15,55)(16,54)(17,56)(18,52)(19,51)(20,53)(21,57)(22,59)
(23,58)(24,64)(25,63)(26,65)(27,61)(28,60)(29,62)(30,66)(31,68)(32,67)(33,73)
(34,72)(35,74)(36,70)(37,69)(38,71);
s3 := Sym(74)!( 3, 6)( 4, 8)( 5, 7)( 9,10)(12,15)(13,17)(14,16)(18,19)(21,24)
(22,26)(23,25)(27,28)(30,33)(31,35)(32,34)(36,37)(39,60)(40,62)(41,61)(42,57)
(43,59)(44,58)(45,64)(46,63)(47,65)(48,69)(49,71)(50,70)(51,66)(52,68)(53,67)
(54,73)(55,72)(56,74);
poly := sub<Sym(74)|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, s1*s2*s1*s2*s1*s2*s1*s2, 
s1*s2*s3*s2*s1*s2*s3*s2, s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 >; 
 

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