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# Polytope of Type {6,6,3}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,6,3}*1944g
if this polytope has a name.
Group : SmallGroup(1944,2344)
Rank : 4
Schlafli Type : {6,6,3}
Number of vertices, edges, etc : 18, 162, 81, 9
Order of s0s1s2s3 : 18
Order of s0s1s2s3s2s1 : 6
Special Properties :
Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
None in this Atlas
Vertex Figure Of :
None in this Atlas
Quotients (Maximal Quotients in Boldface) :
3-fold quotients : {6,6,3}*648b, {6,6,3}*648e
6-fold quotients : {3,6,3}*324a
9-fold quotients : {6,6,3}*216a, {6,6,3}*216b
18-fold quotients : {3,6,3}*108
27-fold quotients : {2,6,3}*72, {6,2,3}*72
54-fold quotients : {3,2,3}*36
81-fold quotients : {2,2,3}*24
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
```s0 := ( 2, 3)( 4, 7)( 5, 9)( 6, 8)(10,19)(11,21)(12,20)(13,25)(14,27)(15,26)
(16,22)(17,24)(18,23)(29,30)(31,34)(32,36)(33,35)(37,46)(38,48)(39,47)(40,52)
(41,54)(42,53)(43,49)(44,51)(45,50)(56,57)(58,61)(59,63)(60,62)(64,73)(65,75)
(66,74)(67,79)(68,81)(69,80)(70,76)(71,78)(72,77);;
s1 := ( 1,10)( 2,12)( 3,11)( 4,16)( 5,18)( 6,17)( 7,13)( 8,15)( 9,14)(20,21)
(22,25)(23,27)(24,26)(28,64)(29,66)(30,65)(31,70)(32,72)(33,71)(34,67)(35,69)
(36,68)(37,55)(38,57)(39,56)(40,61)(41,63)(42,62)(43,58)(44,60)(45,59)(46,73)
(47,75)(48,74)(49,79)(50,81)(51,80)(52,76)(53,78)(54,77);;
s2 := ( 1,28)( 2,29)( 3,30)( 4,35)( 5,36)( 6,34)( 7,33)( 8,31)( 9,32)(10,41)
(11,42)(12,40)(13,39)(14,37)(15,38)(16,43)(17,44)(18,45)(19,54)(20,52)(21,53)
(22,49)(23,50)(24,51)(25,47)(26,48)(27,46)(58,62)(59,63)(60,61)(64,68)(65,69)
(66,67)(73,81)(74,79)(75,80);;
s3 := ( 4, 7)( 5, 8)( 6, 9)(13,16)(14,17)(15,18)(22,25)(23,26)(24,27)(28,55)
(29,56)(30,57)(31,61)(32,62)(33,63)(34,58)(35,59)(36,60)(37,64)(38,65)(39,66)
(40,70)(41,71)(42,72)(43,67)(44,68)(45,69)(46,73)(47,74)(48,75)(49,79)(50,80)
(51,81)(52,76)(53,77)(54,78);;
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*s3*s2*s3*s2*s3,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s2*s0*s1*s2*s1*s0*s1*s2*s0*s1*s2*s1*s0*s1,
s1*s3*s2*s1*s3*s2*s1*s3*s2*s1*s3*s2*s1*s3*s2*s1*s3*s2,
s1*s2*s3*s1*s2*s1*s0*s1*s0*s1*s2*s3*s0*s1*s2*s0*s1*s0*s1*s0 ];;
poly := F / rels;;

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

```
References : None.
to this polytope