Part of the Atlas of Small Regular Polytopes

Polytope of Type {3,6,18,2}

Atlas Canonical Name {3,6,18,2}*1296b

Overview

Group
SmallGroup(1296,2984)
Rank
5
Schläfli Type
{3,6,18,2}
Vertices, edges, …
3, 9, 54, 18, 2
Order of s0s1s2s3s4
18
Order of s0s1s2s3s4s3s2s1
2
Also known as
if this polytope has a name.

Special Properties

  • Degenerate
  • Universal
  • Orientable
  • Flat

Quotients maximal quotients in bold

3-fold

6-fold

9-fold

18-fold

27-fold

Covers minimal covers in bold

None in this atlas.

Representations

Permutation Representation (GAP) 
s0 := ( 4, 7)( 5, 8)( 6, 9)(10,19)(11,20)(12,21)(13,25)(14,26)(15,27)(16,22)(17,23)(18,24)(31,34)(32,35)(33,36)(37,46)(38,47)(39,48)(40,52)(41,53)(42,54)(43,49)(44,50)(45,51)(58,61)(59,62)(60,63)(64,73)(65,74)(66,75)(67,79)(68,80)(69,81)(70,76)(71,77)(72,78);;
s1 := ( 1,13)( 2,14)( 3,15)( 4,10)( 5,11)( 6,12)( 7,16)( 8,17)( 9,18)(19,22)(20,23)(21,24)(28,40)(29,41)(30,42)(31,37)(32,38)(33,39)(34,43)(35,44)(36,45)(46,49)(47,50)(48,51)(55,67)(56,68)(57,69)(58,64)(59,65)(60,66)(61,70)(62,71)(63,72)(73,76)(74,77)(75,78);;
s2 := ( 2, 3)( 4, 7)( 5, 9)( 6, 8)(11,12)(13,16)(14,18)(15,17)(20,21)(22,25)(23,27)(24,26)(28,57)(29,56)(30,55)(31,63)(32,62)(33,61)(34,60)(35,59)(36,58)(37,66)(38,65)(39,64)(40,72)(41,71)(42,70)(43,69)(44,68)(45,67)(46,75)(47,74)(48,73)(49,81)(50,80)(51,79)(52,78)(53,77)(54,76);;
s3 := ( 1,28)( 2,30)( 3,29)( 4,31)( 5,33)( 6,32)( 7,34)( 8,36)( 9,35)(10,37)(11,39)(12,38)(13,40)(14,42)(15,41)(16,43)(17,45)(18,44)(19,46)(20,48)(21,47)(22,49)(23,51)(24,50)(25,52)(26,54)(27,53)(55,57)(58,60)(61,63)(64,66)(67,69)(70,72)(73,75)(76,78)(79,81);;
s4 := (82,83);;
poly := Group([s0,s1,s2,s3,s4]);;
Finitely Presented Group Representation (GAP) 
F := FreeGroup("s0","s1","s2","s3","s4");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;  s4 := F.5;;  
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s4*s4, s0*s2*s0*s2, 
s0*s3*s0*s3, s1*s3*s1*s3, s0*s4*s0*s4, 
s1*s4*s1*s4, s2*s4*s2*s4, s3*s4*s3*s4, 
s0*s1*s0*s1*s0*s1, s1*s2*s3*s2*s1*s2*s3*s2, 
s2*s0*s1*s2*s1*s2*s0*s1*s2*s1, s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 ];;
poly := F / rels;;
Permutation Representation (Magma) 
s0 := Sym(83)!( 4, 7)( 5, 8)( 6, 9)(10,19)(11,20)(12,21)(13,25)(14,26)(15,27)(16,22)(17,23)(18,24)(31,34)(32,35)(33,36)(37,46)(38,47)(39,48)(40,52)(41,53)(42,54)(43,49)(44,50)(45,51)(58,61)(59,62)(60,63)(64,73)(65,74)(66,75)(67,79)(68,80)(69,81)(70,76)(71,77)(72,78);
s1 := Sym(83)!( 1,13)( 2,14)( 3,15)( 4,10)( 5,11)( 6,12)( 7,16)( 8,17)( 9,18)(19,22)(20,23)(21,24)(28,40)(29,41)(30,42)(31,37)(32,38)(33,39)(34,43)(35,44)(36,45)(46,49)(47,50)(48,51)(55,67)(56,68)(57,69)(58,64)(59,65)(60,66)(61,70)(62,71)(63,72)(73,76)(74,77)(75,78);
s2 := Sym(83)!( 2, 3)( 4, 7)( 5, 9)( 6, 8)(11,12)(13,16)(14,18)(15,17)(20,21)(22,25)(23,27)(24,26)(28,57)(29,56)(30,55)(31,63)(32,62)(33,61)(34,60)(35,59)(36,58)(37,66)(38,65)(39,64)(40,72)(41,71)(42,70)(43,69)(44,68)(45,67)(46,75)(47,74)(48,73)(49,81)(50,80)(51,79)(52,78)(53,77)(54,76);
s3 := Sym(83)!( 1,28)( 2,30)( 3,29)( 4,31)( 5,33)( 6,32)( 7,34)( 8,36)( 9,35)(10,37)(11,39)(12,38)(13,40)(14,42)(15,41)(16,43)(17,45)(18,44)(19,46)(20,48)(21,47)(22,49)(23,51)(24,50)(25,52)(26,54)(27,53)(55,57)(58,60)(61,63)(64,66)(67,69)(70,72)(73,75)(76,78)(79,81);
s4 := Sym(83)!(82,83);
poly := sub<Sym(83)|s0,s1,s2,s3,s4>;
Finitely Presented Group Representation (Magma) 
poly<s0,s1,s2,s3,s4> := Group< s0,s1,s2,s3,s4 | s0*s0, s1*s1, s2*s2, 
s3*s3, s4*s4, s0*s2*s0*s2, s0*s3*s0*s3, 
s1*s3*s1*s3, s0*s4*s0*s4, s1*s4*s1*s4, 
s2*s4*s2*s4, s3*s4*s3*s4, s0*s1*s0*s1*s0*s1, 
s1*s2*s3*s2*s1*s2*s3*s2, s2*s0*s1*s2*s1*s2*s0*s1*s2*s1, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 >;