Part of the Atlas of Small Regular Polytopes

Polytope of Type {3,6,12}

Atlas Canonical Name {3,6,12}*1296f

Overview

Group
SmallGroup(1296,3529)
Rank
4
Schläfli Type
{3,6,12}
Vertices, edges, …
3, 27, 108, 36
Order of s0s1s2s3
12
Order of s0s1s2s3s2s1
6
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

27-fold

36-fold

54-fold

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*s3*s2*s1*s2*s3> of order 3

24 facets

3 vertex figures

P/N, where N=<(s2*s1*s2*s3)^2> of order 3

12 facets

3 vertex figures

P/N, where N=<s2*s1*(s3*s2)^2*s1*s3*s2*s3> of order 3

12 facets

3 vertex figures

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

References

None.

to this polytope.