Polytope of Type {8,4,3} with group SmallGroup(768,1086051)

Overview

Group: SmallGroup(768,1086051)
Rank: 4
Schläfli Type: {8,4,3}
Vertices, edges, …: 32, 64, 24, 3
Order of s₀s₁s₂s₃: 6
Order of s₀s₁s₂s₃s₂s₁: 8
Also known as: if this polytope has a name.

Special Properties

Universal
Non-Orientable
Flat

Quotients maximal quotients in bold

4-fold

{4,4,3}*192a

16-fold

{2,4,3}*48

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

3 facets

1 of {8,4}*128b
2 of 2-fold non-regular quotient of {8,4}*256d

16 vertex figures

16 of {4,3}*24

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

3 facets

1 of 2-fold non-regular quotient of {8,4}*128b
2 of 4-fold non-regular quotient of {8,4}*256d

8 vertex figures

8 of {4,3}*24

Representations

Permutation Representation (GAP)

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

Permutation Representation (Magma)

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

References

None.

to this polytope.