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# Polytope of Type {18,6}

Atlas Canonical Name : {18,6}*1944r
if this polytope has a name.
Group : SmallGroup(1944,2345)
Rank : 3
Schlafli Type : {18,6}
Number of vertices, edges, etc : 162, 486, 54
Order of s0s1s2 : 18
Order of s0s1s2s1 : 6
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 : {18,6}*648e, {6,6}*648e
6-fold quotients : {9,6}*324d
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)( 4,17)( 5,16)( 6,18)( 7,12)( 8,11)( 9,10)(19,21)
(23,24)(25,26)(28,67)(29,69)(30,68)(31,71)(32,70)(33,72)(34,66)(35,65)(36,64)
(37,63)(38,62)(39,61)(40,55)(41,57)(42,56)(43,59)(44,58)(45,60)(46,75)(47,74)
(48,73)(49,76)(50,78)(51,77)(52,80)(53,79)(54,81);;
s2 := ( 1,28)( 2,29)( 3,30)( 4,34)( 5,35)( 6,36)( 7,31)( 8,32)( 9,33)(10,37)
(11,38)(12,39)(13,43)(14,44)(15,45)(16,40)(17,41)(18,42)(19,46)(20,47)(21,48)
(22,52)(23,53)(24,54)(25,49)(26,50)(27,51)(58,61)(59,62)(60,63)(67,70)(68,71)
(69,72)(76,79)(77,80)(78,81);;
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, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s2*s0*s1*s2*s1*s2*s0*s1*s0*s1*s2*s0*s1*s2*s1*s2*s0*s1*s0*s1,
s2*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1,
s1*s0*s2*s1*s0*s1*s0*s2*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1*s0 ];;
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)( 4,17)( 5,16)( 6,18)( 7,12)( 8,11)( 9,10)
(19,21)(23,24)(25,26)(28,67)(29,69)(30,68)(31,71)(32,70)(33,72)(34,66)(35,65)
(36,64)(37,63)(38,62)(39,61)(40,55)(41,57)(42,56)(43,59)(44,58)(45,60)(46,75)
(47,74)(48,73)(49,76)(50,78)(51,77)(52,80)(53,79)(54,81);
s2 := Sym(81)!( 1,28)( 2,29)( 3,30)( 4,34)( 5,35)( 6,36)( 7,31)( 8,32)( 9,33)
(10,37)(11,38)(12,39)(13,43)(14,44)(15,45)(16,40)(17,41)(18,42)(19,46)(20,47)
(21,48)(22,52)(23,53)(24,54)(25,49)(26,50)(27,51)(58,61)(59,62)(60,63)(67,70)
(68,71)(69,72)(76,79)(77,80)(78,81);
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, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s2*s0*s1*s2*s1*s2*s0*s1*s0*s1*s2*s0*s1*s2*s1*s2*s0*s1*s0*s1,
s2*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1,
s1*s0*s2*s1*s0*s1*s0*s2*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1*s0 >;

```
References : None.
to this polytope