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

Overview

Group: SmallGroup(972,102)
Rank: 3
Schläfli Type: {9,18}
Vertices, edges, …: 27, 243, 54
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

{9,6}*324c

9-fold

{3,6}*108

27-fold

{3,6}*36

81-fold

{3,2}*12

Covers minimal covers in bold

2-fold

{18,18}*1944f

Irregular Quotients of which this is a minimal cover

Click an entry to reveal its facets and vertex figures.

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

18 facets

18 of {9}*18

9 vertex figures

9 of {18}*36

Representations

Permutation Representation (GAP)

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

Permutation Representation (Magma)

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

References

None.

to this polytope.