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# Polytope of Type {8,4}

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

```
Permutation Representation (Magma) :
```s0 := Sym(16)!( 2, 3)( 4, 6)( 5, 8)( 9,11)(10,12)(13,15);
s1 := Sym(16)!( 1, 2)( 3, 5)( 4, 7)( 6, 9)( 8,10)(11,13)(12,14)(15,16);
s2 := Sym(16)!( 2, 4)( 3, 6)(10,13)(12,15);
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*s2*s1*s0*s1*s2*s1,
s1*s2*s1*s2*s1*s2*s1*s2, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 >;

```
References : None.
to this polytope