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# Polytope of Type {6,12}

Atlas Canonical Name : {6,12}*432e
if this polytope has a name.
Group : SmallGroup(432,530)
Rank : 3
Schlafli Type : {6,12}
Number of vertices, edges, etc : 18, 108, 36
Order of s0s1s2 : 12
Order of s0s1s2s1 : 6
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{6,12,2} of size 864
{6,12,4} of size 1728
Vertex Figure Of :
{2,6,12} of size 864
{4,6,12} of size 1728
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {6,12}*216a
3-fold quotients : {6,4}*144
6-fold quotients : {6,4}*72
27-fold quotients : {2,4}*16
54-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
2-fold covers : {6,24}*864d, {12,12}*864g
3-fold covers : {6,12}*1296j, {6,12}*1296k, {6,12}*1296o
4-fold covers : {6,48}*1728d, {12,12}*1728g, {12,24}*1728h, {24,12}*1728j, {24,12}*1728l, {12,24}*1728n
Permutation Representation (GAP) :
```s0 := ( 1,28)( 2,29)( 3,30)( 4,34)( 5,35)( 6,36)( 7,31)( 8,32)( 9,33)(10,46)
(11,47)(12,48)(13,52)(14,53)(15,54)(16,49)(17,50)(18,51)(19,37)(20,38)(21,39)
(22,43)(23,44)(24,45)(25,40)(26,41)(27,42);;
s1 := ( 1,10)( 2,12)( 3,11)( 4,14)( 5,13)( 6,15)( 7,18)( 8,17)( 9,16)(20,21)
(22,23)(25,27)(28,37)(29,39)(30,38)(31,41)(32,40)(33,42)(34,45)(35,44)(36,43)
(47,48)(49,50)(52,54);;
s2 := ( 2, 3)( 4,10)( 5,12)( 6,11)( 7,19)( 8,21)( 9,20)(13,14)(16,24)(17,23)
(18,22)(25,26)(29,30)(31,37)(32,39)(33,38)(34,46)(35,48)(36,47)(40,41)(43,51)
(44,50)(45,49)(52,53);;
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*s1*s0*s1*s0*s1*s2*s1*s2*s1*s0*s1,
s0*s1*s2*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1*s2*s1 ];;
poly := F / rels;;

```
Permutation Representation (Magma) :
```s0 := Sym(54)!( 1,28)( 2,29)( 3,30)( 4,34)( 5,35)( 6,36)( 7,31)( 8,32)( 9,33)
(10,46)(11,47)(12,48)(13,52)(14,53)(15,54)(16,49)(17,50)(18,51)(19,37)(20,38)
(21,39)(22,43)(23,44)(24,45)(25,40)(26,41)(27,42);
s1 := Sym(54)!( 1,10)( 2,12)( 3,11)( 4,14)( 5,13)( 6,15)( 7,18)( 8,17)( 9,16)
(20,21)(22,23)(25,27)(28,37)(29,39)(30,38)(31,41)(32,40)(33,42)(34,45)(35,44)
(36,43)(47,48)(49,50)(52,54);
s2 := Sym(54)!( 2, 3)( 4,10)( 5,12)( 6,11)( 7,19)( 8,21)( 9,20)(13,14)(16,24)
(17,23)(18,22)(25,26)(29,30)(31,37)(32,39)(33,38)(34,46)(35,48)(36,47)(40,41)
(43,51)(44,50)(45,49)(52,53);
poly := sub<Sym(54)|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*s1*s0*s1*s0*s1*s2*s1*s2*s1*s0*s1,
s0*s1*s2*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1*s2*s1 >;

```
References : None.
to this polytope