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# Polytope of Type {18,6}

Atlas Canonical Name : {18,6}*1944m
if this polytope has a name.
Group : SmallGroup(1944,2340)
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}*648a, {18,6}*648i, {6,6}*648e
6-fold quotients : {9,6}*324a
9-fold quotients : {18,6}*216a, {18,6}*216b, {6,6}*216c, {6,6}*216d
18-fold quotients : {9,6}*108, {3,6}*108
27-fold quotients : {18,2}*72, {6,6}*72a, {6,6}*72b, {6,6}*72c
54-fold quotients : {9,2}*36, {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 := ( 4, 7)( 5, 8)( 6, 9)(10,22)(11,23)(12,24)(13,19)(14,20)(15,21)(16,25)
(17,26)(18,27)(31,34)(32,35)(33,36)(37,49)(38,50)(39,51)(40,46)(41,47)(42,48)
(43,52)(44,53)(45,54)(58,61)(59,62)(60,63)(64,76)(65,77)(66,78)(67,73)(68,74)
(69,75)(70,79)(71,80)(72,81);;
s1 := ( 1,11)( 2,12)( 3,10)( 4,17)( 5,18)( 6,16)( 7,14)( 8,15)( 9,13)(19,22)
(20,23)(21,24)(28,65)(29,66)(30,64)(31,71)(32,72)(33,70)(34,68)(35,69)(36,67)
(37,57)(38,55)(39,56)(40,63)(41,61)(42,62)(43,60)(44,58)(45,59)(46,76)(47,77)
(48,78)(49,73)(50,74)(51,75)(52,79)(53,80)(54,81);;
s2 := ( 1,28)( 2,30)( 3,29)( 4,31)( 5,33)( 6,32)( 7,34)( 8,36)( 9,35)(10,37)
(11,39)(12,38)(13,40)(14,42)(15,41)(16,43)(17,45)(18,44)(19,46)(20,48)(21,47)
(22,49)(23,51)(24,50)(25,52)(26,54)(27,53)(56,57)(59,60)(62,63)(65,66)(68,69)
(71,72)(74,75)(77,78)(80,81);;
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*s0*s1*s2*s1*s2*s0*s1*s2*s1*s0*s1*s2*s1,
s2*s0*s1*s2*s1*s2*s0*s1*s0*s1*s2*s0*s1*s2*s1*s2*s0*s1*s0*s1,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 ];;
poly := F / rels;;

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

```
References : None.
to this polytope