Part of the Atlas of Small Regular Polytopes

Polytope of Type {24,4}

Atlas Canonical Name {24,4}*576a

▶ Play as a twisty puzzle

Overview

Group
SmallGroup(576,5339)
Rank
3
Schläfli Type
{24,4}
Vertices, edges, …
72, 144, 12
Order of s0s1s2
8
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

2-fold

4-fold

8-fold

9-fold

18-fold

36-fold

72-fold

Covers minimal covers in bold

2-fold

3-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*s1)^3*s2*(s1*s0)^2*s1*s2> of order 2

6 facets

36 vertex figures

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

8 facets

24 vertex figures

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

4 facets

24 vertex figures

Representations

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

References

None.

to this polytope.

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