Polytope of Type {18,8}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {18,8}*288
Also Known As : {18,8|2}. if this polytope has another name.
Group : SmallGroup(288,120)
Rank : 3
Schlafli Type : {18,8}
Number of vertices, edges, etc : 18, 72, 8
Order of s₀s₁s₂ : 72
Order of s₀s₁s₂s₁ : 2
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {18,8,2} of size 576
   {18,8,4} of size 1152
   {18,8,4} of size 1152
   {18,8,6} of size 1728
   {18,8,3} of size 1728
Vertex Figure Of :
   {2,18,8} of size 576
   {4,18,8} of size 1152
   {4,18,8} of size 1152
   {6,18,8} of size 1728
   {6,18,8} of size 1728
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {18,4}*144a
   3-fold quotients : {6,8}*96
   4-fold quotients : {18,2}*72
   6-fold quotients : {6,4}*48a
   8-fold quotients : {9,2}*36
   9-fold quotients : {2,8}*32
   12-fold quotients : {6,2}*24
   18-fold quotients : {2,4}*16
   24-fold quotients : {3,2}*12
   36-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   2-fold covers : {36,8}*576a, {18,16}*576
   3-fold covers : {54,8}*864, {18,24}*864a, {18,24}*864b
   4-fold covers : {36,8}*1152a, {72,8}*1152a, {72,8}*1152c, {36,16}*1152a, {36,16}*1152b, {18,32}*1152, {18,8}*1152g
   5-fold covers : {18,40}*1440, {90,8}*1440
   6-fold covers : {108,8}*1728a, {54,16}*1728, {18,48}*1728a, {36,24}*1728b, {36,24}*1728c, {18,48}*1728b
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)(38,39)(40,44)(41,43)(42,45)(47,48)
(49,53)(50,52)(51,54)(56,57)(58,62)(59,61)(60,63)(65,66)(67,71)(68,70)
(69,72);;
s1 := ( 1, 4)( 2, 6)( 3, 5)( 7, 8)(10,13)(11,15)(12,14)(16,17)(19,31)(20,33)
(21,32)(22,28)(23,30)(24,29)(25,35)(26,34)(27,36)(37,58)(38,60)(39,59)(40,55)
(41,57)(42,56)(43,62)(44,61)(45,63)(46,67)(47,69)(48,68)(49,64)(50,66)(51,65)
(52,71)(53,70)(54,72);;
s2 := ( 1,37)( 2,38)( 3,39)( 4,40)( 5,41)( 6,42)( 7,43)( 8,44)( 9,45)(10,46)
(11,47)(12,48)(13,49)(14,50)(15,51)(16,52)(17,53)(18,54)(19,64)(20,65)(21,66)
(22,67)(23,68)(24,69)(25,70)(26,71)(27,72)(28,55)(29,56)(30,57)(31,58)(32,59)
(33,60)(34,61)(35,62)(36,63);;
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*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 ];;
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)(38,39)(40,44)(41,43)(42,45)
(47,48)(49,53)(50,52)(51,54)(56,57)(58,62)(59,61)(60,63)(65,66)(67,71)(68,70)
(69,72);
s1 := Sym(72)!( 1, 4)( 2, 6)( 3, 5)( 7, 8)(10,13)(11,15)(12,14)(16,17)(19,31)
(20,33)(21,32)(22,28)(23,30)(24,29)(25,35)(26,34)(27,36)(37,58)(38,60)(39,59)
(40,55)(41,57)(42,56)(43,62)(44,61)(45,63)(46,67)(47,69)(48,68)(49,64)(50,66)
(51,65)(52,71)(53,70)(54,72);
s2 := Sym(72)!( 1,37)( 2,38)( 3,39)( 4,40)( 5,41)( 6,42)( 7,43)( 8,44)( 9,45)
(10,46)(11,47)(12,48)(13,49)(14,50)(15,51)(16,52)(17,53)(18,54)(19,64)(20,65)
(21,66)(22,67)(23,68)(24,69)(25,70)(26,71)(27,72)(28,55)(29,56)(30,57)(31,58)
(32,59)(33,60)(34,61)(35,62)(36,63);
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*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 >;

References : None.
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