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# Polytope of Type {12,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {12,6}*432i
if this polytope has a name.
Group : SmallGroup(432,756)
Rank : 3
Schlafli Type : {12,6}
Number of vertices, edges, etc : 36, 108, 18
Order of s0s1s2 : 12
Order of s0s1s2s1 : 6
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Self-Petrie
Related Polytopes :
Facet
Vertex Figure
Dual
Petrial
Facet Of :
{12,6,2} of size 864
{12,6,3} of size 1296
{12,6,4} of size 1728
Vertex Figure Of :
{2,12,6} of size 864
{4,12,6} of size 1728
{4,12,6} of size 1728
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {12,6}*216c
3-fold quotients : {4,6}*144
6-fold quotients : {4,6}*72
9-fold quotients : {12,2}*48
18-fold quotients : {6,2}*24
27-fold quotients : {4,2}*16
36-fold quotients : {3,2}*12
54-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
2-fold covers : {24,6}*864h, {12,12}*864k
3-fold covers : {36,6}*1296m, {12,6}*1296o, {36,6}*1296n, {36,6}*1296o, {12,6}*1296t, {12,6}*1296u
4-fold covers : {48,6}*1728h, {12,12}*1728s, {24,12}*1728u, {12,24}*1728v, {12,24}*1728w, {24,12}*1728x, {12,12}*1728aa
Permutation Representation (GAP) :
```s0 := ( 2, 3)( 4,10)( 5,12)( 6,11)( 7,19)( 8,21)( 9,20)(14,15)(16,22)(17,24)
(18,23)(26,27)(29,30)(31,37)(32,39)(33,38)(34,46)(35,48)(36,47)(41,42)(43,49)
(44,51)(45,50)(53,54);;
s1 := ( 1, 2)( 4, 8)( 5, 7)( 6, 9)(10,11)(13,17)(14,16)(15,18)(19,20)(22,26)
(23,25)(24,27)(28,29)(31,35)(32,34)(33,36)(37,38)(40,44)(41,43)(42,45)(46,47)
(49,53)(50,52)(51,54);;
s2 := ( 1,40)( 2,41)( 3,42)( 4,37)( 5,38)( 6,39)( 7,43)( 8,44)( 9,45)(10,31)
(11,32)(12,33)(13,28)(14,29)(15,30)(16,34)(17,35)(18,36)(19,49)(20,50)(21,51)
(22,46)(23,47)(24,48)(25,52)(26,53)(27,54);;
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,
s0*s1*s2*s1*s2*s0*s1*s2*s0*s1*s2*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(54)!( 2, 3)( 4,10)( 5,12)( 6,11)( 7,19)( 8,21)( 9,20)(14,15)(16,22)
(17,24)(18,23)(26,27)(29,30)(31,37)(32,39)(33,38)(34,46)(35,48)(36,47)(41,42)
(43,49)(44,51)(45,50)(53,54);
s1 := Sym(54)!( 1, 2)( 4, 8)( 5, 7)( 6, 9)(10,11)(13,17)(14,16)(15,18)(19,20)
(22,26)(23,25)(24,27)(28,29)(31,35)(32,34)(33,36)(37,38)(40,44)(41,43)(42,45)
(46,47)(49,53)(50,52)(51,54);
s2 := Sym(54)!( 1,40)( 2,41)( 3,42)( 4,37)( 5,38)( 6,39)( 7,43)( 8,44)( 9,45)
(10,31)(11,32)(12,33)(13,28)(14,29)(15,30)(16,34)(17,35)(18,36)(19,49)(20,50)
(21,51)(22,46)(23,47)(24,48)(25,52)(26,53)(27,54);
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, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s0*s1*s2*s1*s2*s0*s1*s2*s0*s1*s2*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