Polytope of Type {18,9} with group SmallGroup(972,103)

Overview

Group: SmallGroup(972,103)
Rank: 3
Schläfli Type: {18,9}
Vertices, edges, …: 54, 243, 27
Order of s₀s₁s₂: 6
Order of s₀s₁s₂s₁: 18
Also known as: if this polytope has a name.

Special Properties

Compact Hyperbolic Quotient
Locally Spherical
Orientable

Quotients maximal quotients in bold

3-fold

9-fold

{6,3}*108

27-fold

{6,3}*36

81-fold

{2,3}*12

Covers minimal covers in bold

2-fold

{18,18}*1944g

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*s0*s2*s1*s0*s1*s2> of order 3

9 facets

9 of {18}*36

18 vertex figures

18 of {9}*18

Representations

Permutation Representation (GAP)

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

Permutation Representation (Magma)

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

References

None.

to this polytope.