Polytope of Type {10,4,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {10,4,6}*720
if this polytope has a name.
Group : SmallGroup(720,784)
Rank : 4
Schlafli Type : {10,4,6}
Number of vertices, edges, etc : 10, 30, 18, 9
Order of s₀s₁s₂s₃ : 20
Order of s₀s₁s₂s₃s₂s₁ : 2
Special Properties :
   Universal
   Non-Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {10,4,6,2} of size 1440
Vertex Figure Of :
   {2,10,4,6} of size 1440
Quotients (Maximal Quotients in Boldface) :
   5-fold quotients : {2,4,6}*144
Covers (Minimal Covers in Boldface) :
   2-fold covers : {20,4,6}*1440, {10,4,6}*1440c
Permutation Representation (GAP) :

s0 := ( 2, 5)( 3, 4)( 7,10)( 8, 9)(12,15)(13,14)(17,20)(18,19)(22,25)(23,24)
(27,30)(28,29)(32,35)(33,34)(37,40)(38,39)(42,45)(43,44);;
s1 := ( 1, 2)( 3, 5)( 6,37)( 7,36)( 8,40)( 9,39)(10,38)(11,27)(12,26)(13,30)
(14,29)(15,28)(16,32)(17,31)(18,35)(19,34)(20,33)(21,22)(23,25)(41,42)
(43,45);;
s2 := (16,41)(17,42)(18,43)(19,44)(20,45)(21,31)(22,32)(23,33)(24,34)(25,35)
(26,36)(27,37)(28,38)(29,39)(30,40);;
s3 := ( 1,21)( 2,22)( 3,23)( 4,24)( 5,25)( 6,16)( 7,17)( 8,18)( 9,19)(10,20)
(11,26)(12,27)(13,28)(14,29)(15,30)(31,36)(32,37)(33,38)(34,39)(35,40);;
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, s0*s1*s2*s1*s0*s1*s2*s1, 
s1*s2*s1*s2*s1*s2*s1*s2, s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
s2*s3*s2*s1*s2*s3*s1*s2*s1*s2*s3*s2*s1, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(45)!( 2, 5)( 3, 4)( 7,10)( 8, 9)(12,15)(13,14)(17,20)(18,19)(22,25)
(23,24)(27,30)(28,29)(32,35)(33,34)(37,40)(38,39)(42,45)(43,44);
s1 := Sym(45)!( 1, 2)( 3, 5)( 6,37)( 7,36)( 8,40)( 9,39)(10,38)(11,27)(12,26)
(13,30)(14,29)(15,28)(16,32)(17,31)(18,35)(19,34)(20,33)(21,22)(23,25)(41,42)
(43,45);
s2 := Sym(45)!(16,41)(17,42)(18,43)(19,44)(20,45)(21,31)(22,32)(23,33)(24,34)
(25,35)(26,36)(27,37)(28,38)(29,39)(30,40);
s3 := Sym(45)!( 1,21)( 2,22)( 3,23)( 4,24)( 5,25)( 6,16)( 7,17)( 8,18)( 9,19)
(10,20)(11,26)(12,27)(13,28)(14,29)(15,30)(31,36)(32,37)(33,38)(34,39)(35,40);
poly := sub<Sym(45)|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, 
s0*s1*s2*s1*s0*s1*s2*s1, s1*s2*s1*s2*s1*s2*s1*s2, 
s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
s2*s3*s2*s1*s2*s3*s1*s2*s1*s2*s3*s2*s1, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 >;

References : None.
to this polytope