Polytope of Type {8,8}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {8,8}*128c
if this polytope has a name.
Group : SmallGroup(128,351)
Rank : 3
Schlafli Type : {8,8}
Number of vertices, edges, etc : 8, 32, 8
Order of s₀s₁s₂ : 8
Order of s₀s₁s₂s₁ : 4
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
Facet Of :
   {8,8,2} of size 256
   {8,8,4} of size 512
   {8,8,4} of size 512
   {8,8,6} of size 768
   {8,8,10} of size 1280
   {8,8,14} of size 1792
Vertex Figure Of :
   {2,8,8} of size 256
   {4,8,8} of size 512
   {4,8,8} of size 512
   {6,8,8} of size 768
   {10,8,8} of size 1280
   {14,8,8} of size 1792
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {8,4}*64a, {4,8}*64b
   4-fold quotients : {4,4}*32, {8,2}*32
   8-fold quotients : {2,4}*16, {4,2}*16
   16-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   2-fold covers : {8,8}*256a, {8,16}*256a, {8,16}*256b, {16,8}*256c, {16,8}*256e
   3-fold covers : {24,8}*384a, {8,24}*384d
   4-fold covers : {16,16}*512c, {16,16}*512f, {16,16}*512i, {16,16}*512j, {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, {32,8}*512a, {32,8}*512c
   5-fold covers : {40,8}*640a, {8,40}*640d
   6-fold covers : {8,24}*768a, {24,8}*768a, {8,48}*768a, {24,16}*768a, {8,48}*768b, {24,16}*768b, {16,24}*768c, {48,8}*768c, {16,24}*768e, {48,8}*768e
   7-fold covers : {56,8}*896a, {8,56}*896d
   9-fold covers : {8,72}*1152a, {72,8}*1152b, {24,24}*1152c, {24,24}*1152f, {24,24}*1152g, {24,8}*1152a, {8,8}*1152b, {8,24}*1152b
   10-fold covers : {8,40}*1280a, {40,8}*1280a, {8,80}*1280a, {40,16}*1280a, {8,80}*1280b, {40,16}*1280b, {16,40}*1280c, {80,8}*1280c, {16,40}*1280e, {80,8}*1280e
   11-fold covers : {8,88}*1408a, {88,8}*1408b
   13-fold covers : {8,104}*1664a, {104,8}*1664b
   14-fold covers : {8,56}*1792a, {56,8}*1792a, {8,112}*1792a, {56,16}*1792a, {8,112}*1792b, {56,16}*1792b, {16,56}*1792c, {112,8}*1792c, {16,56}*1792e, {112,8}*1792e
   15-fold covers : {8,120}*1920a, {120,8}*1920b, {24,40}*1920b, {40,24}*1920c
Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

References : None.
to this polytope