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# Polytope of Type {4,12}

Atlas Canonical Name : {4,12}*96a
Also Known As : {4,12|2}. if this polytope has another name.
Group : SmallGroup(96,89)
Rank : 3
Schlafli Type : {4,12}
Number of vertices, edges, etc : 4, 24, 12
Order of s0s1s2 : 12
Order of s0s1s2s1 : 2
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{4,12,2} of size 192
{4,12,4} of size 384
{4,12,4} of size 384
{4,12,4} of size 384
{4,12,3} of size 384
{4,12,6} of size 576
{4,12,6} of size 576
{4,12,6} of size 576
{4,12,3} of size 576
{4,12,8} of size 768
{4,12,8} of size 768
{4,12,4} of size 768
{4,12,4} of size 768
{4,12,6} of size 768
{4,12,4} of size 768
{4,12,6} of size 768
{4,12,6} of size 864
{4,12,6} of size 864
{4,12,6} of size 864
{4,12,10} of size 960
{4,12,12} of size 1152
{4,12,12} of size 1152
{4,12,12} of size 1152
{4,12,4} of size 1152
{4,12,3} of size 1152
{4,12,6} of size 1152
{4,12,6} of size 1152
{4,12,14} of size 1344
{4,12,18} of size 1728
{4,12,6} of size 1728
{4,12,6} of size 1728
{4,12,18} of size 1728
{4,12,6} of size 1728
{4,12,9} of size 1728
{4,12,3} of size 1728
{4,12,4} of size 1728
{4,12,6} of size 1728
{4,12,6} of size 1728
{4,12,6} of size 1728
{4,12,6} of size 1728
{4,12,4} of size 1728
{4,12,6} of size 1728
{4,12,20} of size 1920
{4,12,15} of size 1920
{4,12,6} of size 1920
{4,12,6} of size 1920
{4,12,10} of size 1920
{4,12,10} of size 1920
{4,12,10} of size 1920
{4,12,10} of size 1920
{4,12,5} of size 1920
Vertex Figure Of :
{2,4,12} of size 192
{4,4,12} of size 384
{6,4,12} of size 576
{3,4,12} of size 576
{8,4,12} of size 768
{8,4,12} of size 768
{4,4,12} of size 768
{6,4,12} of size 864
{10,4,12} of size 960
{12,4,12} of size 1152
{6,4,12} of size 1152
{14,4,12} of size 1344
{5,4,12} of size 1440
{18,4,12} of size 1728
{9,4,12} of size 1728
{4,4,12} of size 1728
{6,4,12} of size 1728
{20,4,12} of size 1920
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {2,12}*48, {4,6}*48a
3-fold quotients : {4,4}*32
4-fold quotients : {2,6}*24
6-fold quotients : {2,4}*16, {4,2}*16
8-fold quotients : {2,3}*12
12-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
2-fold covers : {4,24}*192a, {4,12}*192a, {4,24}*192b, {8,12}*192a, {8,12}*192b
3-fold covers : {4,36}*288a, {12,12}*288a, {12,12}*288b
4-fold covers : {4,24}*384a, {8,24}*384a, {8,24}*384b, {8,12}*384a, {8,24}*384c, {8,24}*384d, {4,48}*384a, {4,48}*384b, {4,12}*384a, {4,24}*384b, {8,12}*384b, {16,12}*384a, {16,12}*384b, {4,12}*384d
5-fold covers : {20,12}*480, {4,60}*480a
6-fold covers : {4,72}*576a, {4,36}*576a, {4,72}*576b, {8,36}*576a, {8,36}*576b, {24,12}*576a, {12,12}*576a, {12,12}*576b, {24,12}*576b, {12,24}*576c, {12,24}*576d, {24,12}*576c, {12,24}*576e, {12,24}*576f, {24,12}*576e
7-fold covers : {28,12}*672, {4,84}*672a
8-fold covers : {8,24}*768a, {8,12}*768a, {8,24}*768b, {4,24}*768a, {8,24}*768c, {8,24}*768d, {16,12}*768a, {4,48}*768a, {16,12}*768b, {4,48}*768b, {8,48}*768a, {16,24}*768a, {8,48}*768b, {16,24}*768b, {16,24}*768c, {8,48}*768c, {8,48}*768d, {16,24}*768d, {16,24}*768e, {8,48}*768e, {8,48}*768f, {16,24}*768f, {32,12}*768a, {4,96}*768a, {32,12}*768b, {4,96}*768b, {4,12}*768a, {4,24}*768b, {8,12}*768b, {8,12}*768c, {8,24}*768e, {4,24}*768c, {4,24}*768d, {8,12}*768d, {8,24}*768f, {8,24}*768g, {8,24}*768h, {8,12}*768s, {4,24}*768i, {4,12}*768d, {8,12}*768t, {4,24}*768j, {8,12}*768u, {4,12}*768e, {4,24}*768k, {8,12}*768w, {4,12}*768f, {4,24}*768l
9-fold covers : {4,108}*864a, {12,36}*864a, {12,36}*864b, {36,12}*864a, {12,12}*864b, {12,12}*864c, {12,12}*864h, {4,12}*864c, {4,12}*864d, {12,12}*864l
10-fold covers : {20,12}*960a, {20,24}*960a, {40,12}*960a, {20,24}*960b, {40,12}*960b, {4,120}*960a, {4,60}*960a, {4,120}*960b, {8,60}*960a, {8,60}*960b
11-fold covers : {44,12}*1056, {4,132}*1056a
12-fold covers : {8,36}*1152a, {4,72}*1152a, {12,24}*1152a, {12,24}*1152b, {24,12}*1152b, {24,12}*1152c, {8,72}*1152a, {8,72}*1152b, {8,72}*1152c, {24,24}*1152a, {24,24}*1152b, {24,24}*1152f, {24,24}*1152g, {24,24}*1152h, {24,24}*1152i, {8,72}*1152d, {24,24}*1152j, {24,24}*1152k, {16,36}*1152a, {4,144}*1152a, {12,48}*1152a, {12,48}*1152b, {48,12}*1152b, {48,12}*1152c, {16,36}*1152b, {4,144}*1152b, {12,48}*1152d, {12,48}*1152e, {48,12}*1152e, {48,12}*1152f, {4,36}*1152a, {4,72}*1152b, {8,36}*1152b, {12,12}*1152b, {12,24}*1152e, {24,12}*1152d, {24,12}*1152e, {12,12}*1152c, {12,24}*1152f, {4,36}*1152d, {12,12}*1152j, {12,12}*1152k, {12,12}*1152n, {12,12}*1152o
13-fold covers : {52,12}*1248, {4,156}*1248a
14-fold covers : {28,12}*1344a, {28,24}*1344a, {56,12}*1344a, {28,24}*1344b, {56,12}*1344b, {4,168}*1344a, {4,84}*1344a, {4,168}*1344b, {8,84}*1344a, {8,84}*1344b
15-fold covers : {20,36}*1440, {4,180}*1440a, {60,12}*1440a, {12,60}*1440b, {12,60}*1440c, {60,12}*1440b
17-fold covers : {68,12}*1632, {4,204}*1632a
18-fold covers : {4,216}*1728a, {4,108}*1728a, {4,216}*1728b, {8,108}*1728a, {8,108}*1728b, {24,36}*1728a, {24,12}*1728a, {12,36}*1728a, {12,36}*1728b, {36,12}*1728a, {12,12}*1728b, {12,12}*1728c, {24,36}*1728b, {24,12}*1728b, {12,72}*1728a, {12,72}*1728b, {72,12}*1728a, {24,36}*1728c, {36,24}*1728c, {12,24}*1728c, {12,24}*1728d, {24,12}*1728d, {12,72}*1728c, {12,72}*1728d, {72,12}*1728c, {24,36}*1728d, {36,24}*1728d, {12,24}*1728e, {12,24}*1728f, {24,12}*1728f, {12,24}*1728o, {24,12}*1728o, {12,24}*1728p, {24,12}*1728p, {12,12}*1728h, {4,24}*1728e, {4,24}*1728f, {8,12}*1728e, {4,24}*1728g, {4,24}*1728h, {8,12}*1728f, {8,12}*1728g, {8,12}*1728h, {4,12}*1728c, {4,12}*1728d, {12,12}*1728t, {12,24}*1728u, {24,12}*1728v, {24,12}*1728w, {12,24}*1728x
19-fold covers : {76,12}*1824, {4,228}*1824a
20-fold covers : {8,60}*1920a, {4,120}*1920a, {40,12}*1920a, {20,24}*1920a, {8,120}*1920a, {8,120}*1920b, {8,120}*1920c, {40,24}*1920a, {40,24}*1920b, {40,24}*1920c, {8,120}*1920d, {40,24}*1920d, {16,60}*1920a, {4,240}*1920a, {80,12}*1920a, {20,48}*1920a, {16,60}*1920b, {4,240}*1920b, {80,12}*1920b, {20,48}*1920b, {4,60}*1920a, {4,120}*1920b, {8,60}*1920b, {40,12}*1920b, {20,24}*1920b, {20,12}*1920a, {20,12}*1920c, {4,60}*1920d
Permutation Representation (GAP) :
```s0 := ( 2, 6)( 3,10)( 8,15)( 9,16)(11,19)(12,20);;
s1 := ( 1, 2)( 3, 7)( 4, 9)( 5, 8)( 6,14)(10,13)(11,18)(12,17)(15,24)(16,23)
(19,22)(20,21);;
s2 := ( 1, 4)( 2,11)( 3, 8)( 6,19)( 7,17)( 9,12)(10,15)(13,21)(14,23)(16,20);;
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, 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(24)!( 2, 6)( 3,10)( 8,15)( 9,16)(11,19)(12,20);
s1 := Sym(24)!( 1, 2)( 3, 7)( 4, 9)( 5, 8)( 6,14)(10,13)(11,18)(12,17)(15,24)
(16,23)(19,22)(20,21);
s2 := Sym(24)!( 1, 4)( 2,11)( 3, 8)( 6,19)( 7,17)( 9,12)(10,15)(13,21)(14,23)
(16,20);
poly := sub<Sym(24)|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, 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