Polytope of Type {4,9}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,9}*144
if this polytope has a name.
Group : SmallGroup(144,109)
Rank : 3
Schlafli Type : {4,9}
Number of vertices, edges, etc : 8, 36, 18
Order of s₀s₁s₂ : 18
Order of s₀s₁s₂s₁ : 4
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {4,9,2} of size 288
   {4,9,4} of size 576
   {4,9,6} of size 864
   {4,9,4} of size 1152
Vertex Figure Of :
   {2,4,9} of size 288
   {4,4,9} of size 576
   {6,4,9} of size 864
   {4,4,9} of size 1152
   {8,4,9} of size 1152
   {10,4,9} of size 1440
   {12,4,9} of size 1728
   {3,4,9} of size 1728
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {4,9}*72
   3-fold quotients : {4,3}*48
   4-fold quotients : {2,9}*36
   6-fold quotients : {4,3}*24
   12-fold quotients : {2,3}*12
Covers (Minimal Covers in Boldface) :
   2-fold covers : {8,9}*288, {4,18}*288
   3-fold covers : {4,27}*432, {12,9}*432
   4-fold covers : {8,9}*576, {4,36}*576b, {4,18}*576b, {4,36}*576c, {8,18}*576b, {8,18}*576c
   5-fold covers : {4,45}*720
   6-fold covers : {8,27}*864, {4,54}*864, {24,9}*864, {12,18}*864a, {12,18}*864b
   7-fold covers : {4,63}*1008
   8-fold covers : {8,9}*1152, {8,18}*1152a, {4,36}*1152d, {8,36}*1152e, {8,36}*1152f, {4,18}*1152a, {8,18}*1152d, {8,18}*1152e, {8,18}*1152f, {8,36}*1152g, {8,36}*1152h, {4,72}*1152c, {4,72}*1152d, {8,18}*1152g, {4,36}*1152e, {4,72}*1152e, {4,18}*1152b, {4,72}*1152f
   9-fold covers : {4,81}*1296, {12,27}*1296, {36,9}*1296, {12,9}*1296c
   10-fold covers : {8,45}*1440, {20,18}*1440, {4,90}*1440
   11-fold covers : {4,99}*1584
   12-fold covers : {8,27}*1728, {4,108}*1728b, {4,54}*1728b, {4,108}*1728c, {8,54}*1728b, {8,54}*1728c, {24,9}*1728, {12,36}*1728e, {12,36}*1728f, {12,18}*1728c, {12,36}*1728g, {24,18}*1728b, {24,18}*1728c, {24,18}*1728d, {24,18}*1728e, {12,18}*1728d, {12,36}*1728h, {12,9}*1728
   13-fold covers : {4,117}*1872
Permutation Representation (GAP) :

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

References : None.
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