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# Polytope of Type {2,6,18}

Atlas Canonical Name : {2,6,18}*648b
if this polytope has a name.
Group : SmallGroup(648,300)
Rank : 4
Schlafli Type : {2,6,18}
Number of vertices, edges, etc : 2, 9, 81, 27
Order of s0s1s2s3 : 18
Order of s0s1s2s3s2s1 : 2
Special Properties :
Degenerate
Universal
Non-Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{2,6,18,2} of size 1296
Vertex Figure Of :
{2,2,6,18} of size 1296
{3,2,6,18} of size 1944
Quotients (Maximal Quotients in Boldface) :
3-fold quotients : {2,6,6}*216
Covers (Minimal Covers in Boldface) :
2-fold covers : {2,6,18}*1296f
3-fold covers : {2,6,18}*1944a, {2,18,18}*1944d, {2,18,18}*1944e, {2,18,18}*1944f, {2,6,18}*1944d, {6,6,18}*1944b
Permutation Representation (GAP) :
```s0 := (1,2);;
s1 := ( 6,10)( 7,11)( 8, 9)(12,21)(13,22)(14,23)(15,28)(16,29)(17,27)(18,26)
(19,24)(20,25)(33,37)(34,38)(35,36)(39,48)(40,49)(41,50)(42,55)(43,56)(44,54)
(45,53)(46,51)(47,52)(60,64)(61,65)(62,63)(66,75)(67,76)(68,77)(69,82)(70,83)
(71,81)(72,80)(73,78)(74,79);;
s2 := ( 3,12)( 4,14)( 5,13)( 6,15)( 7,17)( 8,16)( 9,18)(10,20)(11,19)(22,23)
(25,26)(28,29)(30,67)(31,66)(32,68)(33,70)(34,69)(35,71)(36,73)(37,72)(38,74)
(39,58)(40,57)(41,59)(42,61)(43,60)(44,62)(45,64)(46,63)(47,65)(48,76)(49,75)
(50,77)(51,79)(52,78)(53,80)(54,82)(55,81)(56,83);;
s3 := ( 3,30)( 4,32)( 5,31)( 6,36)( 7,38)( 8,37)( 9,33)(10,35)(11,34)(12,42)
(13,44)(14,43)(15,39)(16,41)(17,40)(18,45)(19,47)(20,46)(21,55)(22,54)(23,56)
(24,52)(25,51)(26,53)(27,49)(28,48)(29,50)(57,58)(60,64)(61,63)(62,65)(66,70)
(67,69)(68,71)(72,73)(75,83)(76,82)(77,81)(78,80);;
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*s1*s2,
s3*s2*s1*s2*s3*s2*s3*s2*s3*s2*s3*s2*s1*s2*s1 ];;
poly := F / rels;;

```
Permutation Representation (Magma) :
```s0 := Sym(83)!(1,2);
s1 := Sym(83)!( 6,10)( 7,11)( 8, 9)(12,21)(13,22)(14,23)(15,28)(16,29)(17,27)
(18,26)(19,24)(20,25)(33,37)(34,38)(35,36)(39,48)(40,49)(41,50)(42,55)(43,56)
(44,54)(45,53)(46,51)(47,52)(60,64)(61,65)(62,63)(66,75)(67,76)(68,77)(69,82)
(70,83)(71,81)(72,80)(73,78)(74,79);
s2 := Sym(83)!( 3,12)( 4,14)( 5,13)( 6,15)( 7,17)( 8,16)( 9,18)(10,20)(11,19)
(22,23)(25,26)(28,29)(30,67)(31,66)(32,68)(33,70)(34,69)(35,71)(36,73)(37,72)
(38,74)(39,58)(40,57)(41,59)(42,61)(43,60)(44,62)(45,64)(46,63)(47,65)(48,76)
(49,75)(50,77)(51,79)(52,78)(53,80)(54,82)(55,81)(56,83);
s3 := Sym(83)!( 3,30)( 4,32)( 5,31)( 6,36)( 7,38)( 8,37)( 9,33)(10,35)(11,34)
(12,42)(13,44)(14,43)(15,39)(16,41)(17,40)(18,45)(19,47)(20,46)(21,55)(22,54)
(23,56)(24,52)(25,51)(26,53)(27,49)(28,48)(29,50)(57,58)(60,64)(61,63)(62,65)
(66,70)(67,69)(68,71)(72,73)(75,83)(76,82)(77,81)(78,80);
poly := sub<Sym(83)|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*s1*s2,
s3*s2*s1*s2*s3*s2*s3*s2*s3*s2*s3*s2*s1*s2*s1 >;

```

to this polytope