Polytope of Type {9,6,6} with group SmallGroup(1944,2340)

Overview

Group: SmallGroup(1944,2340)
Rank: 4
Schläfli Type: {9,6,6}
Vertices, edges, …: 27, 81, 54, 6
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

18-fold

{9,2,3}*108

27-fold

54-fold

{3,2,3}*36

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

6 facets

6 of 3-fold non-regular quotient of {9,6}*324a

9 vertex figures

9 of {6,6}*72a

P/N, where N=<(s0*s1*s2*s1)^2> of order 3

6 facets

6 of 3-fold non-regular quotient of {9,6}*324a

15 vertex figures

6 of {6,6}*72a
9 of {2,6}*24

Representations

Permutation Representation (GAP)

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

References

None.

to this polytope.