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# Polytope of Type {4,4}

Atlas Canonical Name : {4,4}*64
Also Known As : {4,4}(2,2), {4,4}4if this polytope has another name.
Group : SmallGroup(64,138)
Rank : 3
Schlafli Type : {4,4}
Number of vertices, edges, etc : 8, 16, 8
Order of s0s1s2 : 4
Order of s0s1s2s1 : 4
Special Properties :
Toroidal
Locally Spherical
Orientable
Self-Dual
Self-Petrie
Related Polytopes :
Facet
Vertex Figure
Dual
Petrial
Halving Operation
Skewing Operation
Facet Of :
{4,4,2} of size 128
{4,4,3} of size 192
{4,4,4} of size 256
{4,4,6} of size 384
{4,4,3} of size 384
{4,4,6} of size 384
{4,4,6} of size 384
{4,4,4} of size 512
{4,4,8} of size 512
{4,4,8} of size 512
{4,4,9} of size 576
{4,4,10} of size 640
{4,4,12} of size 768
{4,4,3} of size 768
{4,4,6} of size 768
{4,4,12} of size 768
{4,4,12} of size 768
{4,4,6} of size 768
{4,4,14} of size 896
{4,4,15} of size 960
{4,4,18} of size 1152
{4,4,6} of size 1152
{4,4,9} of size 1152
{4,4,18} of size 1152
{4,4,18} of size 1152
{4,4,4} of size 1152
{4,4,20} of size 1280
{4,4,21} of size 1344
{4,4,22} of size 1408
{4,4,26} of size 1664
{4,4,27} of size 1728
{4,4,28} of size 1792
{4,4,30} of size 1920
{4,4,15} of size 1920
{4,4,30} of size 1920
{4,4,30} of size 1920
{4,4,6} of size 1920
Vertex Figure Of :
{2,4,4} of size 128
{3,4,4} of size 192
{4,4,4} of size 256
{6,4,4} of size 384
{3,4,4} of size 384
{6,4,4} of size 384
{6,4,4} of size 384
{4,4,4} of size 512
{8,4,4} of size 512
{8,4,4} of size 512
{9,4,4} of size 576
{10,4,4} of size 640
{12,4,4} of size 768
{3,4,4} of size 768
{6,4,4} of size 768
{12,4,4} of size 768
{12,4,4} of size 768
{6,4,4} of size 768
{14,4,4} of size 896
{15,4,4} of size 960
{18,4,4} of size 1152
{6,4,4} of size 1152
{9,4,4} of size 1152
{18,4,4} of size 1152
{18,4,4} of size 1152
{4,4,4} of size 1152
{20,4,4} of size 1280
{21,4,4} of size 1344
{22,4,4} of size 1408
{26,4,4} of size 1664
{27,4,4} of size 1728
{28,4,4} of size 1792
{30,4,4} of size 1920
{15,4,4} of size 1920
{30,4,4} of size 1920
{30,4,4} of size 1920
{6,4,4} of size 1920
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {4,4}*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, {8,4}*128a, {4,4}*128, {4,8}*128b, {8,4}*128b
3-fold covers : {4,12}*192a, {12,4}*192a
4-fold covers : {8,8}*256a, {4,8}*256a, {8,4}*256a, {8,8}*256b, {8,8}*256c, {8,8}*256d, {4,16}*256a, {16,4}*256a, {4,16}*256b, {16,4}*256b, {4,4}*256, {4,8}*256b, {8,4}*256b, {4,8}*256c, {4,8}*256d, {8,4}*256c, {8,4}*256d, {8,8}*256e, {8,8}*256f, {8,8}*256g, {8,8}*256h
5-fold covers : {4,20}*320, {20,4}*320
6-fold covers : {4,24}*384a, {24,4}*384a, {8,12}*384a, {12,8}*384a, {4,12}*384a, {4,24}*384b, {12,4}*384a, {24,4}*384b, {8,12}*384b, {12,8}*384b
7-fold covers : {4,28}*448, {28,4}*448
8-fold covers : {4,16}*512a, {16,4}*512a, {8,16}*512a, {16,8}*512a, {8,16}*512b, {16,8}*512b, {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, {8,4}*512a, {8,8}*512d, {8,8}*512e, {8,8}*512f, {8,8}*512g, {4,16}*512b, {16,4}*512b, {4,8}*512b, {4,8}*512c, {8,4}*512b, {8,4}*512c, {8,8}*512h, {8,8}*512i, {8,8}*512j, {8,8}*512k, {8,8}*512l, {8,8}*512m, {8,8}*512n, {8,8}*512o, {4,16}*512c, {4,16}*512d, {16,4}*512c, {16,4}*512d, {8,8}*512p, {8,8}*512q, {8,8}*512r, {8,8}*512s, {8,8}*512t, {8,16}*512g, {16,8}*512g, {8,16}*512h, {16,8}*512h, {4,32}*512a, {32,4}*512a, {4,32}*512b, {32,4}*512b, {4,4}*512, {4,16}*512e, {16,4}*512e, {4,8}*512d, {4,16}*512f, {8,4}*512d, {16,4}*512f
9-fold covers : {4,36}*576a, {36,4}*576a, {12,12}*576a, {12,12}*576b, {12,12}*576c, {4,4}*576, {4,12}*576, {12,4}*576
10-fold covers : {4,40}*640a, {40,4}*640a, {8,20}*640a, {20,8}*640a, {4,20}*640a, {4,40}*640b, {20,4}*640a, {40,4}*640b, {8,20}*640b, {20,8}*640b
11-fold covers : {4,44}*704, {44,4}*704
12-fold covers : {8,24}*768a, {24,8}*768a, {8,12}*768a, {8,24}*768b, {12,8}*768a, {24,8}*768b, {4,24}*768a, {24,4}*768a, {8,24}*768c, {24,8}*768c, {8,24}*768d, {24,8}*768d, {12,16}*768a, {16,12}*768a, {4,48}*768a, {48,4}*768a, {12,16}*768b, {16,12}*768b, {4,48}*768b, {48,4}*768b, {4,12}*768a, {4,24}*768b, {12,4}*768a, {24,4}*768b, {8,12}*768b, {12,8}*768b, {8,12}*768c, {8,24}*768e, {12,8}*768c, {24,8}*768e, {4,24}*768c, {4,24}*768d, {24,4}*768c, {24,4}*768d, {8,12}*768d, {8,24}*768f, {12,8}*768d, {24,8}*768f, {8,24}*768g, {24,8}*768g, {8,24}*768h, {24,8}*768h, {4,12}*768d, {12,4}*768d, {12,12}*768b
13-fold covers : {4,52}*832, {52,4}*832
14-fold covers : {4,56}*896a, {56,4}*896a, {8,28}*896a, {28,8}*896a, {4,28}*896, {4,56}*896b, {28,4}*896, {56,4}*896b, {8,28}*896b, {28,8}*896b
15-fold covers : {12,20}*960a, {20,12}*960a, {4,60}*960a, {60,4}*960a
17-fold covers : {4,68}*1088, {68,4}*1088
18-fold covers : {8,36}*1152a, {36,8}*1152a, {4,72}*1152a, {72,4}*1152a, {12,24}*1152a, {12,24}*1152b, {24,12}*1152a, {24,12}*1152b, {12,24}*1152c, {24,12}*1152c, {4,8}*1152a, {4,24}*1152a, {8,4}*1152a, {24,4}*1152a, {8,12}*1152a, {12,8}*1152a, {4,36}*1152a, {4,72}*1152b, {36,4}*1152a, {72,4}*1152b, {8,36}*1152b, {36,8}*1152b, {12,12}*1152a, {12,12}*1152b, {12,24}*1152d, {12,24}*1152e, {24,12}*1152d, {24,12}*1152e, {12,12}*1152c, {12,24}*1152f, {24,12}*1152f, {4,8}*1152b, {4,12}*1152a, {8,4}*1152b, {8,12}*1152b, {12,4}*1152a, {12,8}*1152b, {4,4}*1152, {4,24}*1152b, {24,4}*1152b
19-fold covers : {4,76}*1216, {76,4}*1216
20-fold covers : {8,40}*1280a, {40,8}*1280a, {8,20}*1280a, {8,40}*1280b, {20,8}*1280a, {40,8}*1280b, {4,40}*1280a, {40,4}*1280a, {8,40}*1280c, {40,8}*1280c, {8,40}*1280d, {40,8}*1280d, {16,20}*1280a, {20,16}*1280a, {4,80}*1280a, {80,4}*1280a, {16,20}*1280b, {20,16}*1280b, {4,80}*1280b, {80,4}*1280b, {4,20}*1280a, {4,40}*1280b, {20,4}*1280a, {40,4}*1280b, {8,20}*1280b, {20,8}*1280b, {8,20}*1280c, {8,40}*1280e, {20,8}*1280c, {40,8}*1280e, {4,40}*1280c, {4,40}*1280d, {40,4}*1280c, {40,4}*1280d, {8,20}*1280d, {8,40}*1280f, {20,8}*1280d, {40,8}*1280f, {8,40}*1280g, {40,8}*1280g, {8,40}*1280h, {40,8}*1280h
21-fold covers : {12,28}*1344a, {28,12}*1344a, {4,84}*1344a, {84,4}*1344a
22-fold covers : {8,44}*1408a, {44,8}*1408a, {4,88}*1408a, {88,4}*1408a, {4,44}*1408, {4,88}*1408b, {44,4}*1408, {88,4}*1408b, {8,44}*1408b, {44,8}*1408b
23-fold covers : {4,92}*1472, {92,4}*1472
25-fold covers : {4,100}*1600, {100,4}*1600, {20,20}*1600a, {20,20}*1600b, {20,20}*1600c, {4,4}*1600, {4,20}*1600, {20,4}*1600
26-fold covers : {8,52}*1664a, {52,8}*1664a, {4,104}*1664a, {104,4}*1664a, {4,52}*1664, {4,104}*1664b, {52,4}*1664, {104,4}*1664b, {8,52}*1664b, {52,8}*1664b
27-fold covers : {4,108}*1728a, {108,4}*1728a, {12,36}*1728a, {12,36}*1728b, {36,12}*1728a, {36,12}*1728b, {12,12}*1728a, {12,12}*1728b, {12,12}*1728c, {4,12}*1728a, {4,12}*1728b, {12,4}*1728a, {12,4}*1728b, {12,12}*1728d, {12,12}*1728e, {12,12}*1728f, {12,12}*1728g, {12,12}*1728h, {4,12}*1728c, {4,12}*1728d, {12,4}*1728c, {12,4}*1728d, {12,12}*1728q, {12,12}*1728r, {12,12}*1728s, {12,12}*1728t
28-fold covers : {8,56}*1792a, {56,8}*1792a, {8,28}*1792a, {8,56}*1792b, {28,8}*1792a, {56,8}*1792b, {4,56}*1792a, {56,4}*1792a, {8,56}*1792c, {56,8}*1792c, {8,56}*1792d, {56,8}*1792d, {16,28}*1792a, {28,16}*1792a, {4,112}*1792a, {112,4}*1792a, {16,28}*1792b, {28,16}*1792b, {4,112}*1792b, {112,4}*1792b, {4,28}*1792, {4,56}*1792b, {28,4}*1792, {56,4}*1792b, {8,28}*1792b, {28,8}*1792b, {8,28}*1792c, {8,56}*1792e, {28,8}*1792c, {56,8}*1792e, {4,56}*1792c, {4,56}*1792d, {56,4}*1792c, {56,4}*1792d, {8,28}*1792d, {8,56}*1792f, {28,8}*1792d, {56,8}*1792f, {8,56}*1792g, {56,8}*1792g, {8,56}*1792h, {56,8}*1792h
29-fold covers : {4,116}*1856, {116,4}*1856
30-fold covers : {8,60}*1920a, {60,8}*1920a, {4,120}*1920a, {120,4}*1920a, {12,40}*1920a, {40,12}*1920a, {20,24}*1920a, {24,20}*1920a, {4,60}*1920a, {4,120}*1920b, {60,4}*1920a, {120,4}*1920b, {8,60}*1920b, {60,8}*1920b, {12,40}*1920b, {40,12}*1920b, {20,24}*1920b, {24,20}*1920b, {12,20}*1920a, {20,12}*1920a
31-fold covers : {4,124}*1984, {124,4}*1984
Permutation Representation (GAP) :
```s0 := ( 2, 3)( 4, 6)( 7,10)( 9,12)(11,14)(13,15);;
s1 := ( 1, 2)( 3, 5)( 4, 7)( 6, 9)( 8,11)(10,13)(12,15)(14,16);;
s2 := ( 2, 4)( 3, 6)( 5, 8)( 9,12)(11,15)(13,14);;
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,
s1*s2*s1*s2*s1*s2*s1*s2, s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1 ];;
poly := F / rels;;

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

```
References : None.
to this polytope