Polytope of Type {8,4,3}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {8,4,3}*768a
if this polytope has a name.
Group : SmallGroup(768,1086051)
Rank : 4
Schlafli Type : {8,4,3}
Number of vertices, edges, etc : 32, 64, 24, 3
Order of s₀s₁s₂s₃ : 6
Order of s₀s₁s₂s₃s₂s₁ : 8
Special Properties :
   Universal
   Non-Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   4-fold quotients : {4,4,3}*192a
   16-fold quotients : {2,4,3}*48
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

s0 := ( 5, 6)( 7, 8)( 9,11)(10,12)(13,16)(14,15)(17,22)(18,21)(19,24)(20,23)
(25,32)(26,31)(27,30)(28,29)(33,42)(34,41)(35,44)(36,43)(37,45)(38,46)(39,47)
(40,48)(49,64)(50,63)(51,62)(52,61)(53,60)(54,59)(55,58)(56,57);;
s1 := ( 1,49)( 2,50)( 3,51)( 4,52)( 5,55)( 6,56)( 7,53)( 8,54)( 9,58)(10,57)
(11,60)(12,59)(13,64)(14,63)(15,62)(16,61)(17,33)(18,34)(19,35)(20,36)(21,39)
(22,40)(23,37)(24,38)(25,42)(26,41)(27,44)(28,43)(29,48)(30,47)(31,46)
(32,45);;
s2 := ( 3, 4)( 7, 8)( 9,15)(10,16)(11,14)(12,13)(19,20)(23,24)(25,31)(26,32)
(27,30)(28,29)(33,49)(34,50)(35,52)(36,51)(37,53)(38,54)(39,56)(40,55)(41,63)
(42,64)(43,62)(44,61)(45,60)(46,59)(47,57)(48,58);;
s3 := ( 2, 3)( 5,12)( 6,10)( 7,11)( 8, 9)(14,15)(17,33)(18,35)(19,34)(20,36)
(21,44)(22,42)(23,43)(24,41)(25,40)(26,38)(27,39)(28,37)(29,45)(30,47)(31,46)
(32,48)(50,51)(53,60)(54,58)(55,59)(56,57)(62,63);;
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*s2*s0*s2, 
s0*s3*s0*s3, s1*s3*s1*s3, s2*s3*s2*s3*s2*s3, 
s1*s2*s1*s2*s1*s2*s1*s2, s3*s1*s2*s3*s1*s2*s3*s1*s2, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s2*s0*s1*s0*s1*s0*s2*s1*s2*s1*s0*s1*s2*s1*s0*s1*s0*s1 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(64)!( 5, 6)( 7, 8)( 9,11)(10,12)(13,16)(14,15)(17,22)(18,21)(19,24)
(20,23)(25,32)(26,31)(27,30)(28,29)(33,42)(34,41)(35,44)(36,43)(37,45)(38,46)
(39,47)(40,48)(49,64)(50,63)(51,62)(52,61)(53,60)(54,59)(55,58)(56,57);
s1 := Sym(64)!( 1,49)( 2,50)( 3,51)( 4,52)( 5,55)( 6,56)( 7,53)( 8,54)( 9,58)
(10,57)(11,60)(12,59)(13,64)(14,63)(15,62)(16,61)(17,33)(18,34)(19,35)(20,36)
(21,39)(22,40)(23,37)(24,38)(25,42)(26,41)(27,44)(28,43)(29,48)(30,47)(31,46)
(32,45);
s2 := Sym(64)!( 3, 4)( 7, 8)( 9,15)(10,16)(11,14)(12,13)(19,20)(23,24)(25,31)
(26,32)(27,30)(28,29)(33,49)(34,50)(35,52)(36,51)(37,53)(38,54)(39,56)(40,55)
(41,63)(42,64)(43,62)(44,61)(45,60)(46,59)(47,57)(48,58);
s3 := Sym(64)!( 2, 3)( 5,12)( 6,10)( 7,11)( 8, 9)(14,15)(17,33)(18,35)(19,34)
(20,36)(21,44)(22,42)(23,43)(24,41)(25,40)(26,38)(27,39)(28,37)(29,45)(30,47)
(31,46)(32,48)(50,51)(53,60)(54,58)(55,59)(56,57)(62,63);
poly := sub<Sym(64)|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*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s2*s3*s2*s3*s2*s3, s1*s2*s1*s2*s1*s2*s1*s2, 
s3*s1*s2*s3*s1*s2*s3*s1*s2, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s2*s0*s1*s0*s1*s0*s2*s1*s2*s1*s0*s1*s2*s1*s0*s1*s0*s1 >;

References : None.
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