Polytope of Type {6,18} with group SmallGroup(972,103)

Overview

Group: SmallGroup(972,103)
Rank: 3
Schläfli Type: {6,18}
Vertices, edges, …: 27, 243, 81
Order of s₀s₁s₂: 9
Order of s₀s₁s₂s₁: 6
Also known as: if this polytope has a name.

Special Properties

Compact Hyperbolic Quotient
Locally Spherical
Non-Orientable

Quotients maximal quotients in bold

3-fold

9-fold

{6,6}*108

Covers minimal covers in bold

2-fold

{6,18}*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*(s2*s1)^4*s0*s1*s2> of order 3

27 facets

27 of {6}*12

9 vertex figures

9 of {18}*36

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

27 facets

27 of {6}*12

9 vertex figures

9 of {18}*36

Representations

Permutation Representation (GAP)

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

Permutation Representation (Magma)

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

References

None.

to this polytope.