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# Polytope of Type {6,9,6}

Atlas Canonical Name : {6,9,6}*648
if this polytope has a name.
Group : SmallGroup(648,554)
Rank : 4
Schlafli Type : {6,9,6}
Number of vertices, edges, etc : 6, 27, 27, 6
Order of s0s1s2s3 : 18
Order of s0s1s2s3s2s1 : 2
Special Properties :
Universal
Orientable
Flat
Self-Dual
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{6,9,6,2} of size 1296
{6,9,6,3} of size 1944
Vertex Figure Of :
{2,6,9,6} of size 1296
{3,6,9,6} of size 1944
Quotients (Maximal Quotients in Boldface) :
3-fold quotients : {2,9,6}*216, {6,9,2}*216, {6,3,6}*216
9-fold quotients : {2,9,2}*72, {2,3,6}*72, {6,3,2}*72
27-fold quotients : {2,3,2}*24
Covers (Minimal Covers in Boldface) :
2-fold covers : {6,18,6}*1296d
3-fold covers : {6,9,18}*1944, {18,9,6}*1944, {6,9,6}*1944a, {6,9,6}*1944b, {6,27,6}*1944
Permutation Representation (GAP) :
```s0 := ( 4, 7)( 5, 8)( 6, 9)(13,16)(14,17)(15,18)(22,25)(23,26)(24,27)(31,34)
(32,35)(33,36)(40,43)(41,44)(42,45)(49,52)(50,53)(51,54)(58,61)(59,62)(60,63)
(67,70)(68,71)(69,72)(76,79)(77,80)(78,81);;
s1 := ( 1, 4)( 2, 6)( 3, 5)( 8, 9)(10,22)(11,24)(12,23)(13,19)(14,21)(15,20)
(16,25)(17,27)(18,26)(28,60)(29,59)(30,58)(31,57)(32,56)(33,55)(34,63)(35,62)
(36,61)(37,78)(38,77)(39,76)(40,75)(41,74)(42,73)(43,81)(44,80)(45,79)(46,69)
(47,68)(48,67)(49,66)(50,65)(51,64)(52,72)(53,71)(54,70);;
s2 := ( 1,37)( 2,39)( 3,38)( 4,43)( 5,45)( 6,44)( 7,40)( 8,42)( 9,41)(10,28)
(11,30)(12,29)(13,34)(14,36)(15,35)(16,31)(17,33)(18,32)(19,46)(20,48)(21,47)
(22,52)(23,54)(24,53)(25,49)(26,51)(27,50)(55,66)(56,65)(57,64)(58,72)(59,71)
(60,70)(61,69)(62,68)(63,67)(73,75)(76,81)(77,80)(78,79);;
s3 := (10,19)(11,20)(12,21)(13,22)(14,23)(15,24)(16,25)(17,26)(18,27)(37,46)
(38,47)(39,48)(40,49)(41,50)(42,51)(43,52)(44,53)(45,54)(64,73)(65,74)(66,75)
(67,76)(68,77)(69,78)(70,79)(71,80)(72,81);;
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, s2*s0*s1*s0*s1*s2*s0*s1*s0*s1,
s3*s1*s2*s3*s2*s3*s1*s2*s3*s2, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s1*s2*s3*s2*s1*s2*s1*s2*s3*s2*s1*s2,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 ];;
poly := F / rels;;

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

```
References : None.
to this polytope