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# Polytope of Type {4,3,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,3,6}*288
Also Known As : Dual of 4T4(1,1)if this polytope has another name.
Group : SmallGroup(288,1028)
Rank : 4
Schlafli Type : {4,3,6}
Number of vertices, edges, etc : 8, 12, 18, 6
Order of s0s1s2s3 : 6
Order of s0s1s2s3s2s1 : 2
Special Properties :
Universal
Locally Toroidal
Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{4,3,6,2} of size 576
{4,3,6,3} of size 864
{4,3,6,4} of size 1152
{4,3,6,6} of size 1728
{4,3,6,6} of size 1728
Vertex Figure Of :
{2,4,3,6} of size 576
{4,4,3,6} of size 1152
{6,4,3,6} of size 1728
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {4,3,6}*144
3-fold quotients : {4,3,2}*96
4-fold quotients : {2,3,6}*72
6-fold quotients : {4,3,2}*48
12-fold quotients : {2,3,2}*24
Covers (Minimal Covers in Boldface) :
2-fold covers : {8,3,6}*576, {4,6,6}*576b
3-fold covers : {4,9,6}*864, {4,3,6}*864, {12,3,6}*864
4-fold covers : {8,3,6}*1152, {4,12,6}*1152f, {4,6,6}*1152e, {4,12,6}*1152i, {8,6,6}*1152c, {8,6,6}*1152e, {4,6,12}*1152d, {4,3,6}*1152b, {4,3,12}*1152b
5-fold covers : {4,15,6}*1440b
6-fold covers : {8,9,6}*1728, {8,3,6}*1728, {4,18,6}*1728b, {4,6,6}*1728a, {24,3,6}*1728, {4,6,6}*1728c, {12,6,6}*1728c, {12,6,6}*1728d
Permutation Representation (GAP) :
```s0 := ( 1,38)( 2,37)( 3,40)( 4,39)( 5,42)( 6,41)( 7,44)( 8,43)( 9,46)(10,45)
(11,48)(12,47)(13,50)(14,49)(15,52)(16,51)(17,54)(18,53)(19,56)(20,55)(21,58)
(22,57)(23,60)(24,59)(25,62)(26,61)(27,64)(28,63)(29,66)(30,65)(31,68)(32,67)
(33,70)(34,69)(35,72)(36,71);;
s1 := ( 2, 3)( 5, 9)( 6,11)( 7,10)( 8,12)(13,25)(14,27)(15,26)(16,28)(17,33)
(18,35)(19,34)(20,36)(21,29)(22,31)(23,30)(24,32)(38,39)(41,45)(42,47)(43,46)
(44,48)(49,61)(50,63)(51,62)(52,64)(53,69)(54,71)(55,70)(56,72)(57,65)(58,67)
(59,66)(60,68);;
s2 := ( 1,17)( 2,18)( 3,20)( 4,19)( 5,13)( 6,14)( 7,16)( 8,15)( 9,21)(10,22)
(11,24)(12,23)(25,29)(26,30)(27,32)(28,31)(35,36)(37,53)(38,54)(39,56)(40,55)
(41,49)(42,50)(43,52)(44,51)(45,57)(46,58)(47,60)(48,59)(61,65)(62,66)(63,68)
(64,67)(71,72);;
s3 := ( 5, 9)( 6,10)( 7,11)( 8,12)(17,21)(18,22)(19,23)(20,24)(29,33)(30,34)
(31,35)(32,36)(41,45)(42,46)(43,47)(44,48)(53,57)(54,58)(55,59)(56,60)(65,69)
(66,70)(67,71)(68,72);;
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, s1*s2*s1*s2*s1*s2,
s0*s1*s0*s1*s0*s1*s0*s1, s3*s1*s2*s3*s2*s3*s1*s2*s3*s2 ];;
poly := F / rels;;

```
Permutation Representation (Magma) :
```s0 := Sym(72)!( 1,38)( 2,37)( 3,40)( 4,39)( 5,42)( 6,41)( 7,44)( 8,43)( 9,46)
(10,45)(11,48)(12,47)(13,50)(14,49)(15,52)(16,51)(17,54)(18,53)(19,56)(20,55)
(21,58)(22,57)(23,60)(24,59)(25,62)(26,61)(27,64)(28,63)(29,66)(30,65)(31,68)
(32,67)(33,70)(34,69)(35,72)(36,71);
s1 := Sym(72)!( 2, 3)( 5, 9)( 6,11)( 7,10)( 8,12)(13,25)(14,27)(15,26)(16,28)
(17,33)(18,35)(19,34)(20,36)(21,29)(22,31)(23,30)(24,32)(38,39)(41,45)(42,47)
(43,46)(44,48)(49,61)(50,63)(51,62)(52,64)(53,69)(54,71)(55,70)(56,72)(57,65)
(58,67)(59,66)(60,68);
s2 := Sym(72)!( 1,17)( 2,18)( 3,20)( 4,19)( 5,13)( 6,14)( 7,16)( 8,15)( 9,21)
(10,22)(11,24)(12,23)(25,29)(26,30)(27,32)(28,31)(35,36)(37,53)(38,54)(39,56)
(40,55)(41,49)(42,50)(43,52)(44,51)(45,57)(46,58)(47,60)(48,59)(61,65)(62,66)
(63,68)(64,67)(71,72);
s3 := Sym(72)!( 5, 9)( 6,10)( 7,11)( 8,12)(17,21)(18,22)(19,23)(20,24)(29,33)
(30,34)(31,35)(32,36)(41,45)(42,46)(43,47)(44,48)(53,57)(54,58)(55,59)(56,60)
(65,69)(66,70)(67,71)(68,72);
poly := sub<Sym(72)|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,
s1*s2*s1*s2*s1*s2, s0*s1*s0*s1*s0*s1*s0*s1,
s3*s1*s2*s3*s2*s3*s1*s2*s3*s2 >;

```
References :
1. Theorem 11C7,11C8, McMullen P., Schulte, E.; Abstract Regular Polytopes (\ Cambridge University Press, 2002)

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