Part of the Atlas of Small Regular Polytopes

Polytope of Type {8,3,4}

Atlas Canonical Name {8,3,4}*768a

Overview

Group
SmallGroup(768,1090071)
Rank
4
Schläfli Type
{8,3,4}
Vertices, edges, …
32, 48, 24, 4
Order of s0s1s2s3
6
Order of s0s1s2s3s2s1
2
Also known as
{{8,3}6,{3,4}3}. if this polytope has another name.

Special Properties

  • Universal
  • Non-Orientable
  • Flat

Quotients maximal quotients in bold

4-fold

8-fold

16-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=<(s0*s1)^4> of order 2

4 facets

  • 4 of 2-fold non-regular quotient of {8,3}*192

16 vertex figures

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

4 facets

  • 4 of 2-fold non-regular quotient of {8,3}*192

16 vertex figures

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

4 facets

  • 4 of 4-fold non-regular quotient of {8,3}*192

8 vertex figures

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

4 facets

  • 4 of 4-fold non-regular quotient of {8,3}*192

8 vertex figures

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

4 facets

  • 4 of 4-fold non-regular quotient of {8,3}*192

8 vertex figures

Representations

Permutation Representation (GAP) 
s0 := ( 1,15)( 2,16)( 3,13)( 4,14)( 5,11)( 6,12)( 7, 9)( 8,10)(17,31)(18,32)(19,29)(20,30)(21,27)(22,28)(23,25)(24,26)(33,47)(34,48)(35,45)(36,46)(37,43)(38,44)(39,41)(40,42)(49,63)(50,64)(51,61)(52,62)(53,59)(54,60)(55,57)(56,58);;
s1 := ( 3, 4)( 5, 6)( 9,14)(10,13)(11,15)(12,16)(19,20)(21,22)(25,30)(26,29)(27,31)(28,32)(33,49)(34,50)(35,52)(36,51)(37,54)(38,53)(39,55)(40,56)(41,62)(42,61)(43,63)(44,64)(45,58)(46,57)(47,59)(48,60);;
s2 := ( 2, 3)( 5,10)( 6,12)( 7, 9)( 8,11)(13,16)(17,33)(18,35)(19,34)(20,36)(21,42)(22,44)(23,41)(24,43)(25,39)(26,37)(27,40)(28,38)(29,48)(30,46)(31,47)(32,45)(50,51)(53,58)(54,60)(55,57)(56,59)(61,64);;
s3 := ( 1,17)( 2,18)( 3,19)( 4,20)( 5,21)( 6,22)( 7,23)( 8,24)( 9,25)(10,26)(11,27)(12,28)(13,29)(14,30)(15,31)(16,32)(33,49)(34,50)(35,51)(36,52)(37,53)(38,54)(39,55)(40,56)(41,57)(42,58)(43,59)(44,60)(45,61)(46,62)(47,63)(48,64);;
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, s1*s2*s1*s2*s1*s2, 
s2*s3*s2*s3*s2*s3*s2*s3, s1*s3*s2*s1*s3*s2*s1*s3*s2, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s0*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s1*s2*s1 ];;
poly := F / rels;;
Permutation Representation (Magma) 
s0 := Sym(64)!( 1,15)( 2,16)( 3,13)( 4,14)( 5,11)( 6,12)( 7, 9)( 8,10)(17,31)(18,32)(19,29)(20,30)(21,27)(22,28)(23,25)(24,26)(33,47)(34,48)(35,45)(36,46)(37,43)(38,44)(39,41)(40,42)(49,63)(50,64)(51,61)(52,62)(53,59)(54,60)(55,57)(56,58);
s1 := Sym(64)!( 3, 4)( 5, 6)( 9,14)(10,13)(11,15)(12,16)(19,20)(21,22)(25,30)(26,29)(27,31)(28,32)(33,49)(34,50)(35,52)(36,51)(37,54)(38,53)(39,55)(40,56)(41,62)(42,61)(43,63)(44,64)(45,58)(46,57)(47,59)(48,60);
s2 := Sym(64)!( 2, 3)( 5,10)( 6,12)( 7, 9)( 8,11)(13,16)(17,33)(18,35)(19,34)(20,36)(21,42)(22,44)(23,41)(24,43)(25,39)(26,37)(27,40)(28,38)(29,48)(30,46)(31,47)(32,45)(50,51)(53,58)(54,60)(55,57)(56,59)(61,64);
s3 := Sym(64)!( 1,17)( 2,18)( 3,19)( 4,20)( 5,21)( 6,22)( 7,23)( 8,24)( 9,25)(10,26)(11,27)(12,28)(13,29)(14,30)(15,31)(16,32)(33,49)(34,50)(35,51)(36,52)(37,53)(38,54)(39,55)(40,56)(41,57)(42,58)(43,59)(44,60)(45,61)(46,62)(47,63)(48,64);
poly := sub<Sym(64)|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, 
s1*s2*s1*s2*s1*s2, s2*s3*s2*s3*s2*s3*s2*s3, 
s1*s3*s2*s1*s3*s2*s1*s3*s2, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s0*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s1*s2*s1 >;

References

None.

to this polytope.