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

Overview

Group: SmallGroup(1296,2908)
Rank: 3
Schläfli Type: {4,36}
Vertices, edges, …: 18, 324, 162
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

{4,12}*432b

9-fold

{4,4}*144

18-fold

36-fold

{2,9}*36

54-fold

{2,6}*24

108-fold

{2,3}*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=<(s0*s1)^2> of order 2

90 facets

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

10 vertex figures

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

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

81 facets

81 of {4}*8

9 vertex figures

9 of {36}*72

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

81 facets

81 of {4}*8

9 vertex figures

9 of {36}*72

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

54 facets

54 of {4}*8

6 vertex figures

6 of {36}*72

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

54 facets

54 of {4}*8

6 vertex figures

6 of {36}*72

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

45 facets

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

5 vertex figures

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

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

36 facets

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

4 vertex figures

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

Representations

Permutation Representation (GAP)

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

Permutation Representation (Magma)

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

References

None.

to this polytope.