# Polytope of Type {12,3,4}

Atlas Canonical Name : {12,3,4}*576
if this polytope has a name.
Group : SmallGroup(576,8660)
Rank : 4
Schlafli Type : {12,3,4}
Number of vertices, edges, etc : 24, 36, 12, 4
Order of s0s1s2s3 : 6
Order of s0s1s2s3s2s1 : 2
Special Properties :
Universal
Non-Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{12,3,4,2} of size 1152
Vertex Figure Of :
{2,12,3,4} of size 1152
Quotients (Maximal Quotients in Boldface) :
3-fold quotients : {4,3,4}*192b
4-fold quotients : {6,3,4}*144
6-fold quotients : {4,3,4}*96
12-fold quotients : {2,3,4}*48
Covers (Minimal Covers in Boldface) :
2-fold covers : {24,3,4}*1152, {12,3,4}*1152b, {12,6,4}*1152g, {12,6,4}*1152h
3-fold covers : {12,9,4}*1728, {12,3,4}*1728
Permutation Representation (GAP) :
```s0 := ( 1, 9)( 2,10)( 3,11)( 4,12)( 5,13)( 6,14)( 7,15)( 8,16)(17,41)(18,42)
(19,43)(20,44)(21,45)(22,46)(23,47)(24,48)(25,33)(26,34)(27,35)(28,36)(29,37)
(30,38)(31,39)(32,40);;
s1 := ( 1,17)( 2,19)( 3,18)( 4,20)( 5,25)( 6,27)( 7,26)( 8,28)( 9,21)(10,23)
(11,22)(12,24)(13,29)(14,31)(15,30)(16,32)(34,35)(37,41)(38,43)(39,42)(40,44)
(46,47);;
s2 := ( 3, 4)( 5,13)( 6,14)( 7,16)( 8,15)(11,12)(17,33)(18,34)(19,36)(20,35)
(21,45)(22,46)(23,48)(24,47)(25,41)(26,42)(27,44)(28,43)(29,37)(30,38)(31,40)
(32,39);;
s3 := ( 1, 4)( 2, 3)( 5, 8)( 6, 7)( 9,12)(10,11)(13,16)(14,15)(17,20)(18,19)
(21,24)(22,23)(25,28)(26,27)(29,32)(30,31)(33,36)(34,35)(37,40)(38,39)(41,44)
(42,43)(45,48)(46,47);;
poly := Group([s0,s1,s2,s3]);;

```
Finitely Presented Group Representation (GAP) :
```F := FreeGroup("s0","s1","s2","s3");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s0*s2*s0*s2,
s0*s3*s0*s3, s1*s3*s1*s3, s1*s2*s1*s2*s1*s2,
s2*s3*s2*s3*s2*s3*s2*s3, s1*s3*s2*s1*s3*s2*s1*s3*s2,
s2*s0*s1*s0*s1*s0*s1*s0*s1*s2*s0*s1*s0*s1*s0*s1*s0*s1,
s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1 ];;
poly := F / rels;;

```
Permutation Representation (Magma) :
```s0 := Sym(48)!( 1, 9)( 2,10)( 3,11)( 4,12)( 5,13)( 6,14)( 7,15)( 8,16)(17,41)
(18,42)(19,43)(20,44)(21,45)(22,46)(23,47)(24,48)(25,33)(26,34)(27,35)(28,36)
(29,37)(30,38)(31,39)(32,40);
s1 := Sym(48)!( 1,17)( 2,19)( 3,18)( 4,20)( 5,25)( 6,27)( 7,26)( 8,28)( 9,21)
(10,23)(11,22)(12,24)(13,29)(14,31)(15,30)(16,32)(34,35)(37,41)(38,43)(39,42)
(40,44)(46,47);
s2 := Sym(48)!( 3, 4)( 5,13)( 6,14)( 7,16)( 8,15)(11,12)(17,33)(18,34)(19,36)
(20,35)(21,45)(22,46)(23,48)(24,47)(25,41)(26,42)(27,44)(28,43)(29,37)(30,38)
(31,40)(32,39);
s3 := Sym(48)!( 1, 4)( 2, 3)( 5, 8)( 6, 7)( 9,12)(10,11)(13,16)(14,15)(17,20)
(18,19)(21,24)(22,23)(25,28)(26,27)(29,32)(30,31)(33,36)(34,35)(37,40)(38,39)
(41,44)(42,43)(45,48)(46,47);
poly := sub<Sym(48)|s0,s1,s2,s3>;

```
Finitely Presented Group Representation (Magma) :
```poly<s0,s1,s2,s3> := Group< s0,s1,s2,s3 | s0*s0, s1*s1, s2*s2,
s3*s3, s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3,
s1*s2*s1*s2*s1*s2, s2*s3*s2*s3*s2*s3*s2*s3,
s1*s3*s2*s1*s3*s2*s1*s3*s2, s2*s0*s1*s0*s1*s0*s1*s0*s1*s2*s0*s1*s0*s1*s0*s1*s0*s1,
s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1 >;

```
References : None.
to this polytope