Polytope of Type {6,6,6} with group SmallGroup(1296,3538)

Overview

Group: SmallGroup(1296,3538)
Rank: 4
Schläfli Type: {6,6,6}
Vertices, edges, …: 6, 54, 54, 18
Order of s₀s₁s₂s₃: 6
Order of s₀s₁s₂s₃s₂s₁: 2
Also known as: if this polytope has a name.

Special Properties

Universal
Orientable
Flat

Quotients maximal quotients in bold

3-fold

6-fold

{6,3,6}*216

9-fold

18-fold

27-fold

54-fold

81-fold

{2,2,2}*16

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*s2)^3> of order 2

12 facets

6 of {6,3}*36
6 of {6,6}*72b

6 vertex figures

6 of 2-fold non-regular quotient of {6,6}*216d

P/N, where N=<s1*s2*s3*s2*s1*s3> of order 3

6 facets

6 of {6,6}*72b

6 vertex figures

6 of 3-fold non-regular quotient of {6,6}*216d

Representations

Permutation Representation (GAP)

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

Permutation Representation (Magma)

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

References

None.

to this polytope.