Polytope of Type {4,66} with group SmallGroup(1584,657)

Overview

Group: SmallGroup(1584,657)
Rank: 3
Schläfli Type: {4,66}
Vertices, edges, …: 12, 396, 198
Order of s₀s₁s₂: 44
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

9-fold

{4,22}*176

11-fold

{4,6}*144

18-fold

{2,22}*88

22-fold

{4,6}*72

36-fold

{2,11}*44

99-fold

{4,2}*16

198-fold

{2,2}*8

Covers minimal covers in bold

None in this atlas.

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)^2*s1*s0*(s2*s1)^8*s2> of order 2

99 facets

99 of {4}*8

7 vertex figures

5 of {66}*132
2 of {33}*66

P/N, where N=<(s2*s1)^2*s0*(s2*s1)^3*s0*(s2*s1)^5*s2> of order 2

99 facets

99 of {4}*8

6 vertex figures

6 of {66}*132

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

66 facets

66 of {4}*8

8 vertex figures

2 of {66}*132
6 of {22}*44

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

66 facets

66 of {4}*8

4 vertex figures

4 of {66}*132

P/N, where N=<(s0*s1)^2*(s2*s1*s0)^2*(s1*s2)^2, s0*(s1*s2)^2*s1*s0*(s2*s1)^8*s2> of order 6

33 facets

33 of {4}*8

5 vertex figures

3 of {22}*44
2 of {33}*66

Representations

Permutation Representation (GAP)

s0 := (12,34)(13,35)(14,36)(15,37)(16,38)(17,39)(18,40)(19,41)(20,42)(21,43)(22,44)(23,67)(24,68)(25,69)(26,70)(27,71)(28,72)(29,73)(30,74)(31,75)(32,76)(33,77)(56,78)(57,79)(58,80)(59,81)(60,82)(61,83)(62,84)(63,85)(64,86)(65,87)(66,88);;
s1 := ( 2,11)( 3,10)( 4, 9)( 5, 8)( 6, 7)(13,22)(14,21)(15,20)(16,19)(17,18)(24,33)(25,32)(26,31)(27,30)(28,29)(34,67)(35,77)(36,76)(37,75)(38,74)(39,73)(40,72)(41,71)(42,70)(43,69)(44,68)(45,78)(46,88)(47,87)(48,86)(49,85)(50,84)(51,83)(52,82)(53,81)(54,80)(55,79)(56,89)(57,99)(58,98)(59,97)(60,96)(61,95)(62,94)(63,93)(64,92)(65,91)(66,90);;
s2 := ( 1,46)( 2,45)( 3,55)( 4,54)( 5,53)( 6,52)( 7,51)( 8,50)( 9,49)(10,48)(11,47)(12,35)(13,34)(14,44)(15,43)(16,42)(17,41)(18,40)(19,39)(20,38)(21,37)(22,36)(23,57)(24,56)(25,66)(26,65)(27,64)(28,63)(29,62)(30,61)(31,60)(32,59)(33,58)(67,79)(68,78)(69,88)(70,87)(71,86)(72,85)(73,84)(74,83)(75,82)(76,81)(77,80)(89,90)(91,99)(92,98)(93,97)(94,96);;
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*s2*s1*s2*s0*s1*s2*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*s0*s1*s2*s1, 
s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s0*s1 ];;
poly := F / rels;;

Permutation Representation (Magma)

s0 := Sym(99)!(12,34)(13,35)(14,36)(15,37)(16,38)(17,39)(18,40)(19,41)(20,42)(21,43)(22,44)(23,67)(24,68)(25,69)(26,70)(27,71)(28,72)(29,73)(30,74)(31,75)(32,76)(33,77)(56,78)(57,79)(58,80)(59,81)(60,82)(61,83)(62,84)(63,85)(64,86)(65,87)(66,88);
s1 := Sym(99)!( 2,11)( 3,10)( 4, 9)( 5, 8)( 6, 7)(13,22)(14,21)(15,20)(16,19)(17,18)(24,33)(25,32)(26,31)(27,30)(28,29)(34,67)(35,77)(36,76)(37,75)(38,74)(39,73)(40,72)(41,71)(42,70)(43,69)(44,68)(45,78)(46,88)(47,87)(48,86)(49,85)(50,84)(51,83)(52,82)(53,81)(54,80)(55,79)(56,89)(57,99)(58,98)(59,97)(60,96)(61,95)(62,94)(63,93)(64,92)(65,91)(66,90);
s2 := Sym(99)!( 1,46)( 2,45)( 3,55)( 4,54)( 5,53)( 6,52)( 7,51)( 8,50)( 9,49)(10,48)(11,47)(12,35)(13,34)(14,44)(15,43)(16,42)(17,41)(18,40)(19,39)(20,38)(21,37)(22,36)(23,57)(24,56)(25,66)(26,65)(27,64)(28,63)(29,62)(30,61)(31,60)(32,59)(33,58)(67,79)(68,78)(69,88)(70,87)(71,86)(72,85)(73,84)(74,83)(75,82)(76,81)(77,80)(89,90)(91,99)(92,98)(93,97)(94,96);
poly := sub<Sym(99)|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*s2*s1*s2*s0*s1*s2*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*s0*s1*s2*s1, 
s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s0*s1 >;

References

None.

to this polytope.