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# Polytope of Type {8,8}

Atlas Canonical Name : {8,8}*128a
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
Self-Petrie
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 : {4,8}*64a, {8,4}*64b
4-fold quotients : {4,4}*32, {2,8}*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, {16,8}*256a, {16,8}*256b, {8,16}*256c, {8,16}*256e
3-fold covers : {8,24}*384a, {24,8}*384d
4-fold covers : {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, {8,32}*512a, {8,32}*512c
5-fold covers : {8,40}*640a, {40,8}*640d
6-fold covers : {8,24}*768a, {24,8}*768a, {48,8}*768a, {16,24}*768a, {48,8}*768b, {16,24}*768b, {24,16}*768c, {8,48}*768c, {24,16}*768e, {8,48}*768e
7-fold covers : {8,56}*896a, {56,8}*896d
9-fold covers : {72,8}*1152a, {8,72}*1152b, {24,24}*1152a, {24,24}*1152d, {24,24}*1152i, {8,24}*1152a, {8,8}*1152c, {24,8}*1152b
10-fold covers : {8,40}*1280a, {40,8}*1280a, {80,8}*1280a, {16,40}*1280a, {80,8}*1280b, {16,40}*1280b, {40,16}*1280c, {8,80}*1280c, {40,16}*1280e, {8,80}*1280e
11-fold covers : {88,8}*1408a, {8,88}*1408b
13-fold covers : {104,8}*1664a, {8,104}*1664b
14-fold covers : {8,56}*1792a, {56,8}*1792a, {112,8}*1792a, {16,56}*1792a, {112,8}*1792b, {16,56}*1792b, {56,16}*1792c, {8,112}*1792c, {56,16}*1792e, {8,112}*1792e
15-fold covers : {120,8}*1920a, {8,120}*1920b, {40,24}*1920b, {24,40}*1920c
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 := ( 1,33)( 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);;
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*s0*s1*s2*s0*s1*s0*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)!( 1,33)( 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);
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*s0*s1*s2*s0*s1*s0*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