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Polytope of Type {8,4}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {8,4}*576b
if this polytope has a name.
Group : SmallGroup(576,5357)
Rank : 3
Schlafli Type : {8,4}
Number of vertices, edges, etc : 72, 144, 36
Order of s0s1s2 : 24
Order of s0s1s2s1 : 12
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
Facet Of :
   {8,4,2} of size 1152
Vertex Figure Of :
   {2,8,4} of size 1152
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {4,4}*288
   4-fold quotients : {4,4}*144
   8-fold quotients : {4,4}*72
   9-fold quotients : {8,4}*64b
   18-fold quotients : {4,4}*32
   36-fold quotients : {2,4}*16, {4,2}*16
   72-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   2-fold covers : {8,4}*1152a, {8,8}*1152c, {8,8}*1152d
   3-fold covers : {8,12}*1728c, {24,4}*1728d, {24,4}*1728h, {8,12}*1728f
Permutation Representation (GAP) :
s0 := ( 1,46)( 2,48)( 3,47)( 4,49)( 5,51)( 6,50)( 7,52)( 8,54)( 9,53)(10,37)
(11,39)(12,38)(13,40)(14,42)(15,41)(16,43)(17,45)(18,44)(19,64)(20,66)(21,65)
(22,67)(23,69)(24,68)(25,70)(26,72)(27,71)(28,55)(29,57)(30,56)(31,58)(32,60)
(33,59)(34,61)(35,63)(36,62);;
s1 := ( 2, 4)( 3, 7)( 6, 8)(11,13)(12,16)(15,17)(19,28)(20,31)(21,34)(22,29)
(23,32)(24,35)(25,30)(26,33)(27,36)(37,55)(38,58)(39,61)(40,56)(41,59)(42,62)
(43,57)(44,60)(45,63)(46,64)(47,67)(48,70)(49,65)(50,68)(51,71)(52,66)(53,69)
(54,72);;
s2 := ( 1, 4)( 2, 5)( 3, 6)(10,13)(11,14)(12,15)(19,31)(20,32)(21,33)(22,28)
(23,29)(24,30)(25,34)(26,35)(27,36)(37,40)(38,41)(39,42)(46,49)(47,50)(48,51)
(55,67)(56,68)(57,69)(58,64)(59,65)(60,66)(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, 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*s2*s1*s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1, 
s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s0*s2*s1*s0*s1*s2*s0 ];;
poly := F / rels;;
 
Permutation Representation (Magma) :
s0 := Sym(72)!( 1,46)( 2,48)( 3,47)( 4,49)( 5,51)( 6,50)( 7,52)( 8,54)( 9,53)
(10,37)(11,39)(12,38)(13,40)(14,42)(15,41)(16,43)(17,45)(18,44)(19,64)(20,66)
(21,65)(22,67)(23,69)(24,68)(25,70)(26,72)(27,71)(28,55)(29,57)(30,56)(31,58)
(32,60)(33,59)(34,61)(35,63)(36,62);
s1 := Sym(72)!( 2, 4)( 3, 7)( 6, 8)(11,13)(12,16)(15,17)(19,28)(20,31)(21,34)
(22,29)(23,32)(24,35)(25,30)(26,33)(27,36)(37,55)(38,58)(39,61)(40,56)(41,59)
(42,62)(43,57)(44,60)(45,63)(46,64)(47,67)(48,70)(49,65)(50,68)(51,71)(52,66)
(53,69)(54,72);
s2 := Sym(72)!( 1, 4)( 2, 5)( 3, 6)(10,13)(11,14)(12,15)(19,31)(20,32)(21,33)
(22,28)(23,29)(24,30)(25,34)(26,35)(27,36)(37,40)(38,41)(39,42)(46,49)(47,50)
(48,51)(55,67)(56,68)(57,69)(58,64)(59,65)(60,66)(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, 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*s2*s1*s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1, 
s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s0*s2*s1*s0*s1*s2*s0 >; 
 
References : None.
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