Polytope of Type {18,9} with group SmallGroup(972,102)

Overview

Group: SmallGroup(972,102)
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

{6,9}*324c

9-fold

{6,3}*108

27-fold

{6,3}*36

81-fold

{2,3}*12

Covers minimal covers in bold

2-fold

{18,18}*1944e

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)(28,57)(29,56)(30,55)(31,60)(32,59)(33,58)(34,63)(35,62)(36,61)(37,78)(38,77)(39,76)(40,81)(41,80)(42,79)(43,75)(44,74)(45,73)(46,72)(47,71)(48,70)(49,66)(50,65)(51,64)(52,69)(53,68)(54,67);;
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)(55,57)(58,63)(59,62)(60,61)(64,65)(67,71)(68,70)(69,72)(74,75)(76,79)(77,81)(78,80);;
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)(28,78)(29,77)(30,76)(31,75)(32,74)(33,73)(34,81)(35,80)(36,79)(37,57)(38,56)(39,55)(40,63)(41,62)(42,61)(43,60)(44,59)(45,58)(46,72)(47,71)(48,70)(49,69)(50,68)(51,67)(52,66)(53,65)(54,64);;
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*s2*s0*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*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*s2*s0*s1*s2*s0*s1*s2*s1*s0*s1*s0*s2*s1*s0*s1*s0*s2*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)(28,57)(29,56)(30,55)(31,60)(32,59)(33,58)(34,63)(35,62)(36,61)(37,78)(38,77)(39,76)(40,81)(41,80)(42,79)(43,75)(44,74)(45,73)(46,72)(47,71)(48,70)(49,66)(50,65)(51,64)(52,69)(53,68)(54,67);
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)(55,57)(58,63)(59,62)(60,61)(64,65)(67,71)(68,70)(69,72)(74,75)(76,79)(77,81)(78,80);
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)(28,78)(29,77)(30,76)(31,75)(32,74)(33,73)(34,81)(35,80)(36,79)(37,57)(38,56)(39,55)(40,63)(41,62)(42,61)(43,60)(44,59)(45,58)(46,72)(47,71)(48,70)(49,69)(50,68)(51,67)(52,66)(53,65)(54,64);
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*s2*s0*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*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*s2*s0*s1*s2*s0*s1*s2*s1*s0*s1*s0*s2*s1*s0*s1*s0*s2*s1 >;

References

None.

to this polytope.