Polytope of Type {18,6,3} with group SmallGroup(1944,2341)

Overview

Group: SmallGroup(1944,2341)
Rank: 4
Schläfli Type: {18,6,3}
Vertices, edges, …: 18, 162, 27, 9
Order of s₀s₁s₂s₃: 18
Order of s₀s₁s₂s₃s₂s₁: 2
Also known as: {{18,6|2},{6,3}₆}. if this polytope has another 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

{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)^2> of order 3

5 facets

3 of {18,2}*72
2 of {18,6}*216a

18 vertex figures

18 of 3-fold non-regular quotient of {6,3}*108

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

References

None.

to this polytope.