Polytope of Type {4,6,9} with group SmallGroup(1296,2908)

Overview

Group: SmallGroup(1296,2908)
Rank: 4
Schläfli Type: {4,6,9}
Vertices, edges, …: 12, 36, 81, 9
Order of s₀s₁s₂s₃: 36
Order of s₀s₁s₂s₃s₂s₁: 6
Also known as: if this polytope has a name.

Special Properties

Universal
Orientable
Flat

Quotients maximal quotients in bold

3-fold

{4,6,3}*432b

9-fold

{4,2,9}*144

18-fold

{2,2,9}*72

27-fold

{4,2,3}*48

54-fold

{2,2,3}*24

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

9 facets

9 of 3-fold non-regular quotient of {4,6}*144

4 vertex figures

4 of {6,9}*108

P/N, where N=<s0*s1*s2*s1*s0*s2> of order 3

9 facets

9 of 3-fold non-regular quotient of {4,6}*144

8 vertex figures

2 of {6,9}*108
6 of {2,9}*36

Representations

Permutation Representation (GAP)

s0 := (10,28)(11,29)(12,30)(13,31)(14,32)(15,33)(16,34)(17,35)(18,36)(19,55)(20,56)(21,57)(22,58)(23,59)(24,60)(25,61)(26,62)(27,63)(46,64)(47,65)(48,66)(49,67)(50,68)(51,69)(52,70)(53,71)(54,72);;
s1 := (28,55)(29,56)(30,57)(31,58)(32,59)(33,60)(34,61)(35,62)(36,63)(37,64)(38,65)(39,66)(40,67)(41,68)(42,69)(43,70)(44,71)(45,72)(46,73)(47,74)(48,75)(49,76)(50,77)(51,78)(52,79)(53,80)(54,81);;
s2 := ( 1,37)( 2,39)( 3,38)( 4,44)( 5,43)( 6,45)( 7,41)( 8,40)( 9,42)(10,28)(11,30)(12,29)(13,35)(14,34)(15,36)(16,32)(17,31)(18,33)(19,46)(20,48)(21,47)(22,53)(23,52)(24,54)(25,50)(26,49)(27,51)(55,64)(56,66)(57,65)(58,71)(59,70)(60,72)(61,68)(62,67)(63,69)(74,75)(76,80)(77,79)(78,81);;
s3 := ( 1, 4)( 2, 6)( 3, 5)( 7, 8)(10,22)(11,24)(12,23)(13,19)(14,21)(15,20)(16,26)(17,25)(18,27)(28,58)(29,60)(30,59)(31,55)(32,57)(33,56)(34,62)(35,61)(36,63)(37,76)(38,78)(39,77)(40,73)(41,75)(42,74)(43,80)(44,79)(45,81)(46,67)(47,69)(48,68)(49,64)(50,66)(51,65)(52,71)(53,70)(54,72);;
poly := Group([s0,s1,s2,s3]);;

Finitely Presented Group Representation (GAP)

F := FreeGroup("s0","s1","s2","s3");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;  
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s0*s2*s0*s2, 
s0*s3*s0*s3, s1*s3*s1*s3, s0*s1*s0*s1*s0*s1*s0*s1, 
s1*s2*s3*s1*s2*s1*s2*s3*s1*s2, s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 ];;
poly := F / rels;;

Permutation Representation (Magma)

s0 := Sym(81)!(10,28)(11,29)(12,30)(13,31)(14,32)(15,33)(16,34)(17,35)(18,36)(19,55)(20,56)(21,57)(22,58)(23,59)(24,60)(25,61)(26,62)(27,63)(46,64)(47,65)(48,66)(49,67)(50,68)(51,69)(52,70)(53,71)(54,72);
s1 := Sym(81)!(28,55)(29,56)(30,57)(31,58)(32,59)(33,60)(34,61)(35,62)(36,63)(37,64)(38,65)(39,66)(40,67)(41,68)(42,69)(43,70)(44,71)(45,72)(46,73)(47,74)(48,75)(49,76)(50,77)(51,78)(52,79)(53,80)(54,81);
s2 := Sym(81)!( 1,37)( 2,39)( 3,38)( 4,44)( 5,43)( 6,45)( 7,41)( 8,40)( 9,42)(10,28)(11,30)(12,29)(13,35)(14,34)(15,36)(16,32)(17,31)(18,33)(19,46)(20,48)(21,47)(22,53)(23,52)(24,54)(25,50)(26,49)(27,51)(55,64)(56,66)(57,65)(58,71)(59,70)(60,72)(61,68)(62,67)(63,69)(74,75)(76,80)(77,79)(78,81);
s3 := Sym(81)!( 1, 4)( 2, 6)( 3, 5)( 7, 8)(10,22)(11,24)(12,23)(13,19)(14,21)(15,20)(16,26)(17,25)(18,27)(28,58)(29,60)(30,59)(31,55)(32,57)(33,56)(34,62)(35,61)(36,63)(37,76)(38,78)(39,77)(40,73)(41,75)(42,74)(43,80)(44,79)(45,81)(46,67)(47,69)(48,68)(49,64)(50,66)(51,65)(52,71)(53,70)(54,72);
poly := sub<Sym(81)|s0,s1,s2,s3>;

Finitely Presented Group Representation (Magma)

poly<s0,s1,s2,s3> := Group< s0,s1,s2,s3 | s0*s0, s1*s1, s2*s2, 
s3*s3, s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s0*s1*s0*s1*s0*s1*s0*s1, s1*s2*s3*s1*s2*s1*s2*s3*s1*s2, 
s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 >;

References

None.

to this polytope.