# Polytope of Type {3,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {3,6}*324
Also Known As : {3,6}(3,3)if this polytope has another name.
Group : SmallGroup(324,41)
Rank : 3
Schlafli Type : {3,6}
Number of vertices, edges, etc : 27, 81, 54
Order of s0s1s2 : 18
Order of s0s1s2s1 : 6
Special Properties :
Toroidal
Locally Spherical
Orientable
Related Polytopes :
Facet
Vertex Figure
Dual
Petrial
Facet Of :
{3,6,2} of size 648
{3,6,4} of size 1296
{3,6,6} of size 1944
Vertex Figure Of :
{2,3,6} of size 648
{4,3,6} of size 1296
{6,3,6} of size 1944
Quotients (Maximal Quotients in Boldface) :
3-fold quotients : {3,6}*108
9-fold quotients : {3,6}*36
27-fold quotients : {3,2}*12
Covers (Minimal Covers in Boldface) :
2-fold covers : {6,6}*648d
3-fold covers : {3,18}*972a, {9,6}*972c, {9,6}*972e, {3,6}*972
4-fold covers : {12,6}*1296c, {6,12}*1296d, {3,6}*1296, {3,12}*1296a
5-fold covers : {15,6}*1620
6-fold covers : {6,18}*1944b, {18,6}*1944f, {18,6}*1944i, {6,6}*1944b, {6,6}*1944j
Permutation Representation (GAP) :
```s0 := ( 2, 3)( 4, 6)( 7, 8)(10,19)(11,21)(12,20)(13,24)(14,23)(15,22)(16,26)
(17,25)(18,27)(28,57)(29,56)(30,55)(31,59)(32,58)(33,60)(34,61)(35,63)(36,62)
(37,75)(38,74)(39,73)(40,77)(41,76)(42,78)(43,79)(44,81)(45,80)(46,66)(47,65)
(48,64)(49,68)(50,67)(51,69)(52,70)(53,72)(54,71);;
s1 := ( 1,43)( 2,45)( 3,44)( 4,37)( 5,39)( 6,38)( 7,40)( 8,42)( 9,41)(10,31)
(11,33)(12,32)(13,34)(14,36)(15,35)(16,28)(17,30)(18,29)(19,47)(20,46)(21,48)
(22,50)(23,49)(24,51)(25,53)(26,52)(27,54)(55,72)(56,71)(57,70)(58,66)(59,65)
(60,64)(61,69)(62,68)(63,67)(74,75)(77,78)(80,81);;
s2 := ( 4, 8)( 5, 9)( 6, 7)(10,19)(11,20)(12,21)(13,26)(14,27)(15,25)(16,24)
(17,22)(18,23)(31,35)(32,36)(33,34)(37,46)(38,47)(39,48)(40,53)(41,54)(42,52)
(43,51)(44,49)(45,50)(58,62)(59,63)(60,61)(64,73)(65,74)(66,75)(67,80)(68,81)
(69,79)(70,78)(71,76)(72,77);;
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,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1 ];;
poly := F / rels;;

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

```
References : None.
to this polytope