Polytope of Type {36,4} with group SmallGroup(1296,2908)

Overview

Group: SmallGroup(1296,2908)
Rank: 3
Schläfli Type: {36,4}
Vertices, edges, …: 162, 324, 18
Order of s₀s₁s₂: 18
Order of s₀s₁s₂s₁: 6
Also known as: if this polytope has a name.

Special Properties

Compact Hyperbolic Quotient
Locally Spherical
Orientable

Quotients maximal quotients in bold

3-fold

{12,4}*432b

9-fold

{4,4}*144

18-fold

36-fold

{9,2}*36

54-fold

{6,2}*24

108-fold

{3,2}*12

162-fold

{2,2}*8

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=<(s2*s1*s0)^2*s1*(s2*s1*s0)^2*s2*s1*s2> of order 2

10 facets

2 of {18}*36
8 of {36}*72

90 vertex figures

72 of {4}*8
18 of {2}*4

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

9 facets

9 of {36}*72

81 vertex figures

81 of {4}*8

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

9 facets

9 of {36}*72

81 vertex figures

81 of {4}*8

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

6 facets

6 of {36}*72

54 vertex figures

54 of {4}*8

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

6 facets

6 of {36}*72

54 vertex figures

54 of {4}*8

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

5 facets

1 of {18}*36
4 of {36}*72

45 vertex figures

36 of {4}*8
9 of {2}*4

P/N, where N=<(s1*s0*s1*s2)^2, (s1*s2*s1*s0)^2*(s1*s2)^2> of order 6

4 facets

2 of {18}*36
2 of {36}*72

36 vertex figures

18 of {4}*8
18 of {2}*4

Representations

Permutation Representation (GAP)

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

Permutation Representation (Magma)

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

References

None.

to this polytope.