Polytope of Type {18,3,6} with group SmallGroup(1944,2346)

Overview

Group: SmallGroup(1944,2346)
Rank: 4
Schläfli Type: {18,3,6}
Vertices, edges, …: 54, 81, 27, 6
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

9-fold

27-fold

81-fold

{2,3,2}*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*s2*(s1*s0)^3*s1*s2*s1> of order 3

6 facets

6 of 3-fold non-regular quotient of {18,3}*324

18 vertex figures

18 of {3,6}*36

Representations

Permutation Representation (GAP)

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

Permutation Representation (Magma)

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

References

None.

to this polytope.