Polytope of Type {8,4}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {8,4}*64b
if this polytope has a name.
Group : SmallGroup(64,134)
Rank : 3
Schlafli Type : {8,4}
Number of vertices, edges, etc : 8, 16, 4
Order of s₀s₁s₂ : 8
Order of s₀s₁s₂s₁ : 4
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
   Flat
   Self-Petrie
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
Facet Of :
   {8,4,2} of size 128
   {8,4,4} of size 256
   {8,4,6} of size 384
   {8,4,8} of size 512
   {8,4,4} of size 512
   {8,4,8} of size 512
   {8,4,10} of size 640
   {8,4,12} of size 768
   {8,4,14} of size 896
   {8,4,18} of size 1152
   {8,4,6} of size 1152
   {8,4,20} of size 1280
   {8,4,22} of size 1408
   {8,4,26} of size 1664
   {8,4,28} of size 1792
   {8,4,30} of size 1920
Vertex Figure Of :
   {2,8,4} of size 128
   {4,8,4} of size 256
   {4,8,4} of size 256
   {6,8,4} of size 384
   {8,8,4} of size 512
   {8,8,4} of size 512
   {8,8,4} of size 512
   {8,8,4} of size 512
   {4,8,4} of size 512
   {4,8,4} of size 512
   {10,8,4} of size 640
   {12,8,4} of size 768
   {12,8,4} of size 768
   {3,8,4} of size 768
   {14,8,4} of size 896
   {18,8,4} of size 1152
   {6,8,4} of size 1152
   {20,8,4} of size 1280
   {20,8,4} of size 1280
   {22,8,4} of size 1408
   {26,8,4} of size 1664
   {28,8,4} of size 1792
   {28,8,4} of size 1792
   {30,8,4} of size 1920
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {4,4}*32
   4-fold quotients : {2,4}*16, {4,2}*16
   8-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   2-fold covers : {8,4}*128a, {8,8}*128a, {8,8}*128d
   3-fold covers : {24,4}*192b, {8,12}*192b
   4-fold covers : {8,8}*256a, {8,4}*256a, {8,8}*256d, {16,4}*256a, {16,4}*256b, {16,8}*256a, {16,8}*256b, {8,16}*256c, {8,16}*256e
   5-fold covers : {40,4}*320b, {8,20}*320b
   6-fold covers : {24,4}*384a, {8,24}*384a, {8,12}*384a, {8,24}*384c, {24,8}*384c, {24,8}*384d
   7-fold covers : {56,4}*448b, {8,28}*448b
   8-fold covers : {16,4}*512a, {16,8}*512a, {16,8}*512b, {16,16}*512a, {16,16}*512d, {16,16}*512g, {16,16}*512l, {8,16}*512c, {16,8}*512c, {8,16}*512d, {16,8}*512d, {8,16}*512e, {16,8}*512e, {8,16}*512f, {16,8}*512f, {8,8}*512a, {8,8}*512b, {8,8}*512c, {8,4}*512a, {8,8}*512f, {16,4}*512b, {8,4}*512b, {8,4}*512c, {8,8}*512l, {8,8}*512n, {16,4}*512c, {16,4}*512d, {8,8}*512q, {8,8}*512s, {16,8}*512g, {16,8}*512h, {32,4}*512a, {32,4}*512b, {8,32}*512a, {8,32}*512c
   9-fold covers : {72,4}*576b, {8,36}*576b, {24,12}*576a, {24,12}*576e, {24,12}*576f, {8,4}*576b, {24,4}*576b, {8,12}*576b
   10-fold covers : {40,4}*640a, {8,40}*640a, {8,20}*640a, {8,40}*640c, {40,8}*640c, {40,8}*640d
   11-fold covers : {88,4}*704b, {8,44}*704b
   12-fold covers : {8,24}*768a, {24,8}*768a, {8,12}*768a, {24,4}*768a, {8,24}*768c, {24,8}*768d, {16,12}*768a, {48,4}*768a, {16,12}*768b, {48,4}*768b, {48,8}*768a, {16,24}*768a, {48,8}*768b, {16,24}*768b, {24,16}*768c, {8,48}*768c, {24,16}*768e, {8,48}*768e, {24,4}*768j, {8,12}*768w, {24,12}*768e
   13-fold covers : {104,4}*832b, {8,52}*832b
   14-fold covers : {56,4}*896a, {8,56}*896a, {8,28}*896a, {8,56}*896c, {56,8}*896c, {56,8}*896d
   15-fold covers : {24,20}*960b, {40,12}*960b, {120,4}*960b, {8,60}*960b
   17-fold covers : {8,68}*1088b, {136,4}*1088b
   18-fold covers : {8,36}*1152a, {72,4}*1152a, {24,12}*1152a, {24,12}*1152b, {24,12}*1152c, {8,4}*1152a, {24,4}*1152a, {8,12}*1152a, {72,8}*1152a, {8,72}*1152b, {24,24}*1152a, {24,24}*1152d, {24,24}*1152i, {8,24}*1152a, {8,8}*1152c, {24,8}*1152b, {8,72}*1152d, {72,8}*1152d, {24,24}*1152j, {24,24}*1152k, {24,24}*1152l, {8,8}*1152d, {8,24}*1152d, {24,8}*1152d
   19-fold covers : {8,76}*1216b, {152,4}*1216b
   20-fold covers : {8,40}*1280a, {40,8}*1280a, {8,20}*1280a, {40,4}*1280a, {8,40}*1280c, {40,8}*1280d, {16,20}*1280a, {80,4}*1280a, {16,20}*1280b, {80,4}*1280b, {80,8}*1280a, {16,40}*1280a, {80,8}*1280b, {16,40}*1280b, {40,16}*1280c, {8,80}*1280c, {40,16}*1280e, {8,80}*1280e
   21-fold covers : {24,28}*1344b, {56,12}*1344b, {168,4}*1344b, {8,84}*1344b
   22-fold covers : {8,44}*1408a, {88,4}*1408a, {88,8}*1408a, {8,88}*1408b, {8,88}*1408d, {88,8}*1408d
   23-fold covers : {8,92}*1472b, {184,4}*1472b
   25-fold covers : {200,4}*1600b, {8,100}*1600b, {40,20}*1600a, {40,20}*1600e, {40,20}*1600f, {8,4}*1600b, {40,4}*1600b, {8,20}*1600b
   26-fold covers : {8,52}*1664a, {104,4}*1664a, {104,8}*1664a, {8,104}*1664b, {8,104}*1664d, {104,8}*1664d
   27-fold covers : {216,4}*1728b, {8,108}*1728b, {24,36}*1728a, {24,12}*1728a, {72,12}*1728c, {72,12}*1728d, {24,36}*1728d, {24,12}*1728e, {24,12}*1728f, {8,12}*1728c, {24,4}*1728c, {24,4}*1728d, {24,12}*1728k, {24,12}*1728l, {8,12}*1728d, {24,12}*1728m, {24,12}*1728n, {24,12}*1728p, {24,4}*1728g, {24,4}*1728h, {8,12}*1728f, {24,12}*1728r, {8,12}*1728h, {24,12}*1728t, {24,12}*1728w, {24,12}*1728x
   28-fold covers : {8,56}*1792a, {56,8}*1792a, {8,28}*1792a, {56,4}*1792a, {8,56}*1792c, {56,8}*1792d, {16,28}*1792a, {112,4}*1792a, {16,28}*1792b, {112,4}*1792b, {112,8}*1792a, {16,56}*1792a, {112,8}*1792b, {16,56}*1792b, {56,16}*1792c, {8,112}*1792c, {56,16}*1792e, {8,112}*1792e
   29-fold covers : {8,116}*1856b, {232,4}*1856b
   30-fold covers : {8,60}*1920a, {120,4}*1920a, {40,12}*1920a, {24,20}*1920a, {120,8}*1920a, {8,120}*1920b, {40,24}*1920b, {24,40}*1920c, {8,120}*1920d, {120,8}*1920d, {24,40}*1920d, {40,24}*1920d
   31-fold covers : {8,124}*1984b, {248,4}*1984b
Permutation Representation (GAP) :

s0 := ( 2, 3)( 4, 6)( 5, 8)( 7,10)(11,14)(13,15);;
s1 := ( 1, 2)( 3, 5)( 4, 7)( 6, 9)( 8,11)(10,13)(12,15)(14,16);;
s2 := ( 2, 4)( 3, 6)( 5, 8)( 9,12)(11,15)(13,14);;
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, 
s2*s0*s1*s0*s1*s2*s0*s1*s0*s1 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(16)!( 2, 3)( 4, 6)( 5, 8)( 7,10)(11,14)(13,15);
s1 := Sym(16)!( 1, 2)( 3, 5)( 4, 7)( 6, 9)( 8,11)(10,13)(12,15)(14,16);
s2 := Sym(16)!( 2, 4)( 3, 6)( 5, 8)( 9,12)(11,15)(13,14);
poly := sub<Sym(16)|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, 
s2*s0*s1*s0*s1*s2*s0*s1*s0*s1 >;

References : None.
to this polytope