Polytope of Type {5,10}
Play with this polytope as a twisty puzzle
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {5,10}*1600
if this polytope has a name.
Group : SmallGroup(1600,10261)
Rank : 3
Schlafli Type : {5,10}
Number of vertices, edges, etc : 80, 400, 160
Order of s0s1s2 : 20
Order of s0s1s2s1 : 10
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Related Polytopes :
Facet
Vertex Figure
Dual
Petrial
Facet Of :
None in this Atlas
Vertex Figure Of :
None in this Atlas
Quotients (Maximal Quotients in Boldface) :
5-fold quotients : {5,10}*320b
10-fold quotients : {5,5}*160
16-fold quotients : {5,10}*100
80-fold quotients : {5,2}*20
Covers (Minimal Covers in Boldface) :
None in this atlas.
Irregular Quotients (of which this is a minimal cover):
P/N, where N=<s0*s1*s2*s1*s0*s1*s0*s2*s1*s0*s1*s2*s1*s0*s1*s0*s2*s1> of order 2.
80 facets:
80 of {5}*10
40 vertex figures:
40 of {10}*20
P/N, where N=<s1*s0*s1*s2*s1*s0*s2*s1*s2*s1*s2*s1> of order 2.
80 facets:
80 of {5}*10
40 vertex figures:
40 of {10}*20
P/N, where N=<s0*s1*s0*s1*s0*s2*s1*s0*s2*s1*s0*s1*s2*s1*s2*s1> of order 2.
80 facets:
80 of {5}*10
40 vertex figures:
40 of {10}*20
P/N, where N=<s0*s1*s0*s1*s2*s1*s0*s2*s1*s2*s1*s2*s1*s2> of order 2.
80 facets:
80 of {5}*10
40 vertex figures:
40 of {10}*20
P/N, where N=<s1*s0*s1*s0*s2*s1*s0*s2*s1*s0*s1*s2*s1*s2*s1*s2> of order 2.
80 facets:
80 of {5}*10
48 vertex figures:
32 of {10}*20
16 of {5}*10
P/N, where N=<s0*s1*s2*s1*s0*s2*s1*s2*s1*s2, s0*s1*s0*s2*s1*s2*s1*s0*s1*s0*s2*s1*s2*s1> of order 4.
40 facets:
40 of {5}*10
20 vertex figures:
20 of {10}*20
P/N, where N=<s0*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s1> of order 4.
40 facets:
40 of {5}*10
20 vertex figures:
20 of {10}*20
P/N, where N=<s1*s0*s1*s2*s1*s0*s2*s1*s2*s1*s2*s1*s2> of order 4.
40 facets:
40 of {5}*10
20 vertex figures:
20 of {10}*20
P/N, where N=<s0*s1*s2*s1*s0*s2*s1*s2*s1*s2, s1*s0*s1*s0*s2*s1*s0*s1*s0*s2*s1*s0*s1*s2*s1> of order 4.
40 facets:
40 of {5}*10
20 vertex figures:
20 of {10}*20
P/N, where N=<s1*s0*s2*s1*s2*s1*s0*s2*s1*s2, s0*s1*s0*s1*s0*s2*s1*s0*s2*s1*s0*s1*s2*s1*s2*s1> of order 4.
40 facets:
40 of {5}*10
20 vertex figures:
20 of {10}*20
P/N, where N=<s1*s0*s1*s2*s1*s0*s2*s1*s2*s1*s2*s1, s0*s1*s0*s1*s2*s1*s0*s2*s1*s2*s1*s2*s1*s0> of order 4.
40 facets:
40 of {5}*10
20 vertex figures:
20 of {10}*20
P/N, where N=<s0*s1*s0*s1*s2*s1*s0*s2*s1*s2*s1*s2*s1*s2, s0*s2*s1*s0*s1*s2*s1*s0*s2*s1*s2*s1*s2*s1> of order 4.
40 facets:
40 of {5}*10
20 vertex figures:
20 of {10}*20
P/N, where N=<s0*s1*s0*s1*s2*s1*s0*s2*s1*s0*s1*s2*s1*s2, s0*s1*s0*s1*s0*s2*s1*s0*s2*s1*s0*s1*s2*s1*s2*s1> of order 4.
40 facets:
40 of {5}*10
20 vertex figures:
20 of {10}*20
P/N, where N=<s1*s0*s1*s2*s1*s0*s2*s1*s2*s1*s2*s1, s0*s1*s0*s1*s2*s1*s0*s2*s1*s2*s1*s2*s1*s2> of order 4.
40 facets:
40 of {5}*10
24 vertex figures:
16 of {10}*20
8 of {5}*10
P/N, where N=<s0*s1*s0*s1*s0*s2*s1*s0*s2*s1*s0*s1*s2*s1*s2*s1, s0*s1*s0*s2*s1*s0*s1*s2*s1*s0*s1*s0*s2*s1*s0*s1*s2*s1> of order 4.
40 facets:
40 of {5}*10
20 vertex figures:
20 of {10}*20
P/N, where N=<s1*s0*s1*s2*s1*s0*s2*s1*s2*s1*s2*s1, s0*s1*s2*s1*s0*s1*s0*s2*s1*s0*s1*s2*s1*s0*s1*s0*s2*s1> of order 4.
40 facets:
40 of {5}*10
20 vertex figures:
20 of {10}*20
P/N, where N=<s1*s0*s2*s1*s2*s1*s0*s2*s1*s2, s1*s0*s1*s2*s1*s0*s2*s1*s2*s1*s2*s1> of order 4.
40 facets:
40 of {5}*10
20 vertex figures:
20 of {10}*20
P/N, where N=<s0*s1*s0*s1*s2*s1*s0*s2*s1*s2*s1*s2*s1*s2, s0*s1*s0*s1*s0*s2*s1*s0*s1*s2*s1*s0*s1*s0*s2*s1*s0*s1> of order 4.
40 facets:
40 of {5}*10
20 vertex figures:
20 of {10}*20
P/N, where N=<s0*s1*s0*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s2> of order 5.
32 facets:
32 of {5}*10
20 vertex figures:
15 of {10}*20
5 of {2}*4
P/N, where N=<s0*s1*s2*s1*s0*s2*s1*s2*s1, s1*s0*s2*s1*s2*s1*s0*s2*s1*s2> of order 8.
20 facets:
20 of {5}*10
10 vertex figures:
10 of {10}*20
P/N, where N=<s1*s0*s2*s1*s2*s1*s0*s2*s1*s2, s1*s0*s1*s2*s1*s0*s2*s1*s2*s1*s2*s1*s2> of order 8.
20 facets:
20 of {5}*10
10 vertex figures:
10 of {10}*20
P/N, where N=<s1*s0*s2*s1*s2*s1*s0*s2*s1*s2, s0*s1*s0*s1*s2*s1*s0*s2*s1*s0*s1*s2*s1, s0*s1*s0*s1*s0*s2*s1*s0*s2*s1*s0*s1*s2*s1*s2*s1> of order 8.
20 facets:
20 of {5}*10
10 vertex figures:
10 of {10}*20
P/N, where N=<s1*s0*s2*s1*s2*s1*s0*s2*s1*s2, s1*s0*s1*s2*s1*s0*s2*s1*s2*s1*s2*s1, s0*s1*s0*s1*s2*s1*s0*s2*s1*s2*s1*s2*s1*s0> of order 8.
20 facets:
20 of {5}*10
10 vertex figures:
10 of {10}*20
P/N, where N=<s0*s2*s1*s2*s1*s0*s2*s1*s2*s1, s0*s1*s0*s1*s2*s1*s0*s2*s1*s0*s1*s2*s1*s2, s0*s1*s2*s1*s0*s2*s1*s0*s1*s2*s1*s0*s2*s1> of order 8.
20 facets:
20 of {5}*10
10 vertex figures:
10 of {10}*20
P/N, where N=<s0*s1*s2*s1*s0*s2*s1*s2*s1*s2, s1*s0*s1*s2*s1*s0*s2*s1*s2*s1*s2*s1, s0*s1*s0*s1*s2*s1*s0*s2*s1*s0*s1*s2*s1*s2> of order 8.
20 facets:
20 of {5}*10
10 vertex figures:
10 of {10}*20
P/N, where N=<s0*s1*s2*s1*s0*s2*s1*s2*s1, s1*s0*s2*s1*s2*s1*s0*s2*s1*s2, s1*s0*s1*s2*s1*s0*s2*s1*s2*s1*s2*s1*s2> of order 16.
10 facets:
10 of {5}*10
5 vertex figures:
5 of {10}*20
P/N, where N=<s0*s2*s1*s2*s1*s0*s2*s1*s2*s1, s1*s0*s1*s2*s1*s0*s2*s1*s2*s1*s2*s1, s0*s1*s0*s1*s0*s2*s1*s0*s1*s0*s2*s1*s2> of order 16.
10 facets:
10 of {5}*10
5 vertex figures:
5 of {10}*20
P/N, where N=<s0*s1*s2*s1*s0*s2*s1*s2*s1, s1*s0*s2*s1*s2*s1*s0*s2*s1*s2, s1*s0*s1*s2*s1*s0*s2*s1*s2*s1*s2*s1> of order 16.
10 facets:
10 of {5}*10
5 vertex figures:
5 of {10}*20
Permutation Representation (GAP) :
s0 := ( 3, 4)( 5, 6)( 9,16)(10,15)(11,13)(12,14)(17,65)(18,66)(19,68)(20,67)(21,70)(22,69)(23,71)(24,72)(25,80)(26,79)(27,77)(28,78)(29,75)(30,76)(31,74)(32,73)(33,49)(34,50)(35,52)(36,51)(37,54)(38,53)(39,55)(40,56)(41,64)(42,63)(43,61)(44,62)(45,59)(46,60)(47,58)(48,57);;
s1 := ( 1,17)( 2,26)( 3,27)( 4,20)( 5,32)( 6,23)( 7,22)( 8,29)( 9,25)(10,18)(11,19)(12,28)(13,24)(14,31)(15,30)(16,21)(33,65)(34,74)(35,75)(36,68)(37,80)(38,71)(39,70)(40,77)(41,73)(42,66)(43,67)(44,76)(45,72)(46,79)(47,78)(48,69)(50,58)(51,59)(53,64)(54,55)(56,61)(62,63);;
s2 := ( 1, 2)( 7, 8)( 9,15)(10,16)(11,14)(12,13)(17,18)(23,24)(25,31)(26,32)(27,30)(28,29)(33,34)(39,40)(41,47)(42,48)(43,46)(44,45)(49,50)(55,56)(57,63)(58,64)(59,62)(60,61)(65,66)(71,72)(73,79)(74,80)(75,78)(76,77);;
poly := Group([s0,s1,s2]);;
Finitely Presented Group Representation (GAP) :
F := FreeGroup("s0","s1","s2");;
s0 := F.1;; s1 := F.2;; s2 := F.3;;
rels := [ s0*s0, s1*s1, s2*s2, s0*s2*s0*s2, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s0*s1*s2*s0*s1*s2*s1*s2*s0*s1*s0*s1*s2*s0*s1*s2*s1*s2*s0*s1,
s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1,
s0*s1*s2*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s2*s1 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(80)!( 3, 4)( 5, 6)( 9,16)(10,15)(11,13)(12,14)(17,65)(18,66)(19,68)(20,67)(21,70)(22,69)(23,71)(24,72)(25,80)(26,79)(27,77)(28,78)(29,75)(30,76)(31,74)(32,73)(33,49)(34,50)(35,52)(36,51)(37,54)(38,53)(39,55)(40,56)(41,64)(42,63)(43,61)(44,62)(45,59)(46,60)(47,58)(48,57);
s1 := Sym(80)!( 1,17)( 2,26)( 3,27)( 4,20)( 5,32)( 6,23)( 7,22)( 8,29)( 9,25)(10,18)(11,19)(12,28)(13,24)(14,31)(15,30)(16,21)(33,65)(34,74)(35,75)(36,68)(37,80)(38,71)(39,70)(40,77)(41,73)(42,66)(43,67)(44,76)(45,72)(46,79)(47,78)(48,69)(50,58)(51,59)(53,64)(54,55)(56,61)(62,63);
s2 := Sym(80)!( 1, 2)( 7, 8)( 9,15)(10,16)(11,14)(12,13)(17,18)(23,24)(25,31)(26,32)(27,30)(28,29)(33,34)(39,40)(41,47)(42,48)(43,46)(44,45)(49,50)(55,56)(57,63)(58,64)(59,62)(60,61)(65,66)(71,72)(73,79)(74,80)(75,78)(76,77);
poly := sub<Sym(80)|s0,s1,s2>;
Finitely Presented Group Representation (Magma) :
poly<s0,s1,s2> := Group< s0,s1,s2 | s0*s0, s1*s1, s2*s2,
s0*s2*s0*s2, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s0*s1*s2*s0*s1*s2*s1*s2*s0*s1*s0*s1*s2*s0*s1*s2*s1*s2*s0*s1,
s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1,
s0*s1*s2*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s2*s1 >;
References : None.
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