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# Polytope of Type {2,4,18}

Atlas Canonical Name : {2,4,18}*288b
if this polytope has a name.
Group : SmallGroup(288,835)
Rank : 4
Schlafli Type : {2,4,18}
Number of vertices, edges, etc : 2, 4, 36, 18
Order of s0s1s2s3 : 18
Order of s0s1s2s3s2s1 : 2
Special Properties :
Degenerate
Universal
Non-Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{2,4,18,2} of size 576
{2,4,18,4} of size 1152
{2,4,18,4} of size 1152
{2,4,18,4} of size 1152
{2,4,18,6} of size 1728
{2,4,18,6} of size 1728
Vertex Figure Of :
{2,2,4,18} of size 576
{3,2,4,18} of size 864
{4,2,4,18} of size 1152
{5,2,4,18} of size 1440
{6,2,4,18} of size 1728
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {2,4,9}*144
3-fold quotients : {2,4,6}*96c
6-fold quotients : {2,4,3}*48
Covers (Minimal Covers in Boldface) :
2-fold covers : {2,4,36}*576b, {2,4,36}*576c, {2,4,18}*576
3-fold covers : {2,4,54}*864b
4-fold covers : {4,4,18}*1152b, {2,4,18}*1152a, {2,4,72}*1152c, {2,4,72}*1152d, {2,4,36}*1152b, {4,4,18}*1152d, {2,4,18}*1152b, {2,4,36}*1152c, {2,8,18}*1152b, {2,8,18}*1152c
5-fold covers : {2,4,90}*1440b
6-fold covers : {2,4,108}*1728b, {2,4,108}*1728c, {2,4,54}*1728, {6,4,18}*1728b, {2,12,18}*1728a, {2,12,18}*1728b
Permutation Representation (GAP) :
```s0 := (1,2);;
s1 := ( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)
(23,24)(25,26)(27,28)(29,30)(31,32)(33,34)(35,36)(37,38)(39,40)(41,42)(43,44)
(45,46)(47,48)(49,50)(51,52)(53,54)(55,56)(57,58)(59,60)(61,62)(63,64)(65,66)
(67,68)(69,70)(71,72)(73,74);;
s2 := ( 4, 5)( 7,11)( 8,13)( 9,12)(10,14)(15,31)(16,33)(17,32)(18,34)(19,27)
(20,29)(21,28)(22,30)(23,35)(24,37)(25,36)(26,38)(40,41)(43,47)(44,49)(45,48)
(46,50)(51,67)(52,69)(53,68)(54,70)(55,63)(56,65)(57,64)(58,66)(59,71)(60,73)
(61,72)(62,74);;
s3 := ( 3,51)( 4,52)( 5,54)( 6,53)( 7,59)( 8,60)( 9,62)(10,61)(11,55)(12,56)
(13,58)(14,57)(15,39)(16,40)(17,42)(18,41)(19,47)(20,48)(21,50)(22,49)(23,43)
(24,44)(25,46)(26,45)(27,67)(28,68)(29,70)(30,69)(31,63)(32,64)(33,66)(34,65)
(35,71)(36,72)(37,74)(38,73);;
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*s1*s0*s1,
s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3,
s1*s2*s1*s2*s1*s2*s1*s2, s1*s2*s3*s2*s1*s2*s3*s1*s2,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 ];;
poly := F / rels;;

```
Permutation Representation (Magma) :
```s0 := Sym(74)!(1,2);
s1 := Sym(74)!( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)
(21,22)(23,24)(25,26)(27,28)(29,30)(31,32)(33,34)(35,36)(37,38)(39,40)(41,42)
(43,44)(45,46)(47,48)(49,50)(51,52)(53,54)(55,56)(57,58)(59,60)(61,62)(63,64)
(65,66)(67,68)(69,70)(71,72)(73,74);
s2 := Sym(74)!( 4, 5)( 7,11)( 8,13)( 9,12)(10,14)(15,31)(16,33)(17,32)(18,34)
(19,27)(20,29)(21,28)(22,30)(23,35)(24,37)(25,36)(26,38)(40,41)(43,47)(44,49)
(45,48)(46,50)(51,67)(52,69)(53,68)(54,70)(55,63)(56,65)(57,64)(58,66)(59,71)
(60,73)(61,72)(62,74);
s3 := Sym(74)!( 3,51)( 4,52)( 5,54)( 6,53)( 7,59)( 8,60)( 9,62)(10,61)(11,55)
(12,56)(13,58)(14,57)(15,39)(16,40)(17,42)(18,41)(19,47)(20,48)(21,50)(22,49)
(23,43)(24,44)(25,46)(26,45)(27,67)(28,68)(29,70)(30,69)(31,63)(32,64)(33,66)
(34,65)(35,71)(36,72)(37,74)(38,73);
poly := sub<Sym(74)|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*s1*s0*s1, s0*s2*s0*s2, s0*s3*s0*s3,
s1*s3*s1*s3, s1*s2*s1*s2*s1*s2*s1*s2,
s1*s2*s3*s2*s1*s2*s3*s1*s2, s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 >;

```

to this polytope