Questions?
See the FAQ
or other info.

# Polytope of Type {4,4,4}

Atlas Canonical Name : {4,4,4}*128
Also Known As : 2T4(2,0)(2,0), {{4,4|2},{4,4|2}}. if this polytope has another name.
Group : SmallGroup(128,1135)
Rank : 4
Schlafli Type : {4,4,4}
Number of vertices, edges, etc : 4, 8, 8, 4
Order of s0s1s2s3 : 4
Order of s0s1s2s3s2s1 : 2
Special Properties :
Universal
Locally Toroidal
Orientable
Flat
Self-Dual
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{4,4,4,2} of size 256
{4,4,4,4} of size 512
{4,4,4,6} of size 768
{4,4,4,3} of size 768
{4,4,4,6} of size 1152
{4,4,4,10} of size 1280
{4,4,4,14} of size 1792
{4,4,4,5} of size 1920
Vertex Figure Of :
{2,4,4,4} of size 256
{4,4,4,4} of size 512
{6,4,4,4} of size 768
{3,4,4,4} of size 768
{6,4,4,4} of size 1152
{10,4,4,4} of size 1280
{14,4,4,4} of size 1792
{5,4,4,4} of size 1920
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {2,4,4}*64, {4,4,2}*64, {4,2,4}*64
4-fold quotients : {2,2,4}*32, {2,4,2}*32, {4,2,2}*32
8-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
2-fold covers : {4,4,8}*256a, {8,4,4}*256a, {4,4,8}*256b, {8,4,4}*256b, {4,8,4}*256a, {4,4,4}*256a, {4,4,4}*256b, {4,8,4}*256b, {4,8,4}*256c, {4,8,4}*256d
3-fold covers : {4,12,4}*384a, {4,4,12}*384, {12,4,4}*384
4-fold covers : {8,4,8}*512a, {8,4,8}*512b, {4,4,4}*512a, {4,8,8}*512a, {8,8,4}*512a, {4,8,8}*512b, {8,8,4}*512b, {4,4,8}*512a, {8,4,4}*512a, {4,8,8}*512c, {8,8,4}*512c, {4,8,8}*512d, {8,8,4}*512d, {4,8,8}*512e, {4,8,8}*512f, {8,8,4}*512e, {8,8,4}*512f, {4,8,8}*512g, {8,8,4}*512g, {4,8,8}*512h, {8,8,4}*512h, {4,4,8}*512b, {8,4,4}*512b, {4,4,8}*512c, {8,4,4}*512c, {4,8,4}*512a, {4,8,4}*512b, {4,8,4}*512c, {4,8,4}*512d, {8,4,8}*512c, {8,4,8}*512d, {4,4,16}*512a, {16,4,4}*512a, {4,4,16}*512b, {16,4,4}*512b, {4,4,4}*512b, {4,4,4}*512c, {4,8,4}*512e, {4,8,4}*512f, {4,8,4}*512g, {4,8,4}*512h, {4,4,8}*512d, {8,4,4}*512d, {4,16,4}*512a, {4,16,4}*512b, {4,16,4}*512c, {4,16,4}*512d
5-fold covers : {4,20,4}*640, {4,4,20}*640, {20,4,4}*640
6-fold covers : {8,4,12}*768a, {12,4,8}*768a, {4,12,8}*768a, {8,12,4}*768a, {4,4,24}*768a, {24,4,4}*768a, {8,4,12}*768b, {12,4,8}*768b, {4,12,8}*768b, {8,12,4}*768b, {4,4,24}*768b, {24,4,4}*768b, {4,8,12}*768a, {12,8,4}*768a, {4,24,4}*768a, {4,4,12}*768a, {12,4,4}*768a, {4,12,4}*768a, {4,12,4}*768b, {4,4,12}*768b, {12,4,4}*768b, {4,8,12}*768b, {12,8,4}*768b, {4,24,4}*768b, {4,24,4}*768c, {4,8,12}*768c, {12,8,4}*768c, {4,8,12}*768d, {12,8,4}*768d, {4,24,4}*768d
7-fold covers : {4,28,4}*896, {4,4,28}*896, {28,4,4}*896
9-fold covers : {4,4,36}*1152, {36,4,4}*1152, {4,36,4}*1152a, {4,12,12}*1152a, {4,12,12}*1152b, {12,12,4}*1152a, {12,12,4}*1152b, {4,12,12}*1152c, {12,12,4}*1152c, {12,4,12}*1152, {4,4,4}*1152a, {4,4,4}*1152b, {4,12,4}*1152a, {4,12,4}*1152b, {4,4,12}*1152, {12,4,4}*1152
10-fold covers : {8,4,20}*1280a, {20,4,8}*1280a, {4,20,8}*1280a, {8,20,4}*1280a, {4,4,40}*1280a, {40,4,4}*1280a, {8,4,20}*1280b, {20,4,8}*1280b, {4,20,8}*1280b, {8,20,4}*1280b, {4,4,40}*1280b, {40,4,4}*1280b, {4,8,20}*1280a, {20,8,4}*1280a, {4,40,4}*1280a, {4,4,20}*1280a, {20,4,4}*1280a, {4,20,4}*1280a, {4,20,4}*1280b, {4,4,20}*1280b, {20,4,4}*1280b, {4,8,20}*1280b, {20,8,4}*1280b, {4,40,4}*1280b, {4,40,4}*1280c, {4,8,20}*1280c, {20,8,4}*1280c, {4,8,20}*1280d, {20,8,4}*1280d, {4,40,4}*1280d
11-fold covers : {4,4,44}*1408, {44,4,4}*1408, {4,44,4}*1408
13-fold covers : {4,4,52}*1664, {52,4,4}*1664, {4,52,4}*1664
14-fold covers : {8,4,28}*1792a, {28,4,8}*1792a, {4,28,8}*1792a, {8,28,4}*1792a, {4,4,56}*1792a, {56,4,4}*1792a, {8,4,28}*1792b, {28,4,8}*1792b, {4,28,8}*1792b, {8,28,4}*1792b, {4,4,56}*1792b, {56,4,4}*1792b, {4,8,28}*1792a, {28,8,4}*1792a, {4,56,4}*1792a, {4,4,28}*1792a, {28,4,4}*1792a, {4,28,4}*1792a, {4,28,4}*1792b, {4,4,28}*1792b, {28,4,4}*1792b, {4,8,28}*1792b, {28,8,4}*1792b, {4,56,4}*1792b, {4,56,4}*1792c, {4,8,28}*1792c, {28,8,4}*1792c, {4,8,28}*1792d, {28,8,4}*1792d, {4,56,4}*1792d
15-fold covers : {4,4,60}*1920, {60,4,4}*1920, {4,60,4}*1920a, {4,20,12}*1920, {12,20,4}*1920, {4,12,20}*1920a, {20,12,4}*1920a, {12,4,20}*1920, {20,4,12}*1920
Permutation Representation (GAP) :
```s0 := ( 1, 9)( 2,10)( 3,11)( 4,12)( 5,13)( 6,14)( 7,15)( 8,16)(17,25)(18,26)
(19,27)(20,28)(21,29)(22,30)(23,31)(24,32)(33,41)(34,42)(35,43)(36,44)(37,45)
(38,46)(39,47)(40,48)(49,57)(50,58)(51,59)(52,60)(53,61)(54,62)(55,63)
(56,64);;
s1 := ( 9,13)(10,14)(11,15)(12,16)(17,19)(18,20)(21,23)(22,24)(25,31)(26,32)
(27,29)(28,30)(33,37)(34,38)(35,39)(36,40)(49,55)(50,56)(51,53)(52,54)(57,59)
(58,60)(61,63)(62,64);;
s2 := ( 1,17)( 2,18)( 3,19)( 4,20)( 5,21)( 6,22)( 7,23)( 8,24)( 9,25)(10,26)
(11,27)(12,28)(13,29)(14,30)(15,31)(16,32)(33,49)(34,50)(35,51)(36,52)(37,53)
(38,54)(39,55)(40,56)(41,57)(42,58)(43,59)(44,60)(45,61)(46,62)(47,63)
(48,64);;
s3 := ( 1,41)( 2,42)( 3,43)( 4,44)( 5,45)( 6,46)( 7,47)( 8,48)( 9,33)(10,34)
(11,35)(12,36)(13,37)(14,38)(15,39)(16,40)(17,58)(18,57)(19,60)(20,59)(21,62)
(22,61)(23,64)(24,63)(25,50)(26,49)(27,52)(28,51)(29,54)(30,53)(31,56)
(32,55);;
poly := Group([s0,s1,s2,s3]);;

```
Finitely Presented Group Representation (GAP) :
```F := FreeGroup("s0","s1","s2","s3");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s0*s2*s0*s2,
s0*s3*s0*s3, s1*s3*s1*s3, 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*s3*s2*s1*s2*s3*s2, s2*s3*s2*s3*s2*s3*s2*s3 ];;
poly := F / rels;;

```
Permutation Representation (Magma) :
```s0 := Sym(64)!( 1, 9)( 2,10)( 3,11)( 4,12)( 5,13)( 6,14)( 7,15)( 8,16)(17,25)
(18,26)(19,27)(20,28)(21,29)(22,30)(23,31)(24,32)(33,41)(34,42)(35,43)(36,44)
(37,45)(38,46)(39,47)(40,48)(49,57)(50,58)(51,59)(52,60)(53,61)(54,62)(55,63)
(56,64);
s1 := Sym(64)!( 9,13)(10,14)(11,15)(12,16)(17,19)(18,20)(21,23)(22,24)(25,31)
(26,32)(27,29)(28,30)(33,37)(34,38)(35,39)(36,40)(49,55)(50,56)(51,53)(52,54)
(57,59)(58,60)(61,63)(62,64);
s2 := Sym(64)!( 1,17)( 2,18)( 3,19)( 4,20)( 5,21)( 6,22)( 7,23)( 8,24)( 9,25)
(10,26)(11,27)(12,28)(13,29)(14,30)(15,31)(16,32)(33,49)(34,50)(35,51)(36,52)
(37,53)(38,54)(39,55)(40,56)(41,57)(42,58)(43,59)(44,60)(45,61)(46,62)(47,63)
(48,64);
s3 := Sym(64)!( 1,41)( 2,42)( 3,43)( 4,44)( 5,45)( 6,46)( 7,47)( 8,48)( 9,33)
(10,34)(11,35)(12,36)(13,37)(14,38)(15,39)(16,40)(17,58)(18,57)(19,60)(20,59)
(21,62)(22,61)(23,64)(24,63)(25,50)(26,49)(27,52)(28,51)(29,54)(30,53)(31,56)
(32,55);
poly := sub<Sym(64)|s0,s1,s2,s3>;

```
Finitely Presented Group Representation (Magma) :
```poly<s0,s1,s2,s3> := Group< s0,s1,s2,s3 | s0*s0, s1*s1, s2*s2,
s3*s3, s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3,
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*s3*s2*s1*s2*s3*s2,
s2*s3*s2*s3*s2*s3*s2*s3 >;

```
References :
1. Theorem 10C12, McMullen P., Schulte, E.; Abstract Regular Polytopes (Camb\ ridge University Press, 2002)

to this polytope