Polytope of Type {3,2,8,8}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {3,2,8,8}*768a
if this polytope has a name.
Group : SmallGroup(768,145169)
Rank : 5
Schlafli Type : {3,2,8,8}
Number of vertices, edges, etc : 3, 3, 8, 32, 8
Order of s₀s₁s₂s₃s₄ : 24
Order of s₀s₁s₂s₃s₄s₃s₂s₁ : 2
Special Properties :
   Degenerate
   Universal
   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) :
   2-fold quotients : {3,2,4,8}*384a, {3,2,8,4}*384b
   4-fold quotients : {3,2,4,4}*192, {3,2,2,8}*192
   8-fold quotients : {3,2,2,4}*96, {3,2,4,2}*96
   16-fold quotients : {3,2,2,2}*48
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

s0 := (2,3);;
s1 := (1,2);;
s2 := ( 4,20)( 5,21)( 6,22)( 7,23)( 8,25)( 9,24)(10,27)(11,26)(12,29)(13,28)
(14,31)(15,30)(16,32)(17,33)(18,34)(19,35)(36,52)(37,53)(38,54)(39,55)(40,57)
(41,56)(42,59)(43,58)(44,61)(45,60)(46,63)(47,62)(48,64)(49,65)(50,66)
(51,67);;
s3 := ( 8, 9)(10,11)(12,14)(13,15)(16,19)(17,18)(20,24)(21,25)(22,26)(23,27)
(28,34)(29,35)(30,32)(31,33)(36,44)(37,45)(38,46)(39,47)(40,49)(41,48)(42,51)
(43,50)(52,65)(53,64)(54,67)(55,66)(56,61)(57,60)(58,63)(59,62);;
s4 := ( 4,36)( 5,37)( 6,38)( 7,39)( 8,41)( 9,40)(10,43)(11,42)(12,46)(13,47)
(14,44)(15,45)(16,51)(17,50)(18,49)(19,48)(20,52)(21,53)(22,54)(23,55)(24,57)
(25,56)(26,59)(27,58)(28,62)(29,63)(30,60)(31,61)(32,67)(33,66)(34,65)
(35,64);;
poly := Group([s0,s1,s2,s3,s4]);;

Finitely Presented Group Representation (GAP) :

F := FreeGroup("s0","s1","s2","s3","s4");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;  s4 := F.5;;  
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s4*s4, s0*s2*s0*s2, 
s1*s2*s1*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s0*s4*s0*s4, s1*s4*s1*s4, s2*s4*s2*s4, 
s0*s1*s0*s1*s0*s1, s4*s2*s3*s2*s3*s4*s2*s3*s2*s3, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(67)!(2,3);
s1 := Sym(67)!(1,2);
s2 := Sym(67)!( 4,20)( 5,21)( 6,22)( 7,23)( 8,25)( 9,24)(10,27)(11,26)(12,29)
(13,28)(14,31)(15,30)(16,32)(17,33)(18,34)(19,35)(36,52)(37,53)(38,54)(39,55)
(40,57)(41,56)(42,59)(43,58)(44,61)(45,60)(46,63)(47,62)(48,64)(49,65)(50,66)
(51,67);
s3 := Sym(67)!( 8, 9)(10,11)(12,14)(13,15)(16,19)(17,18)(20,24)(21,25)(22,26)
(23,27)(28,34)(29,35)(30,32)(31,33)(36,44)(37,45)(38,46)(39,47)(40,49)(41,48)
(42,51)(43,50)(52,65)(53,64)(54,67)(55,66)(56,61)(57,60)(58,63)(59,62);
s4 := Sym(67)!( 4,36)( 5,37)( 6,38)( 7,39)( 8,41)( 9,40)(10,43)(11,42)(12,46)
(13,47)(14,44)(15,45)(16,51)(17,50)(18,49)(19,48)(20,52)(21,53)(22,54)(23,55)
(24,57)(25,56)(26,59)(27,58)(28,62)(29,63)(30,60)(31,61)(32,67)(33,66)(34,65)
(35,64);
poly := sub<Sym(67)|s0,s1,s2,s3,s4>;

Finitely Presented Group Representation (Magma) :

poly<s0,s1,s2,s3,s4> := Group< s0,s1,s2,s3,s4 | s0*s0, s1*s1, s2*s2, 
s3*s3, s4*s4, s0*s2*s0*s2, s1*s2*s1*s2, 
s0*s3*s0*s3, s1*s3*s1*s3, s0*s4*s0*s4, 
s1*s4*s1*s4, s2*s4*s2*s4, s0*s1*s0*s1*s0*s1, 
s4*s2*s3*s2*s3*s4*s2*s3*s2*s3, s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4 >;

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