Polytope of Type {18,6} with group SmallGroup(216,101)

Overview

Group: SmallGroup(216,101)
Rank: 3
Schläfli Type: {18,6}
Vertices, edges, …: 18, 54, 6
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
Flat

Related Polytopes

Constructions

Facet of

Vertex figure of

Quotients maximal quotients in bold

2-fold

{9,6}*108

3-fold

6-fold

9-fold

{6,2}*24

18-fold

{3,2}*12

27-fold

{2,2}*8

Covers minimal covers in bold

2-fold

3-fold

4-fold

5-fold

6-fold

7-fold

8-fold

9-fold

Irregular Quotients of which this is a minimal cover

None.

Representations

Permutation Representation (GAP)

s0 := ( 2, 3)( 4, 7)( 5, 9)( 6, 8)(10,20)(11,19)(12,21)(13,26)(14,25)(15,27)(16,23)(17,22)(18,24)(29,30)(31,34)(32,36)(33,35)(37,47)(38,46)(39,48)(40,53)(41,52)(42,54)(43,50)(44,49)(45,51);;
s1 := ( 1,40)( 2,42)( 3,41)( 4,37)( 5,39)( 6,38)( 7,43)( 8,45)( 9,44)(10,31)(11,33)(12,32)(13,28)(14,30)(15,29)(16,34)(17,36)(18,35)(19,50)(20,49)(21,51)(22,47)(23,46)(24,48)(25,53)(26,52)(27,54);;
s2 := ( 4, 7)( 5, 8)( 6, 9)(13,16)(14,17)(15,18)(22,25)(23,26)(24,27)(31,34)(32,35)(33,36)(40,43)(41,44)(42,45)(49,52)(50,53)(51,54);;
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, s2*s0*s1*s2*s1*s2*s0*s1*s2*s1, 
s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2*s1, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*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(54)!( 2, 3)( 4, 7)( 5, 9)( 6, 8)(10,20)(11,19)(12,21)(13,26)(14,25)(15,27)(16,23)(17,22)(18,24)(29,30)(31,34)(32,36)(33,35)(37,47)(38,46)(39,48)(40,53)(41,52)(42,54)(43,50)(44,49)(45,51);
s1 := Sym(54)!( 1,40)( 2,42)( 3,41)( 4,37)( 5,39)( 6,38)( 7,43)( 8,45)( 9,44)(10,31)(11,33)(12,32)(13,28)(14,30)(15,29)(16,34)(17,36)(18,35)(19,50)(20,49)(21,51)(22,47)(23,46)(24,48)(25,53)(26,52)(27,54);
s2 := Sym(54)!( 4, 7)( 5, 8)( 6, 9)(13,16)(14,17)(15,18)(22,25)(23,26)(24,27)(31,34)(32,35)(33,36)(40,43)(41,44)(42,45)(49,52)(50,53)(51,54);
poly := sub<Sym(54)|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, s2*s0*s1*s2*s1*s2*s0*s1*s2*s1, 
s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2*s1, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 >;

References