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# Polytope of Type {4,12}

Atlas Canonical Name : {4,12}*384a
if this polytope has a name.
Group : SmallGroup(384,1706)
Rank : 3
Schlafli Type : {4,12}
Number of vertices, edges, etc : 16, 96, 48
Order of s0s1s2 : 24
Order of s0s1s2s1 : 4
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{4,12,2} of size 768
Vertex Figure Of :
{2,4,12} of size 768
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {4,12}*192a
3-fold quotients : {4,4}*128
4-fold quotients : {4,12}*96a
6-fold quotients : {4,4}*64
8-fold quotients : {2,12}*48, {4,6}*48a
12-fold quotients : {4,4}*32
16-fold quotients : {2,6}*24
24-fold quotients : {2,4}*16, {4,2}*16
32-fold quotients : {2,3}*12
48-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
2-fold covers : {8,12}*768a, {4,24}*768a, {4,12}*768a, {8,12}*768c, {4,24}*768c
3-fold covers : {4,36}*1152a, {12,12}*1152b, {12,12}*1152c
5-fold covers : {4,60}*1920a, {20,12}*1920a
Permutation Representation (GAP) :
```s0 := ( 1,25)( 2,26)( 3,27)( 4,28)( 5,29)( 6,30)( 7,31)( 8,32)( 9,33)(10,34)
(11,35)(12,36)(13,37)(14,38)(15,39)(16,40)(17,41)(18,42)(19,43)(20,44)(21,45)
(22,46)(23,47)(24,48)(49,73)(50,74)(51,75)(52,76)(53,77)(54,78)(55,79)(56,80)
(57,81)(58,82)(59,83)(60,84)(61,85)(62,86)(63,87)(64,88)(65,89)(66,90)(67,91)
(68,92)(69,93)(70,94)(71,95)(72,96);;
s1 := ( 2, 3)( 5, 6)( 7,10)( 8,12)( 9,11)(14,15)(17,18)(19,22)(20,24)(21,23)
(25,37)(26,39)(27,38)(28,40)(29,42)(30,41)(31,46)(32,48)(33,47)(34,43)(35,45)
(36,44)(50,51)(53,54)(55,58)(56,60)(57,59)(62,63)(65,66)(67,70)(68,72)(69,71)
(73,85)(74,87)(75,86)(76,88)(77,90)(78,89)(79,94)(80,96)(81,95)(82,91)(83,93)
(84,92);;
s2 := ( 1,50)( 2,49)( 3,51)( 4,53)( 5,52)( 6,54)( 7,56)( 8,55)( 9,57)(10,59)
(11,58)(12,60)(13,71)(14,70)(15,72)(16,68)(17,67)(18,69)(19,65)(20,64)(21,66)
(22,62)(23,61)(24,63)(25,74)(26,73)(27,75)(28,77)(29,76)(30,78)(31,80)(32,79)
(33,81)(34,83)(35,82)(36,84)(37,95)(38,94)(39,96)(40,92)(41,91)(42,93)(43,89)
(44,88)(45,90)(46,86)(47,85)(48,87);;
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*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1,
s0*s1*s2*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1*s2*s1,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 ];;
poly := F / rels;;

```
Permutation Representation (Magma) :
```s0 := Sym(96)!( 1,25)( 2,26)( 3,27)( 4,28)( 5,29)( 6,30)( 7,31)( 8,32)( 9,33)
(10,34)(11,35)(12,36)(13,37)(14,38)(15,39)(16,40)(17,41)(18,42)(19,43)(20,44)
(21,45)(22,46)(23,47)(24,48)(49,73)(50,74)(51,75)(52,76)(53,77)(54,78)(55,79)
(56,80)(57,81)(58,82)(59,83)(60,84)(61,85)(62,86)(63,87)(64,88)(65,89)(66,90)
(67,91)(68,92)(69,93)(70,94)(71,95)(72,96);
s1 := Sym(96)!( 2, 3)( 5, 6)( 7,10)( 8,12)( 9,11)(14,15)(17,18)(19,22)(20,24)
(21,23)(25,37)(26,39)(27,38)(28,40)(29,42)(30,41)(31,46)(32,48)(33,47)(34,43)
(35,45)(36,44)(50,51)(53,54)(55,58)(56,60)(57,59)(62,63)(65,66)(67,70)(68,72)
(69,71)(73,85)(74,87)(75,86)(76,88)(77,90)(78,89)(79,94)(80,96)(81,95)(82,91)
(83,93)(84,92);
s2 := Sym(96)!( 1,50)( 2,49)( 3,51)( 4,53)( 5,52)( 6,54)( 7,56)( 8,55)( 9,57)
(10,59)(11,58)(12,60)(13,71)(14,70)(15,72)(16,68)(17,67)(18,69)(19,65)(20,64)
(21,66)(22,62)(23,61)(24,63)(25,74)(26,73)(27,75)(28,77)(29,76)(30,78)(31,80)
(32,79)(33,81)(34,83)(35,82)(36,84)(37,95)(38,94)(39,96)(40,92)(41,91)(42,93)
(43,89)(44,88)(45,90)(46,86)(47,85)(48,87);
poly := sub<Sym(96)|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*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1,
s0*s1*s2*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1*s2*s1,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 >;

```
References : None.
to this polytope