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# Polytope of Type {4,12}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,12}*576
Also Known As : {4,12}4if this polytope has another name.
Group : SmallGroup(576,5296)
Rank : 3
Schlafli Type : {4,12}
Number of vertices, edges, etc : 24, 144, 72
Order of s0s1s2 : 4
Order of s0s1s2s1 : 12
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Self-Petrie
Related Polytopes :
Facet
Vertex Figure
Dual
Petrial
Halving Operation
Facet Of :
{4,12,2} of size 1152
Vertex Figure Of :
{2,4,12} of size 1152
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {4,12}*288
4-fold quotients : {4,6}*144
8-fold quotients : {4,6}*72
9-fold quotients : {4,4}*64
18-fold quotients : {4,4}*32
36-fold quotients : {2,4}*16, {4,2}*16
72-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
2-fold covers : {4,24}*1152a, {8,12}*1152a, {4,12}*1152a, {8,12}*1152b, {4,24}*1152b
3-fold covers : {4,12}*1728a, {12,12}*1728d, {12,12}*1728e, {4,12}*1728c, {12,12}*1728r, {12,12}*1728s
Permutation Representation (GAP) :
```s0 := ( 4, 7)( 5, 8)( 6, 9)(13,16)(14,17)(15,18)(19,28)(20,29)(21,30)(22,34)
(23,35)(24,36)(25,31)(26,32)(27,33)(40,43)(41,44)(42,45)(49,52)(50,53)(51,54)
(55,64)(56,65)(57,66)(58,70)(59,71)(60,72)(61,67)(62,68)(63,69);;
s1 := ( 2, 4)( 3, 7)( 6, 8)(11,13)(12,16)(15,17)(20,22)(21,25)(24,26)(29,31)
(30,34)(33,35)(37,55)(38,58)(39,61)(40,56)(41,59)(42,62)(43,57)(44,60)(45,63)
(46,64)(47,67)(48,70)(49,65)(50,68)(51,71)(52,66)(53,69)(54,72);;
s2 := ( 1,47)( 2,46)( 3,48)( 4,53)( 5,52)( 6,54)( 7,50)( 8,49)( 9,51)(10,38)
(11,37)(12,39)(13,44)(14,43)(15,45)(16,41)(17,40)(18,42)(19,65)(20,64)(21,66)
(22,71)(23,70)(24,72)(25,68)(26,67)(27,69)(28,56)(29,55)(30,57)(31,62)(32,61)
(33,63)(34,59)(35,58)(36,60);;
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,
s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*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(72)!( 4, 7)( 5, 8)( 6, 9)(13,16)(14,17)(15,18)(19,28)(20,29)(21,30)
(22,34)(23,35)(24,36)(25,31)(26,32)(27,33)(40,43)(41,44)(42,45)(49,52)(50,53)
(51,54)(55,64)(56,65)(57,66)(58,70)(59,71)(60,72)(61,67)(62,68)(63,69);
s1 := Sym(72)!( 2, 4)( 3, 7)( 6, 8)(11,13)(12,16)(15,17)(20,22)(21,25)(24,26)
(29,31)(30,34)(33,35)(37,55)(38,58)(39,61)(40,56)(41,59)(42,62)(43,57)(44,60)
(45,63)(46,64)(47,67)(48,70)(49,65)(50,68)(51,71)(52,66)(53,69)(54,72);
s2 := Sym(72)!( 1,47)( 2,46)( 3,48)( 4,53)( 5,52)( 6,54)( 7,50)( 8,49)( 9,51)
(10,38)(11,37)(12,39)(13,44)(14,43)(15,45)(16,41)(17,40)(18,42)(19,65)(20,64)
(21,66)(22,71)(23,70)(24,72)(25,68)(26,67)(27,69)(28,56)(29,55)(30,57)(31,62)
(32,61)(33,63)(34,59)(35,58)(36,60);
poly := sub<Sym(72)|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,
s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*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