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# Polytope of Type {8,8}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {8,8}*128b
Also Known As : {8,8|2}. if this polytope has another 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 : 2
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Flat
Self-Dual
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
{8,8,3} 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
{3,8,8} of size 768
{10,8,8} of size 1280
{14,8,8} of size 1792
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {4,8}*64a, {8,4}*64a
4-fold quotients : {4,4}*32, {2,8}*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}*256d, {16,8}*256d, {8,16}*256f, {16,8}*256f
3-fold covers : {8,24}*384b, {24,8}*384b
4-fold covers : {16,16}*512b, {16,16}*512e, {16,16}*512h, {16,16}*512k, {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, {8,32}*512b, {32,8}*512b, {8,32}*512d, {32,8}*512d
5-fold covers : {8,40}*640b, {40,8}*640b
6-fold covers : {8,24}*768a, {24,8}*768a, {8,48}*768d, {48,8}*768d, {16,24}*768d, {24,16}*768d, {8,48}*768f, {48,8}*768f, {16,24}*768f, {24,16}*768f
7-fold covers : {8,56}*896b, {56,8}*896b
9-fold covers : {8,72}*1152c, {72,8}*1152c, {24,24}*1152b, {24,24}*1152e, {24,24}*1152h, {8,8}*1152a, {8,24}*1152c, {24,8}*1152c
10-fold covers : {8,40}*1280a, {40,8}*1280a, {8,80}*1280d, {80,8}*1280d, {16,40}*1280d, {40,16}*1280d, {8,80}*1280f, {80,8}*1280f, {16,40}*1280f, {40,16}*1280f
11-fold covers : {8,88}*1408c, {88,8}*1408c
13-fold covers : {8,104}*1664c, {104,8}*1664c
14-fold covers : {8,56}*1792a, {56,8}*1792a, {8,112}*1792d, {112,8}*1792d, {16,56}*1792d, {56,16}*1792d, {8,112}*1792f, {112,8}*1792f, {16,56}*1792f, {56,16}*1792f
15-fold covers : {8,120}*1920c, {120,8}*1920c, {24,40}*1920a, {40,24}*1920a
Permutation Representation (GAP) :
s0 := ( 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);;
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 := ( 1,49)( 2,50)( 3,51)( 4,52)( 5,53)( 6,54)( 7,55)( 8,56)( 9,60)(10,59)
(11,58)(12,57)(13,64)(14,63)(15,62)(16,61)(17,33)(18,34)(19,35)(20,36)(21,37)
(22,38)(23,39)(24,40)(25,44)(26,43)(27,42)(28,41)(29,48)(30,47)(31,46)
(32,45);;
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, s0*s1*s2*s1*s0*s1*s2*s1,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 ];;
poly := F / rels;;

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

References : None.
to this polytope