Polytope of Type {18,4} with group SmallGroup(648,252)

Overview

Group: SmallGroup(648,252)
Rank: 3
Schläfli Type: {18,4}
Vertices, edges, …: 81, 162, 18
Order of s₀s₁s₂: 4
Order of s₀s₁s₂s₁: 18
Also known as: if this polytope has a name.

Special Properties

Compact Hyperbolic Quotient
Locally Spherical
Non-Orientable

Quotients maximal quotients in bold

9-fold

{6,4}*72

Covers minimal covers in bold

2-fold

{18,4}*1296a

3-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)^6> of order 3

12 facets

9 of {6}*12
3 of {18}*36

27 vertex figures

27 of {4}*8

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

6 facets

6 of {18}*36

27 vertex figures

27 of {4}*8

Representations

Permutation Representation (GAP)

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

References

None.

to this polytope.