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# Polytope of Type {12,4}

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