Polytope of Type {8,6,3}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {8,6,3}*384
Also Known As : {{8,6|2},{6,3}₄}. if this polytope has another name.
Group : SmallGroup(384,18032)
Rank : 4
Schlafli Type : {8,6,3}
Number of vertices, edges, etc : 8, 32, 12, 4
Order of s₀s₁s₂s₃ : 8
Order of s₀s₁s₂s₃s₂s₁ : 2
Special Properties :
   Universal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {8,6,3,2} of size 768
   {8,6,3,3} of size 1920
Vertex Figure Of :
   {2,8,6,3} of size 768
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {4,6,3}*192
   4-fold quotients : {2,6,3}*96
   8-fold quotients : {2,3,3}*48
Covers (Minimal Covers in Boldface) :
   2-fold covers : {8,12,3}*768, {16,6,3}*768, {8,6,6}*768
   3-fold covers : {24,6,3}*1152, {8,6,3}*1152
   5-fold covers : {40,6,3}*1920, {8,6,15}*1920
Permutation Representation (GAP) :

s0 := (25,37)(26,38)(27,39)(28,40)(29,41)(30,42)(31,43)(32,44)(33,45)(34,46)
(35,47)(36,48)(49,73)(50,74)(51,75)(52,76)(53,77)(54,78)(55,79)(56,80)(57,81)
(58,82)(59,83)(60,84)(61,85)(62,86)(63,87)(64,88)(65,89)(66,90)(67,91)(68,92)
(69,93)(70,94)(71,95)(72,96);;
s1 := ( 1,49)( 2,51)( 3,50)( 4,52)( 5,57)( 6,59)( 7,58)( 8,60)( 9,53)(10,55)
(11,54)(12,56)(13,61)(14,63)(15,62)(16,64)(17,69)(18,71)(19,70)(20,72)(21,65)
(22,67)(23,66)(24,68)(25,85)(26,87)(27,86)(28,88)(29,93)(30,95)(31,94)(32,96)
(33,89)(34,91)(35,90)(36,92)(37,73)(38,75)(39,74)(40,76)(41,81)(42,83)(43,82)
(44,84)(45,77)(46,79)(47,78)(48,80);;
s2 := ( 1, 5)( 2, 6)( 3, 8)( 4, 7)(11,12)(13,17)(14,18)(15,20)(16,19)(23,24)
(25,29)(26,30)(27,32)(28,31)(35,36)(37,41)(38,42)(39,44)(40,43)(47,48)(49,53)
(50,54)(51,56)(52,55)(59,60)(61,65)(62,66)(63,68)(64,67)(71,72)(73,77)(74,78)
(75,80)(76,79)(83,84)(85,89)(86,90)(87,92)(88,91)(95,96);;
s3 := ( 1, 4)( 5,12)( 6,10)( 7,11)( 8, 9)(13,16)(17,24)(18,22)(19,23)(20,21)
(25,28)(29,36)(30,34)(31,35)(32,33)(37,40)(41,48)(42,46)(43,47)(44,45)(49,52)
(53,60)(54,58)(55,59)(56,57)(61,64)(65,72)(66,70)(67,71)(68,69)(73,76)(77,84)
(78,82)(79,83)(80,81)(85,88)(89,96)(90,94)(91,95)(92,93);;
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, 
s0*s1*s2*s1*s0*s1*s2*s1, s1*s3*s2*s1*s3*s2*s1*s3*s2*s1*s3*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 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(96)!(25,37)(26,38)(27,39)(28,40)(29,41)(30,42)(31,43)(32,44)(33,45)
(34,46)(35,47)(36,48)(49,73)(50,74)(51,75)(52,76)(53,77)(54,78)(55,79)(56,80)
(57,81)(58,82)(59,83)(60,84)(61,85)(62,86)(63,87)(64,88)(65,89)(66,90)(67,91)
(68,92)(69,93)(70,94)(71,95)(72,96);
s1 := Sym(96)!( 1,49)( 2,51)( 3,50)( 4,52)( 5,57)( 6,59)( 7,58)( 8,60)( 9,53)
(10,55)(11,54)(12,56)(13,61)(14,63)(15,62)(16,64)(17,69)(18,71)(19,70)(20,72)
(21,65)(22,67)(23,66)(24,68)(25,85)(26,87)(27,86)(28,88)(29,93)(30,95)(31,94)
(32,96)(33,89)(34,91)(35,90)(36,92)(37,73)(38,75)(39,74)(40,76)(41,81)(42,83)
(43,82)(44,84)(45,77)(46,79)(47,78)(48,80);
s2 := Sym(96)!( 1, 5)( 2, 6)( 3, 8)( 4, 7)(11,12)(13,17)(14,18)(15,20)(16,19)
(23,24)(25,29)(26,30)(27,32)(28,31)(35,36)(37,41)(38,42)(39,44)(40,43)(47,48)
(49,53)(50,54)(51,56)(52,55)(59,60)(61,65)(62,66)(63,68)(64,67)(71,72)(73,77)
(74,78)(75,80)(76,79)(83,84)(85,89)(86,90)(87,92)(88,91)(95,96);
s3 := Sym(96)!( 1, 4)( 5,12)( 6,10)( 7,11)( 8, 9)(13,16)(17,24)(18,22)(19,23)
(20,21)(25,28)(29,36)(30,34)(31,35)(32,33)(37,40)(41,48)(42,46)(43,47)(44,45)
(49,52)(53,60)(54,58)(55,59)(56,57)(61,64)(65,72)(66,70)(67,71)(68,69)(73,76)
(77,84)(78,82)(79,83)(80,81)(85,88)(89,96)(90,94)(91,95)(92,93);
poly := sub<Sym(96)|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, s0*s1*s2*s1*s0*s1*s2*s1, 
s1*s3*s2*s1*s3*s2*s1*s3*s2*s1*s3*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 >;

References : None.
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