Part of the Atlas of Small Regular Polytopes

Polytope of Type {6,15}

Atlas Canonical Name {6,15}*960

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Overview

Group
SmallGroup(960,5762)
Rank
3
Schläfli Type
{6,15}
Vertices, edges, …
32, 240, 80
Order of s0s1s2
40
Order of s0s1s2s1
6
Also known as
if this polytope has a name.

Special Properties

  • Compact Hyperbolic Quotient
  • Locally Spherical
  • Orientable

Quotients maximal quotients in bold

4-fold

5-fold

20-fold

40-fold

48-fold

Covers minimal covers in bold

2-fold

Irregular Quotients of which this is a minimal cover

Click an entry to reveal its facets and vertex figures.

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

40 facets

16 vertex figures

P/N, where N=<s0*s1*s0*s2*(s1*s0)^2*s2*s1> of order 4

20 facets

8 vertex figures

P/N, where N=<s0*s1*s0*s2*s1*(s0*(s2*s1)^2)^2> of order 4

20 facets

8 vertex figures

Representations

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

References

None.

to this polytope.

Twisty Puzzle