Polytope of Type {5,10} with group SmallGroup(1600,10261)

Overview

Group: SmallGroup(1600,10261)
Rank: 3
Schläfli Type: {5,10}
Vertices, edges, …: 80, 400, 160
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

{5,10}*320b

10-fold

{5,5}*160

16-fold

{5,10}*100

80-fold

{5,2}*20

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*s2*(s1*s0)^2*s2*s1)^2> 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)^3> of order 2

80 facets

80 of {5}*10

40 vertex figures

40 of {10}*20

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

80 facets

80 of {5}*10

40 vertex figures

40 of {10}*20

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

80 facets

80 of {5}*10

40 vertex figures

40 of {10}*20

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

40 facets

40 of {5}*10

20 vertex figures

20 of {10}*20

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

40 facets

40 of {5}*10

20 vertex figures

20 of {10}*20

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

40 facets

40 of {5}*10

20 vertex figures

20 of {10}*20

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

40 facets

40 of {5}*10

20 vertex figures

20 of {10}*20

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

40 facets

40 of {5}*10

20 vertex figures

20 of {10}*20

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

20 facets

20 of {5}*10

10 vertex figures

10 of {10}*20

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

20 facets

20 of {5}*10

10 vertex figures

10 of {10}*20

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

10 facets

10 of {5}*10

5 vertex figures

5 of {10}*20

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

10 facets

10 of {5}*10

5 vertex figures

5 of {10}*20

Representations

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.

to this polytope.

Overview

Special Properties

Related Polytopes

Quotients maximal quotients in bold

Covers minimal covers in bold

Irregular Quotients of which this is a minimal cover

Representations

References

Twisty Puzzle