Part of the Atlas of Small Regular Polytopes

Polytope of Type {12,4}

Atlas Canonical Name {12,4}*384a

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Overview

Group
SmallGroup(384,1706)
Rank
3
Schläfli Type
{12,4}
Vertices, edges, …
48, 96, 16
Order of s0s1s2
24
Order of s0s1s2s1
4
Also known as
if this polytope has a name.

Special Properties

  • Compact Hyperbolic Quotient
  • Locally Spherical
  • Orientable

Quotients maximal quotients in bold

2-fold

3-fold

4-fold

6-fold

8-fold

12-fold

16-fold

24-fold

32-fold

48-fold

Covers minimal covers in bold

2-fold

3-fold

5-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*s2)^2> of order 2

8 facets

30 vertex figures

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

8 facets

24 vertex figures

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

8 facets

24 vertex figures

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

8 facets

24 vertex figures

P/N, where N=<(s0*s1)^6> of order 2

10 facets

24 vertex figures

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

4 facets

12 vertex figures

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

4 facets

15 vertex figures

P/N, where N=<(s1*s0*s1*s2)^2, (s0*s1)^6> of order 4

6 facets

12 vertex figures

P/N, where N=<(s1*s0*s1*s2)^2, (s1*s0)^3*s2*(s1*s0)^2*s2*s1*s2> of order 4

4 facets

12 vertex figures

P/N, where N=<(s1*s0*s1*s2)^2, (s1*s0)^3*s2*(s1*s0)^2*s2*s1> of order 4

4 facets

12 vertex figures

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

4 facets

18 vertex figures

P/N, where N=<(s0*s1)^6, s1*s0*s1*(s2*s1*s0)^2*(s1*s2)^2> of order 4

5 facets

12 vertex figures

Representations

Permutation Representation (GAP) 
s0 := ( 1,49)( 2,51)( 3,50)( 4,52)( 5,54)( 6,53)( 7,55)( 8,57)( 9,56)(10,58)(11,60)(12,59)(13,70)(14,72)(15,71)(16,67)(17,69)(18,68)(19,64)(20,66)(21,65)(22,61)(23,63)(24,62)(25,73)(26,75)(27,74)(28,76)(29,78)(30,77)(31,79)(32,81)(33,80)(34,82)(35,84)(36,83)(37,94)(38,96)(39,95)(40,91)(41,93)(42,92)(43,88)(44,90)(45,89)(46,85)(47,87)(48,86);;
s1 := ( 1, 2)( 4, 5)( 7,11)( 8,10)( 9,12)(13,14)(16,17)(19,23)(20,22)(21,24)(25,38)(26,37)(27,39)(28,41)(29,40)(30,42)(31,47)(32,46)(33,48)(34,44)(35,43)(36,45)(49,50)(52,53)(55,59)(56,58)(57,60)(61,62)(64,65)(67,71)(68,70)(69,72)(73,86)(74,85)(75,87)(76,89)(77,88)(78,90)(79,95)(80,94)(81,96)(82,92)(83,91)(84,93);;
s2 := ( 1,25)( 2,26)( 3,27)( 4,28)( 5,29)( 6,30)( 7,31)( 8,32)( 9,33)(10,34)(11,35)(12,36)(13,37)(14,38)(15,39)(16,40)(17,41)(18,42)(19,43)(20,44)(21,45)(22,46)(23,47)(24,48)(49,73)(50,74)(51,75)(52,76)(53,77)(54,78)(55,79)(56,80)(57,81)(58,82)(59,83)(60,84)(61,85)(62,86)(63,87)(64,88)(65,89)(66,90)(67,91)(68,92)(69,93)(70,94)(71,95)(72,96);;
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, s1*s2*s1*s2*s1*s2*s1*s2, 
s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s2*s1, 
s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 ];;
poly := F / rels;;
Permutation Representation (Magma) 
s0 := Sym(96)!( 1,49)( 2,51)( 3,50)( 4,52)( 5,54)( 6,53)( 7,55)( 8,57)( 9,56)(10,58)(11,60)(12,59)(13,70)(14,72)(15,71)(16,67)(17,69)(18,68)(19,64)(20,66)(21,65)(22,61)(23,63)(24,62)(25,73)(26,75)(27,74)(28,76)(29,78)(30,77)(31,79)(32,81)(33,80)(34,82)(35,84)(36,83)(37,94)(38,96)(39,95)(40,91)(41,93)(42,92)(43,88)(44,90)(45,89)(46,85)(47,87)(48,86);
s1 := Sym(96)!( 1, 2)( 4, 5)( 7,11)( 8,10)( 9,12)(13,14)(16,17)(19,23)(20,22)(21,24)(25,38)(26,37)(27,39)(28,41)(29,40)(30,42)(31,47)(32,46)(33,48)(34,44)(35,43)(36,45)(49,50)(52,53)(55,59)(56,58)(57,60)(61,62)(64,65)(67,71)(68,70)(69,72)(73,86)(74,85)(75,87)(76,89)(77,88)(78,90)(79,95)(80,94)(81,96)(82,92)(83,91)(84,93);
s2 := Sym(96)!( 1,25)( 2,26)( 3,27)( 4,28)( 5,29)( 6,30)( 7,31)( 8,32)( 9,33)(10,34)(11,35)(12,36)(13,37)(14,38)(15,39)(16,40)(17,41)(18,42)(19,43)(20,44)(21,45)(22,46)(23,47)(24,48)(49,73)(50,74)(51,75)(52,76)(53,77)(54,78)(55,79)(56,80)(57,81)(58,82)(59,83)(60,84)(61,85)(62,86)(63,87)(64,88)(65,89)(66,90)(67,91)(68,92)(69,93)(70,94)(71,95)(72,96);
poly := sub<Sym(96)|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, s1*s2*s1*s2*s1*s2*s1*s2, 
s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s2*s1, 
s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 >;

References

None.

to this polytope.

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