Part of the Atlas of Small Regular Polytopes

Polytope of Type {3,6}

Atlas Canonical Name {3,6}*900

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Overview

Group
SmallGroup(900,95)
Rank
3
Schläfli Type
{3,6}
Vertices, edges, …
75, 225, 150
Order of s0s1s2
30
Order of s0s1s2s1
6
Also known as
{3,6}(5,5). if this polytope has another name.

Special Properties

  • Toroidal
  • Locally Spherical
  • Orientable

Quotients maximal quotients in bold

3-fold

25-fold

75-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=<(s0*s2*s1)^9*s2> of order 3

50 facets

27 vertex figures

P/N, where N=<(s0*(s2*s1)^2)^2> of order 5

30 facets

15 vertex figures

P/N, where N=<(s0*s2*s1)^6> of order 5

30 facets

15 vertex figures

Representations

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

References

None.

to this polytope.

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