Polytope of Type {12,12} with group SmallGroup(1296,2909)

Overview

Group: SmallGroup(1296,2909)
Rank: 3
Schläfli Type: {12,12}
Vertices, edges, …: 54, 324, 54
Order of s₀s₁s₂: 6
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

{12,4}*432b

9-fold

{4,4}*144

18-fold

{4,4}*72

54-fold

{6,2}*24

108-fold

{3,2}*12

162-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*s1)^3> of order 2

27 facets

27 of {12}*24

27 vertex figures

27 of {12}*24

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

27 facets

27 of {12}*24

27 vertex figures

27 of {12}*24

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

30 facets

24 of {12}*24
6 of {6}*12

30 vertex figures

24 of {12}*24
6 of {6}*12

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

18 facets

18 of {12}*24

18 vertex figures

18 of {12}*24

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

18 facets

18 of {12}*24

18 vertex figures

18 of {12}*24

P/N, where N=<s0*(s1*s0*s2)^2*s1*s2, (s0*s2*s1)^3> of order 4

15 facets

12 of {12}*24
3 of {6}*12

15 vertex figures

12 of {12}*24
3 of {6}*12

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

12 facets

6 of {6}*12
6 of {12}*24

12 vertex figures

6 of {12}*24
6 of {6}*12

Representations

Permutation Representation (GAP)

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

Permutation Representation (Magma)

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

References

None.

to this polytope.