Polytope of Type {6,6,3} with group SmallGroup(1944,2346)

Overview

Group: SmallGroup(1944,2346)
Rank: 4
Schläfli Type: {6,6,3}
Vertices, edges, …: 6, 162, 81, 27
Order of s₀s₁s₂s₃: 18
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

9-fold

27-fold

54-fold

{3,2,3}*36

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

9 facets

9 of {6,6}*72a

6 vertex figures

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

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

11 facets

3 of {6,2}*24
8 of {6,6}*72a

6 vertex figures

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

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

References

None.

to this polytope.