# Polytope of Type {6,12}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,12}*648
if this polytope has a name.
Group : SmallGroup(648,547)
Rank : 3
Schlafli Type : {6,12}
Number of vertices, edges, etc : 27, 162, 54
Order of s0s1s2 : 12
Order of s0s1s2s1 : 6
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Non-Orientable
Related Polytopes :
Facet
Vertex Figure
Dual
Petrial
Facet Of :
{6,12,2} of size 1296
Vertex Figure Of :
{2,6,12} of size 1296
Quotients (Maximal Quotients in Boldface) :
3-fold quotients : {6,4}*216, {6,12}*216a, {6,12}*216b, {6,12}*216c
9-fold quotients : {6,4}*72
Covers (Minimal Covers in Boldface) :
2-fold covers : {6,12}*1296o
3-fold covers : {6,36}*1944
Permutation Representation (GAP) :
```s0 := (10,21)(11,19)(12,20)(13,24)(14,22)(15,23)(16,27)(17,25)(18,26)(28,55)
(29,56)(30,57)(31,58)(32,59)(33,60)(34,61)(35,62)(36,63)(37,75)(38,73)(39,74)
(40,78)(41,76)(42,77)(43,81)(44,79)(45,80)(46,65)(47,66)(48,64)(49,68)(50,69)
(51,67)(52,71)(53,72)(54,70);;
s1 := ( 1,28)( 2,30)( 3,29)( 4,34)( 5,36)( 6,35)( 7,31)( 8,33)( 9,32)(11,12)
(13,16)(14,18)(15,17)(19,75)(20,74)(21,73)(22,81)(23,80)(24,79)(25,78)(26,77)
(27,76)(37,64)(38,66)(39,65)(40,70)(41,72)(42,71)(43,67)(44,69)(45,68)(46,48)
(49,54)(50,53)(51,52)(56,57)(58,61)(59,63)(60,62);;
s2 := ( 1, 4)( 2, 6)( 3, 5)( 8, 9)(10,22)(11,24)(12,23)(13,19)(14,21)(15,20)
(16,25)(17,27)(18,26)(28,41)(29,40)(30,42)(31,38)(32,37)(33,39)(34,44)(35,43)
(36,45)(46,50)(47,49)(48,51)(52,53)(55,76)(56,78)(57,77)(58,73)(59,75)(60,74)
(61,79)(62,81)(63,80)(64,67)(65,69)(66,68)(71,72);;
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,
s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2,
s0*s1*s2*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1*s2*s1 ];;
poly := F / rels;;

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

```
References : None.
to this polytope