Polytope of Type {6,6,6} with group SmallGroup(1944,2344)

Overview

Group: SmallGroup(1944,2344)
Rank: 4
Schläfli Type: {6,6,6}
Vertices, edges, …: 6, 81, 81, 27
Order of s₀s₁s₂s₃: 18
Order of s₀s₁s₂s₃s₂s₁: 2
Also known as: if this polytope has a name.

Special Properties

Universal
Non-Orientable
Flat

Quotients maximal quotients in bold

3-fold

9-fold

{2,6,6}*216

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*s3*s2*s1*(s3*s2)^2*s3> of order 3

9 facets

9 of {6,6}*72b

6 vertex figures

6 of 3-fold non-regular quotient of {6,6}*324b

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

9 facets

9 of {6,6}*72b

6 vertex figures

6 of 3-fold non-regular quotient of {6,6}*324b

Representations

Permutation Representation (GAP)

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

Permutation Representation (Magma)

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

References

None.

to this polytope.