Polytope of Type {9,6} with group SmallGroup(324,40) - Atlas of Small Regular Polytopes

Overview

Group: SmallGroup(324,40)
Rank: 3
Schläfli Type: {9,6}
Vertices, edges, …: 27, 81, 18
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

{3,6}*108

9-fold

{3,6}*36

27-fold

{3,2}*12

Covers minimal covers in bold

2-fold

{18,6}*648e

3-fold

4-fold

5-fold

{45,6}*1620d

6-fold

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*s2*s1*s0*(s2*s1)^2*s2> of order 3

6 facets

6 of {9}*18

9 vertex figures

9 of {6}*12

Representations

Permutation Representation (GAP)

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

References

None.

to this polytope.