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# Polytope of Type {6,18}

Atlas Canonical Name : {6,18}*1944u
if this polytope has a name.
Group : SmallGroup(1944,2346)
Rank : 3
Schlafli Type : {6,18}
Number of vertices, edges, etc : 54, 486, 162
Order of s0s1s2 : 6
Order of s0s1s2s1 : 18
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Related Polytopes :
Facet
Vertex Figure
Dual
Petrial
Facet Of :
None in this Atlas
Vertex Figure Of :
None in this Atlas
Quotients (Maximal Quotients in Boldface) :
3-fold quotients : {6,18}*648h, {6,6}*648e
6-fold quotients : {3,18}*324
9-fold quotients : {6,6}*216c, {6,6}*216d
18-fold quotients : {3,6}*108
27-fold quotients : {6,6}*72a, {6,6}*72b, {6,6}*72c
54-fold quotients : {3,6}*36, {6,3}*36
81-fold quotients : {2,6}*24, {6,2}*24
162-fold quotients : {2,3}*12, {3,2}*12
243-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
```s0 := ( 2, 3)( 5, 6)( 8, 9)(10,20)(11,19)(12,21)(13,23)(14,22)(15,24)(16,26)
(17,25)(18,27)(29,30)(32,33)(35,36)(37,47)(38,46)(39,48)(40,50)(41,49)(42,51)
(43,53)(44,52)(45,54)(56,57)(59,60)(62,63)(64,74)(65,73)(66,75)(67,77)(68,76)
(69,78)(70,80)(71,79)(72,81);;
s1 := ( 1,13)( 2,15)( 3,14)( 4,18)( 5,17)( 6,16)( 7,11)( 8,10)( 9,12)(20,21)
(22,24)(25,26)(28,67)(29,69)(30,68)(31,72)(32,71)(33,70)(34,65)(35,64)(36,66)
(37,62)(38,61)(39,63)(40,55)(41,57)(42,56)(43,60)(44,59)(45,58)(46,73)(47,75)
(48,74)(49,78)(50,77)(51,76)(52,80)(53,79)(54,81);;
s2 := ( 1,28)( 2,30)( 3,29)( 4,34)( 5,36)( 6,35)( 7,31)( 8,33)( 9,32)(10,47)
(11,46)(12,48)(13,53)(14,52)(15,54)(16,50)(17,49)(18,51)(19,38)(20,37)(21,39)
(22,44)(23,43)(24,45)(25,41)(26,40)(27,42)(56,57)(58,61)(59,63)(60,62)(64,74)
(65,73)(66,75)(67,80)(68,79)(69,81)(70,77)(71,76)(72,78);;
poly := Group([s0,s1,s2]);;

```
Finitely Presented Group Representation (GAP) :
```F := FreeGroup("s0","s1","s2");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;
rels := [ s0*s0, s1*s1, s2*s2, s0*s2*s0*s2, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1,
s2*s0*s1*s2*s1*s0*s1*s2*s1*s2*s0*s1*s2*s1*s0*s1*s2*s1,
s2*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s0*s2*s1*s0*s1*s2*s1*s0*s1*s0*s1,
s2*s0*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s0*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1 ];;
poly := F / rels;;

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

```
Finitely Presented Group Representation (Magma) :
```poly<s0,s1,s2> := Group< s0,s1,s2 | s0*s0, s1*s1, s2*s2,
s0*s2*s0*s2, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1,
s2*s0*s1*s2*s1*s0*s1*s2*s1*s2*s0*s1*s2*s1*s0*s1*s2*s1,
s2*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s0*s2*s1*s0*s1*s2*s1*s0*s1*s0*s1,
s2*s0*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s0*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1 >;

```
References : None.
to this polytope