Part of the Atlas of Small Regular Polytopes

Polytope of Type {10,6}

Atlas Canonical Name {10,6}*1620

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Overview

Group
SmallGroup(1620,422)
Rank
3
Schläfli Type
{10,6}
Vertices, edges, …
135, 405, 81
Order of s0s1s2
5
Order of s0s1s2s1
6
Also known as
if this polytope has a name.

Special Properties

  • Compact Hyperbolic Quotient
  • Locally Spherical
  • Non-Orientable

Quotients maximal quotients in bold

No regular quotients.

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*s2> of order 3

27 facets

63 vertex figures

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

27 facets

45 vertex figures

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

27 facets

45 vertex figures

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

27 facets

45 vertex figures

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

27 facets

45 vertex figures

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

27 facets

45 vertex figures

P/N, where N=<s0*s1*s2*s1*s0*s2, (s0*s1)^2*s2*(s1*s0)^2*s2> of order 9

9 facets

27 vertex figures

P/N, where N=<(s1*s2)^2, s0*s1*s0*(s2*s1)^2*s0*s2*s1*s0*s2> of order 9

9 facets

21 vertex figures

P/N, where N=<s0*s1*s2*s1*s0*s2, (s0*s1)^3*s2*(s1*s0)^3*s2> of order 9

9 facets

21 vertex figures

P/N, where N=<(s1*s2)^2, s0*s1*s2*(s1*s0)^2*(s2*s1)^2*s0*s1*s2> of order 9

9 facets

21 vertex figures

P/N, where N=<(s0*s1)^3*s2*(s1*s0)^3*s2, (s0*s1)^2*s2*(s1*s0)^2*(s2*s1)^2*s2> of order 9

9 facets

15 vertex figures

P/N, where N=<s0*s1*s0*(s2*s1)^2*s0*s2*s1*s0*s2, (s0*s1)^2*s0*s2*(s1*s0)^2*s2*s1*s2> of order 9

9 facets

15 vertex figures

P/N, where N=<(s1*s0*s1*s2)^2, s0*s1*s0*(s2*s1)^2*s0*s2*s1*s0*s2> of order 9

9 facets

15 vertex figures

Representations

Permutation Representation (GAP) 
s0 := ( 4,49)( 5,50)( 6,51)( 7,70)( 8,71)( 9,72)(10,58)(11,59)(12,60)(13,25)(14,26)(15,27)(16,37)(17,38)(18,39)(19,34)(20,35)(21,36)(22,73)(23,74)(24,75)(28,40)(29,41)(30,42)(31,61)(32,62)(33,63)(43,76)(44,77)(45,78)(46,64)(47,65)(48,66)(55,79)(56,80)(57,81);;
s1 := ( 2,59)( 3,36)( 4,61)( 5,29)( 7,31)( 9,57)(10,26)(11,75)(12,49)(13,77)(14,54)(15,19)(16,47)(17,24)(18,79)(20,70)(21,38)(22,66)(23,40)(25,45)(27,68)(30,63)(32,56)(34,58)(37,53)(39,76)(41,81)(42,46)(43,74)(44,51)(48,65)(50,67)(52,72)(64,80)(69,73)(71,78);;
s2 := ( 1, 2)( 4, 8)( 5, 7)( 6, 9)(10,20)(11,19)(12,21)(13,26)(14,25)(15,27)(16,23)(17,22)(18,24)(28,56)(29,55)(30,57)(31,62)(32,61)(33,63)(34,59)(35,58)(36,60)(37,74)(38,73)(39,75)(40,80)(41,79)(42,81)(43,77)(44,76)(45,78)(46,65)(47,64)(48,66)(49,71)(50,70)(51,72)(52,68)(53,67)(54,69);;
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*s1*s2, 
s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2, 
s0*s1*s2*s1*s2*s0*s1*s2*s1*s0*s1*s2*s1*s2*s0*s1*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(81)!( 4,49)( 5,50)( 6,51)( 7,70)( 8,71)( 9,72)(10,58)(11,59)(12,60)(13,25)(14,26)(15,27)(16,37)(17,38)(18,39)(19,34)(20,35)(21,36)(22,73)(23,74)(24,75)(28,40)(29,41)(30,42)(31,61)(32,62)(33,63)(43,76)(44,77)(45,78)(46,64)(47,65)(48,66)(55,79)(56,80)(57,81);
s1 := Sym(81)!( 2,59)( 3,36)( 4,61)( 5,29)( 7,31)( 9,57)(10,26)(11,75)(12,49)(13,77)(14,54)(15,19)(16,47)(17,24)(18,79)(20,70)(21,38)(22,66)(23,40)(25,45)(27,68)(30,63)(32,56)(34,58)(37,53)(39,76)(41,81)(42,46)(43,74)(44,51)(48,65)(50,67)(52,72)(64,80)(69,73)(71,78);
s2 := Sym(81)!( 1, 2)( 4, 8)( 5, 7)( 6, 9)(10,20)(11,19)(12,21)(13,26)(14,25)(15,27)(16,23)(17,22)(18,24)(28,56)(29,55)(30,57)(31,62)(32,61)(33,63)(34,59)(35,58)(36,60)(37,74)(38,73)(39,75)(40,80)(41,79)(42,81)(43,77)(44,76)(45,78)(46,65)(47,64)(48,66)(49,71)(50,70)(51,72)(52,68)(53,67)(54,69);
poly := sub<Sym(81)|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*s1*s2, 
s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2, 
s0*s1*s2*s1*s2*s0*s1*s2*s1*s0*s1*s2*s1*s2*s0*s1*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.

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