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

Overview

Group: SmallGroup(1296,2908)
Rank: 4
Schläfli Type: {9,6,4}
Vertices, edges, …: 9, 81, 36, 12
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

{3,6,4}*432b

9-fold

{9,2,4}*144

18-fold

{9,2,2}*72

27-fold

{3,2,4}*48

54-fold

{3,2,2}*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*s3*s2*s1*s2*s3> of order 3

8 facets

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

9 vertex figures

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

P/N, where N=<(s2*s1*s2*s3)^2> of order 3

4 facets

4 of {9,6}*108

9 vertex figures

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

Representations

Permutation Representation (GAP)

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

References

None.

to this polytope.