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Polytope of Type {2,8,4}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,8,4}*1152a
if this polytope has a name.
Group : SmallGroup(1152,97552)
Rank : 4
Schlafli Type : {2,8,4}
Number of vertices, edges, etc : 2, 72, 144, 36
Order of s₀s₁s₂s₃ : 24
Order of s₀s₁s₂s₃s₂s₁ : 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 : {2,4,4}*576
   4-fold quotients : {2,4,4}*288
   8-fold quotients : {2,4,4}*144
   9-fold quotients : {2,8,4}*128a
   18-fold quotients : {2,4,4}*64, {2,8,2}*64
   36-fold quotients : {2,2,4}*32, {2,4,2}*32
   72-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

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