Part of the Atlas of Small Regular Polytopes

Polytope of Type {10,5}

Atlas Canonical Name {10,5}*1600

▶ Play as a twisty puzzle

Overview

Group
SmallGroup(1600,10261)
Rank
3
Schläfli Type
{10,5}
Vertices, edges, …
160, 400, 80
Order of s0s1s2
20
Order of s0s1s2s1
10
Also known as
if this polytope has a name.

Special Properties

  • Compact Hyperbolic Quotient
  • Locally Spherical
  • Orientable

Quotients maximal quotients in bold

5-fold

10-fold

16-fold

80-fold

Covers minimal covers in bold

None in this atlas.

Irregular Quotients of which this is a minimal cover

Click an entry to reveal its facets and vertex figures.

P/N, where N=<(s0*s1)^3*s0*s2*(s1*s0)^4*s2*s1> of order 2

40 facets

80 vertex figures

P/N, where N=<(s1*s0)^3*s2*s1*s0*s1*s2*s1> of order 2

40 facets

80 vertex figures

P/N, where N=<(s1*s0)^2*s1*(s2*s1*s0)^2*(s1*s2)^2*s1> of order 2

40 facets

80 vertex figures

P/N, where N=<(s0*s1)^2*s2*s1*s0*s1*s2*s1> of order 2

40 facets

80 vertex figures

P/N, where N=<(s0*s1)^5> of order 2

48 facets

80 vertex figures

P/N, where N=<(s0*s1)^2*s0*s2*s1*s0*s1*s2, (s0*s1*s0*(s2*s1)^2)^2> of order 4

20 facets

40 vertex figures

P/N, where N=<s1*(s2*s1*s0)^3*s2*s1*s2> of order 4

20 facets

40 vertex figures

P/N, where N=<(s0*s1)^3*s0*s2*s1*s0*s1*s2*s1> of order 4

20 facets

40 vertex figures

P/N, where N=<(s0*s1)^2*s0*s2*s1*s0*s1*s2, s0*(s1*s2)^2*(s1*s0)^2*(s2*s1)^2> of order 4

20 facets

40 vertex figures

P/N, where N=<s0*s1*s0*s2*(s1*s0)^2*s2*s1, (s1*s0)^2*s1*(s2*s1*s0)^2*(s1*s2)^2*s1> of order 4

20 facets

40 vertex figures

P/N, where N=<(s1*s0)^3*s2*s1*s0*s1*s2*s1, (s0*s1)^3*s0*s2*s1*s0*s1*s2*s1*s0> of order 4

20 facets

40 vertex figures

P/N, where N=<(s0*s1)^2*s2*s1*s0*s1*s2*s1, (s0*s1*s2*s1)^2*s0*s1> of order 4

20 facets

40 vertex figures

P/N, where N=<(s0*s1)^2*(s2*s1*s0)^2*(s1*s2)^2, (s1*s0)^2*s1*(s2*s1*s0)^2*(s1*s2)^2*s1> of order 4

20 facets

40 vertex figures

P/N, where N=<(s0*s1)^5, (s0*s1)^2*s2*s1*s0*s1*s2*s1> of order 4

24 facets

40 vertex figures

P/N, where N=<(s1*s0)^2*s1*(s2*s1*s0)^2*(s1*s2)^2*s1, (s0*s1)^3*s0*s2*(s1*s0)^4*s2*s1> of order 4

20 facets

40 vertex figures

P/N, where N=<(s1*s0)^3*s2*s1*s0*s1*s2*s1, (s0*s1)^3*s0*s2*(s1*s0)^4*s2*s1> of order 4

20 facets

40 vertex figures

P/N, where N=<s0*s1*s0*s2*(s1*s0)^2*s2*s1, (s1*s0)^3*s2*s1*s0*s1*s2*s1> of order 4

20 facets

40 vertex figures

P/N, where N=<(s0*s1)^2*s2*s1*s0*s1*s2*s1, (s1*s0)^3*s1*s2*(s1*s0)^2*s2*s1> of order 4

20 facets

40 vertex figures

P/N, where N=<(s0*s1)^2> of order 5

20 facets

32 vertex figures

P/N, where N=<(s1*s0)^2*s2*s1*s0*s1*s2, s0*s1*s0*s2*(s1*s0)^2*s2*s1> of order 8

10 facets

20 vertex figures

P/N, where N=<s0*s1*s0*s2*(s1*s0)^2*s2*s1, (s0*s1)^3*s0*s2*s1*s0*s1*s2*s1> of order 8

10 facets

20 vertex figures

P/N, where N=<s0*s1*s0*s2*(s1*s0)^2*s2*s1, s1*s0*s1*(s2*s1*s0)^2*(s1*s2)^2, (s1*s0)^2*s1*(s2*s1*s0)^2*(s1*s2)^2*s1> of order 8

10 facets

20 vertex figures

P/N, where N=<s0*s1*s0*s2*(s1*s0)^2*s2*s1, (s1*s0)^3*s2*s1*s0*s1*s2*s1, (s0*s1)^3*s0*s2*s1*s0*s1*s2*s1*s0> of order 8

10 facets

20 vertex figures

P/N, where N=<s0*s2*(s1*s0)^2*s2*s1*s0*s1, (s0*s1)^2*(s2*s1*s0)^2*(s1*s2)^2, s0*s1*(s2*s1*s0)^2*s1*s2*s1*s0*s2*s1> of order 8

10 facets

20 vertex figures

P/N, where N=<(s0*s1)^2*s0*s2*s1*s0*s1*s2, (s1*s0)^3*s2*s1*s0*s1*s2*s1, (s0*s1)^3*s0*s2*s1*s0*s1*s2*s1*s0> of order 8

10 facets

20 vertex figures

P/N, where N=<(s1*s0)^2*s2*s1*s0*s1*s2, s0*s1*s0*s2*(s1*s0)^2*s2*s1, (s0*s1)^3*s0*s2*s1*s0*s1*s2*s1> of order 16

5 facets

10 vertex figures

P/N, where N=<s0*s2*(s1*s0)^2*s2*s1*s0*s1, (s1*s0)^3*s2*s1*s0*s1*s2*s1, (s0*s1)^4*s2*s1*s0*s1*s2> of order 16

5 facets

10 vertex figures

P/N, where N=<(s1*s0)^2*s2*s1*s0*s1*s2, s0*s1*s0*s2*(s1*s0)^2*s2*s1, (s1*s0)^3*s2*s1*s0*s1*s2*s1> of order 16

5 facets

10 vertex figures

Representations

Permutation Representation (GAP) 
s0 := ( 3, 4)( 5, 6)( 9,16)(10,15)(11,13)(12,14)(19,20)(21,22)(25,32)(26,31)(27,29)(28,30)(35,36)(37,38)(41,48)(42,47)(43,45)(44,46)(51,52)(53,54)(57,64)(58,63)(59,61)(60,62)(67,68)(69,70)(73,80)(74,79)(75,77)(76,78);;
s1 := ( 2,10)( 3,11)( 5,16)( 6, 7)( 8,13)(14,15)(17,65)(18,74)(19,75)(20,68)(21,80)(22,71)(23,70)(24,77)(25,73)(26,66)(27,67)(28,76)(29,72)(30,79)(31,78)(32,69)(33,49)(34,58)(35,59)(36,52)(37,64)(38,55)(39,54)(40,61)(41,57)(42,50)(43,51)(44,60)(45,56)(46,63)(47,62)(48,53);;
s2 := ( 1,18)( 2,17)( 3,19)( 4,20)( 5,21)( 6,22)( 7,24)( 8,23)( 9,31)(10,32)(11,30)(12,29)(13,28)(14,27)(15,25)(16,26)(33,66)(34,65)(35,67)(36,68)(37,69)(38,70)(39,72)(40,71)(41,79)(42,80)(43,78)(44,77)(45,76)(46,75)(47,73)(48,74)(49,50)(55,56)(57,63)(58,64)(59,62)(60,61);;
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, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s0*s1*s0*s1*s0*s2*s1*s2*s1*s0*s2*s1*s0*s1*s0*s1*s2*s1*s2*s1, 
s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*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(80)!( 3, 4)( 5, 6)( 9,16)(10,15)(11,13)(12,14)(19,20)(21,22)(25,32)(26,31)(27,29)(28,30)(35,36)(37,38)(41,48)(42,47)(43,45)(44,46)(51,52)(53,54)(57,64)(58,63)(59,61)(60,62)(67,68)(69,70)(73,80)(74,79)(75,77)(76,78);
s1 := Sym(80)!( 2,10)( 3,11)( 5,16)( 6, 7)( 8,13)(14,15)(17,65)(18,74)(19,75)(20,68)(21,80)(22,71)(23,70)(24,77)(25,73)(26,66)(27,67)(28,76)(29,72)(30,79)(31,78)(32,69)(33,49)(34,58)(35,59)(36,52)(37,64)(38,55)(39,54)(40,61)(41,57)(42,50)(43,51)(44,60)(45,56)(46,63)(47,62)(48,53);
s2 := Sym(80)!( 1,18)( 2,17)( 3,19)( 4,20)( 5,21)( 6,22)( 7,24)( 8,23)( 9,31)(10,32)(11,30)(12,29)(13,28)(14,27)(15,25)(16,26)(33,66)(34,65)(35,67)(36,68)(37,69)(38,70)(39,72)(40,71)(41,79)(42,80)(43,78)(44,77)(45,76)(46,75)(47,73)(48,74)(49,50)(55,56)(57,63)(58,64)(59,62)(60,61);
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, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s0*s1*s0*s1*s0*s2*s1*s2*s1*s0*s2*s1*s0*s1*s0*s1*s2*s1*s2*s1, 
s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*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.

Twisty Puzzle