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# Polytope of Type {4,8}

Atlas Canonical Name : {4,8}*64a
Also Known As : {4,8|2}. if this polytope has another name.
Group : SmallGroup(64,128)
Rank : 3
Schlafli Type : {4,8}
Number of vertices, edges, etc : 4, 16, 8
Order of s0s1s2 : 8
Order of s0s1s2s1 : 2
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{4,8,2} of size 128
{4,8,4} of size 256
{4,8,4} of size 256
{4,8,6} of size 384
{4,8,3} of size 384
{4,8,8} of size 512
{4,8,8} of size 512
{4,8,8} of size 512
{4,8,8} of size 512
{4,8,4} of size 512
{4,8,4} of size 512
{4,8,10} of size 640
{4,8,12} of size 768
{4,8,12} of size 768
{4,8,3} of size 768
{4,8,6} of size 768
{4,8,6} of size 768
{4,8,14} of size 896
{4,8,18} of size 1152
{4,8,6} of size 1152
{4,8,9} of size 1152
{4,8,20} of size 1280
{4,8,20} of size 1280
{4,8,3} of size 1344
{4,8,4} of size 1344
{4,8,22} of size 1408
{4,8,26} of size 1664
{4,8,28} of size 1792
{4,8,28} of size 1792
{4,8,30} of size 1920
{4,8,15} of size 1920
{4,8,5} of size 1920
Vertex Figure Of :
{2,4,8} of size 128
{4,4,8} of size 256
{6,4,8} of size 384
{3,4,8} of size 384
{8,4,8} of size 512
{4,4,8} of size 512
{8,4,8} of size 512
{6,4,8} of size 576
{10,4,8} of size 640
{12,4,8} of size 768
{6,4,8} of size 768
{14,4,8} of size 896
{5,4,8} of size 960
{18,4,8} of size 1152
{6,4,8} of size 1152
{4,4,8} of size 1152
{9,4,8} of size 1152
{20,4,8} of size 1280
{22,4,8} of size 1408
{10,4,8} of size 1600
{26,4,8} of size 1664
{6,4,8} of size 1728
{28,4,8} of size 1792
{30,4,8} of size 1920
{15,4,8} of size 1920
{5,4,8} of size 1920
{10,4,8} of size 1920
{10,4,8} of size 1920
{6,4,8} of size 1920
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {4,4}*32, {2,8}*32
4-fold quotients : {2,4}*16, {4,2}*16
8-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
2-fold covers : {4,8}*128a, {8,8}*128a, {8,8}*128b, {4,16}*128a, {4,16}*128b
3-fold covers : {4,24}*192a, {12,8}*192a
4-fold covers : {8,8}*256a, {4,8}*256a, {8,8}*256c, {4,16}*256a, {4,16}*256b, {16,8}*256a, {16,8}*256b, {8,16}*256c, {8,16}*256d, {16,8}*256d, {8,16}*256e, {8,16}*256f, {16,8}*256f, {4,32}*256a, {4,32}*256b
5-fold covers : {4,40}*320a, {20,8}*320a
6-fold covers : {4,24}*384a, {8,24}*384a, {8,24}*384b, {24,8}*384b, {12,8}*384a, {24,8}*384d, {4,48}*384a, {4,48}*384b, {12,16}*384a, {12,16}*384b
7-fold covers : {4,56}*448a, {28,8}*448a
8-fold covers : {4,16}*512a, {8,16}*512a, {8,16}*512b, {16,16}*512a, {16,16}*512b, {16,16}*512d, {16,16}*512e, {16,16}*512g, {16,16}*512h, {16,16}*512k, {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, {4,8}*512a, {8,8}*512e, {4,16}*512b, {4,8}*512b, {4,8}*512c, {8,8}*512j, {8,8}*512k, {4,16}*512c, {4,16}*512d, {8,8}*512p, {8,8}*512r, {8,16}*512g, {8,16}*512h, {4,32}*512a, {4,32}*512b, {8,32}*512a, {8,32}*512b, {32,8}*512b, {8,32}*512c, {8,32}*512d, {32,8}*512d, {4,64}*512a, {4,64}*512b
9-fold covers : {4,72}*576a, {36,8}*576a, {12,24}*576b, {12,24}*576c, {12,24}*576d, {12,8}*576a, {4,8}*576a, {4,24}*576a
10-fold covers : {4,40}*640a, {8,40}*640a, {8,40}*640b, {40,8}*640b, {20,8}*640a, {40,8}*640d, {4,80}*640a, {4,80}*640b, {20,16}*640a, {20,16}*640b
11-fold covers : {4,88}*704a, {44,8}*704a
12-fold covers : {8,24}*768a, {24,8}*768a, {12,8}*768a, {4,24}*768a, {24,8}*768c, {8,24}*768d, {12,16}*768a, {4,48}*768a, {12,16}*768b, {4,48}*768b, {48,8}*768a, {16,24}*768a, {48,8}*768b, {16,24}*768b, {24,16}*768c, {8,48}*768c, {8,48}*768d, {48,8}*768d, {16,24}*768d, {24,16}*768d, {24,16}*768e, {8,48}*768e, {8,48}*768f, {48,8}*768f, {16,24}*768f, {24,16}*768f, {12,32}*768a, {4,96}*768a, {12,32}*768b, {4,96}*768b, {4,24}*768i, {12,8}*768u, {12,24}*768c
13-fold covers : {4,104}*832a, {52,8}*832a
14-fold covers : {4,56}*896a, {8,56}*896a, {8,56}*896b, {56,8}*896b, {28,8}*896a, {56,8}*896d, {4,112}*896a, {4,112}*896b, {28,16}*896a, {28,16}*896b
15-fold covers : {20,24}*960a, {12,40}*960a, {4,120}*960a, {60,8}*960a
17-fold covers : {68,8}*1088a, {4,136}*1088a
18-fold covers : {36,8}*1152a, {4,72}*1152a, {12,24}*1152a, {12,24}*1152b, {12,24}*1152c, {4,8}*1152a, {4,24}*1152a, {12,8}*1152a, {72,8}*1152a, {8,72}*1152b, {8,72}*1152c, {72,8}*1152c, {24,24}*1152a, {24,24}*1152b, {24,24}*1152d, {24,24}*1152e, {24,24}*1152h, {24,24}*1152i, {8,8}*1152a, {8,24}*1152a, {8,8}*1152c, {8,24}*1152c, {24,8}*1152b, {24,8}*1152c, {36,16}*1152a, {4,144}*1152a, {12,48}*1152a, {12,48}*1152b, {12,48}*1152c, {4,16}*1152a, {4,48}*1152a, {12,16}*1152a, {36,16}*1152b, {4,144}*1152b, {12,48}*1152d, {12,48}*1152e, {12,48}*1152f, {4,16}*1152b, {4,48}*1152b, {12,16}*1152b
19-fold covers : {76,8}*1216a, {4,152}*1216a
20-fold covers : {8,40}*1280a, {40,8}*1280a, {20,8}*1280a, {4,40}*1280a, {40,8}*1280c, {8,40}*1280d, {20,16}*1280a, {4,80}*1280a, {20,16}*1280b, {4,80}*1280b, {80,8}*1280a, {16,40}*1280a, {80,8}*1280b, {16,40}*1280b, {40,16}*1280c, {8,80}*1280c, {8,80}*1280d, {80,8}*1280d, {16,40}*1280d, {40,16}*1280d, {40,16}*1280e, {8,80}*1280e, {8,80}*1280f, {80,8}*1280f, {16,40}*1280f, {40,16}*1280f, {20,32}*1280a, {4,160}*1280a, {20,32}*1280b, {4,160}*1280b
21-fold covers : {28,24}*1344a, {12,56}*1344a, {4,168}*1344a, {84,8}*1344a
22-fold covers : {44,8}*1408a, {4,88}*1408a, {88,8}*1408a, {8,88}*1408b, {8,88}*1408c, {88,8}*1408c, {44,16}*1408a, {4,176}*1408a, {44,16}*1408b, {4,176}*1408b
23-fold covers : {92,8}*1472a, {4,184}*1472a
25-fold covers : {4,200}*1600a, {100,8}*1600a, {20,40}*1600b, {20,40}*1600c, {20,40}*1600d, {20,8}*1600a, {4,8}*1600a, {4,40}*1600a
26-fold covers : {52,8}*1664a, {4,104}*1664a, {104,8}*1664a, {8,104}*1664b, {8,104}*1664c, {104,8}*1664c, {52,16}*1664a, {4,208}*1664a, {52,16}*1664b, {4,208}*1664b
27-fold covers : {4,216}*1728a, {108,8}*1728a, {36,24}*1728b, {12,24}*1728b, {12,72}*1728a, {12,72}*1728b, {36,24}*1728c, {12,24}*1728c, {12,24}*1728d, {12,8}*1728a, {12,24}*1728g, {12,24}*1728h, {4,24}*1728a, {4,24}*1728b, {12,8}*1728b, {12,24}*1728i, {12,24}*1728j, {12,24}*1728o, {4,24}*1728e, {4,24}*1728f, {12,8}*1728e, {12,24}*1728q, {12,8}*1728g, {12,24}*1728s, {12,24}*1728u, {12,24}*1728v
28-fold covers : {8,56}*1792a, {56,8}*1792a, {28,8}*1792a, {4,56}*1792a, {56,8}*1792c, {8,56}*1792d, {28,16}*1792a, {4,112}*1792a, {28,16}*1792b, {4,112}*1792b, {112,8}*1792a, {16,56}*1792a, {112,8}*1792b, {16,56}*1792b, {56,16}*1792c, {8,112}*1792c, {8,112}*1792d, {112,8}*1792d, {16,56}*1792d, {56,16}*1792d, {56,16}*1792e, {8,112}*1792e, {8,112}*1792f, {112,8}*1792f, {16,56}*1792f, {56,16}*1792f, {28,32}*1792a, {4,224}*1792a, {28,32}*1792b, {4,224}*1792b
29-fold covers : {116,8}*1856a, {4,232}*1856a
30-fold covers : {60,8}*1920a, {4,120}*1920a, {12,40}*1920a, {20,24}*1920a, {120,8}*1920a, {8,120}*1920b, {8,120}*1920c, {120,8}*1920c, {24,40}*1920a, {40,24}*1920a, {40,24}*1920b, {24,40}*1920c, {60,16}*1920a, {4,240}*1920a, {12,80}*1920a, {20,48}*1920a, {60,16}*1920b, {4,240}*1920b, {12,80}*1920b, {20,48}*1920b
31-fold covers : {124,8}*1984a, {4,248}*1984a
Permutation Representation (GAP) :
```s0 := ( 2, 4)( 3, 6)(10,13)(12,15);;
s1 := ( 1, 2)( 3, 5)( 4, 7)( 6, 9)( 8,10)(11,13)(12,14)(15,16);;
s2 := ( 2, 3)( 4, 6)( 5, 8)( 9,11)(10,12)(13,15);;
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*s0*s1*s0*s1*s0*s1,
s0*s1*s2*s1*s0*s1*s2*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(16)!( 2, 4)( 3, 6)(10,13)(12,15);
s1 := Sym(16)!( 1, 2)( 3, 5)( 4, 7)( 6, 9)( 8,10)(11,13)(12,14)(15,16);
s2 := Sym(16)!( 2, 3)( 4, 6)( 5, 8)( 9,11)(10,12)(13,15);
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, s0*s1*s0*s1*s0*s1*s0*s1,
s0*s1*s2*s1*s0*s1*s2*s1, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 >;

```
References : None.
to this polytope