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# Polytope of Type {4,6}

Atlas Canonical Name : {4,6}*48a
Also Known As : {4,6|2}. if this polytope has another name.
Group : SmallGroup(48,38)
Rank : 3
Schlafli Type : {4,6}
Number of vertices, edges, etc : 4, 12, 6
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 :
{4,6,2} of size 96
{4,6,3} of size 144
{4,6,4} of size 192
{4,6,3} of size 192
{4,6,4} of size 192
{4,6,6} of size 288
{4,6,6} of size 288
{4,6,6} of size 288
{4,6,8} of size 384
{4,6,4} of size 384
{4,6,6} of size 384
{4,6,9} of size 432
{4,6,3} of size 432
{4,6,5} of size 480
{4,6,5} of size 480
{4,6,10} of size 480
{4,6,12} of size 576
{4,6,12} of size 576
{4,6,12} of size 576
{4,6,3} of size 576
{4,6,4} of size 576
{4,6,14} of size 672
{4,6,15} of size 720
{4,6,16} of size 768
{4,6,4} of size 768
{4,6,3} of size 768
{4,6,4} of size 768
{4,6,12} of size 768
{4,6,6} of size 768
{4,6,8} of size 768
{4,6,8} of size 768
{4,6,12} of size 768
{4,6,18} of size 864
{4,6,6} of size 864
{4,6,6} of size 864
{4,6,18} of size 864
{4,6,6} of size 864
{4,6,6} of size 864
{4,6,20} of size 960
{4,6,4} of size 960
{4,6,6} of size 960
{4,6,5} of size 960
{4,6,10} of size 960
{4,6,10} of size 960
{4,6,5} of size 960
{4,6,10} of size 960
{4,6,10} of size 960
{4,6,15} of size 960
{4,6,21} of size 1008
{4,6,22} of size 1056
{4,6,24} of size 1152
{4,6,24} of size 1152
{4,6,24} of size 1152
{4,6,8} of size 1152
{4,6,6} of size 1152
{4,6,12} of size 1152
{4,6,12} of size 1152
{4,6,6} of size 1152
{4,6,3} of size 1200
{4,6,26} of size 1248
{4,6,9} of size 1296
{4,6,27} of size 1296
{4,6,9} of size 1296
{4,6,9} of size 1296
{4,6,9} of size 1296
{4,6,3} of size 1296
{4,6,28} of size 1344
{4,6,4} of size 1344
{4,6,21} of size 1344
{4,6,30} of size 1440
{4,6,30} of size 1440
{4,6,30} of size 1440
{4,6,33} of size 1584
{4,6,34} of size 1632
{4,6,36} of size 1728
{4,6,12} of size 1728
{4,6,36} of size 1728
{4,6,12} of size 1728
{4,6,12} of size 1728
{4,6,9} of size 1728
{4,6,3} of size 1728
{4,6,4} of size 1728
{4,6,12} of size 1728
{4,6,12} of size 1728
{4,6,12} of size 1728
{4,6,4} of size 1728
{4,6,12} of size 1728
{4,6,12} of size 1728
{4,6,38} of size 1824
{4,6,39} of size 1872
{4,6,40} of size 1920
{4,6,20} of size 1920
{4,6,30} of size 1920
{4,6,10} of size 1920
{4,6,4} of size 1920
{4,6,6} of size 1920
{4,6,10} of size 1920
{4,6,5} of size 1920
{4,6,10} of size 1920
Vertex Figure Of :
{2,4,6} of size 96
{4,4,6} of size 192
{6,4,6} of size 288
{3,4,6} of size 288
{8,4,6} of size 384
{8,4,6} of size 384
{4,4,6} of size 384
{6,4,6} of size 432
{10,4,6} of size 480
{12,4,6} of size 576
{6,4,6} of size 576
{14,4,6} of size 672
{5,4,6} of size 720
{8,4,6} of size 768
{16,4,6} of size 768
{16,4,6} of size 768
{4,4,6} of size 768
{8,4,6} of size 768
{18,4,6} of size 864
{9,4,6} of size 864
{4,4,6} of size 864
{6,4,6} of size 864
{20,4,6} of size 960
{22,4,6} of size 1056
{24,4,6} of size 1152
{24,4,6} of size 1152
{12,4,6} of size 1152
{12,4,6} of size 1152
{6,4,6} of size 1152
{12,4,6} of size 1152
{10,4,6} of size 1200
{26,4,6} of size 1248
{6,4,6} of size 1296
{28,4,6} of size 1344
{30,4,6} of size 1440
{5,4,6} of size 1440
{6,4,6} of size 1440
{10,4,6} of size 1440
{10,4,6} of size 1440
{15,4,6} of size 1440
{34,4,6} of size 1632
{36,4,6} of size 1728
{18,4,6} of size 1728
{4,4,6} of size 1728
{12,4,6} of size 1728
{38,4,6} of size 1824
{40,4,6} of size 1920
{40,4,6} of size 1920
{20,4,6} of size 1920
{5,4,6} of size 1920
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {2,6}*24
3-fold quotients : {4,2}*16
4-fold quotients : {2,3}*12
6-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
2-fold covers : {4,12}*96a, {8,6}*96
3-fold covers : {4,18}*144a, {12,6}*144a, {12,6}*144c
4-fold covers : {4,24}*192a, {4,12}*192a, {4,24}*192b, {8,12}*192a, {8,12}*192b, {16,6}*192, {4,6}*192b
5-fold covers : {20,6}*240a, {4,30}*240a
6-fold covers : {4,36}*288a, {8,18}*288, {24,6}*288a, {12,12}*288a, {12,12}*288b, {24,6}*288c
7-fold covers : {28,6}*336a, {4,42}*336a
8-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, {32,6}*384, {4,12}*384d, {8,6}*384f, {8,6}*384g, {4,12}*384e, {4,6}*384b
9-fold covers : {4,54}*432a, {36,6}*432a, {12,18}*432a, {12,6}*432b, {12,18}*432b, {12,6}*432c, {12,6}*432g, {4,6}*432b
10-fold covers : {40,6}*480, {20,12}*480, {4,60}*480a, {8,30}*480
11-fold covers : {44,6}*528a, {4,66}*528a
12-fold covers : {4,72}*576a, {4,36}*576a, {4,72}*576b, {8,36}*576a, {8,36}*576b, {16,18}*576, {48,6}*576a, {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, {48,6}*576c, {4,18}*576b, {12,12}*576d, {12,6}*576b, {12,6}*576e, {12,6}*576f
13-fold covers : {52,6}*624a, {4,78}*624a
14-fold covers : {56,6}*672, {28,12}*672, {4,84}*672a, {8,42}*672
15-fold covers : {20,18}*720a, {4,90}*720a, {60,6}*720a, {12,30}*720b, {60,6}*720b, {12,30}*720c
16-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, {64,6}*768, {8,6}*768j, {8,12}*768o, {8,12}*768p, {4,6}*768a, {8,12}*768s, {4,24}*768i, {4,12}*768d, {8,12}*768t, {4,24}*768j, {8,12}*768u, {4,12}*768e, {4,24}*768k, {8,6}*768k, {8,12}*768w, {4,12}*768f, {4,24}*768l, {8,6}*768l, {16,6}*768b, {16,6}*768c
17-fold covers : {68,6}*816a, {4,102}*816a
18-fold covers : {4,108}*864a, {8,54}*864, {72,6}*864a, {24,18}*864a, {24,6}*864b, {12,36}*864a, {12,36}*864b, {36,12}*864a, {12,12}*864b, {12,12}*864c, {24,18}*864b, {24,6}*864c, {24,6}*864f, {12,12}*864h, {4,12}*864c, {4,12}*864d, {8,6}*864b, {12,12}*864l
19-fold covers : {76,6}*912a, {4,114}*912a
20-fold covers : {80,6}*960, {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, {16,30}*960, {20,6}*960e, {4,30}*960b
21-fold covers : {28,18}*1008a, {4,126}*1008a, {84,6}*1008a, {12,42}*1008b, {84,6}*1008b, {12,42}*1008c
22-fold covers : {88,6}*1056, {44,12}*1056, {4,132}*1056a, {8,66}*1056
23-fold covers : {92,6}*1104a, {4,138}*1104a
24-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, {32,18}*1152, {96,6}*1152a, {96,6}*1152c, {4,36}*1152d, {8,18}*1152f, {8,18}*1152g, {4,36}*1152e, {4,18}*1152b, {12,24}*1152i, {12,24}*1152k, {24,6}*1152d, {24,12}*1152o, {24,12}*1152q, {24,6}*1152h, {12,6}*1152d, {12,12}*1152i, {12,12}*1152j, {12,12}*1152k, {12,12}*1152n, {12,12}*1152o, {24,6}*1152j, {24,6}*1152k, {12,6}*1152e, {24,6}*1152l, {12,12}*1152p, {12,12}*1152r, {12,6}*1152f, {24,6}*1152m
25-fold covers : {100,6}*1200a, {4,150}*1200a, {20,6}*1200a, {20,6}*1200b, {20,30}*1200a, {20,30}*1200b, {20,30}*1200c, {4,30}*1200b
26-fold covers : {104,6}*1248, {52,12}*1248, {4,156}*1248a, {8,78}*1248
27-fold covers : {4,162}*1296a, {36,18}*1296a, {12,18}*1296a, {36,6}*1296b, {12,54}*1296a, {108,6}*1296a, {12,6}*1296a, {12,6}*1296b, {12,18}*1296b, {36,6}*1296f, {12,18}*1296c, {36,6}*1296g, {36,18}*1296c, {12,18}*1296e, {12,54}*1296b, {12,18}*1296f, {12,18}*1296g, {12,18}*1296h, {12,6}*1296d, {36,6}*1296h, {36,6}*1296l, {12,18}*1296l, {12,6}*1296g, {12,6}*1296h, {12,6}*1296i, {4,18}*1296b, {4,6}*1296a, {12,6}*1296j, {12,6}*1296k, {12,6}*1296s, {12,6}*1296t
28-fold covers : {112,6}*1344, {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, {16,42}*1344, {28,6}*1344e, {4,42}*1344b
29-fold covers : {116,6}*1392a, {4,174}*1392a
30-fold covers : {40,18}*1440, {20,36}*1440, {4,180}*1440a, {8,90}*1440, {120,6}*1440a, {60,12}*1440a, {24,30}*1440b, {120,6}*1440b, {12,60}*1440b, {12,60}*1440c, {60,12}*1440b, {24,30}*1440c
31-fold covers : {124,6}*1488a, {4,186}*1488a
33-fold covers : {44,18}*1584a, {4,198}*1584a, {132,6}*1584a, {12,66}*1584b, {132,6}*1584b, {12,66}*1584c
34-fold covers : {136,6}*1632, {68,12}*1632, {4,204}*1632a, {8,102}*1632
35-fold covers : {28,30}*1680a, {20,42}*1680a, {140,6}*1680a, {4,210}*1680a
36-fold covers : {4,216}*1728a, {4,108}*1728a, {4,216}*1728b, {8,108}*1728a, {8,108}*1728b, {16,54}*1728, {144,6}*1728a, {48,18}*1728a, {48,6}*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, {48,18}*1728b, {48,6}*1728c, {4,54}*1728b, {48,6}*1728f, {12,24}*1728o, {24,12}*1728o, {12,24}*1728p, {24,12}*1728p, {12,12}*1728h, {12,36}*1728c, {36,6}*1728b, {12,18}*1728b, {36,12}*1728e, {12,18}*1728c, {12,12}*1728j, {12,6}*1728b, {12,18}*1728d, {12,6}*1728e, {12,6}*1728f, {4,24}*1728e, {4,24}*1728f, {8,12}*1728e, {4,24}*1728g, {4,24}*1728h, {8,12}*1728f, {16,6}*1728b, {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, {12,12}*1728v, {12,6}*1728h, {12,6}*1728i, {4,6}*1728
37-fold covers : {148,6}*1776a, {4,222}*1776a
38-fold covers : {152,6}*1824, {76,12}*1824, {4,228}*1824a, {8,114}*1824
39-fold covers : {52,18}*1872a, {4,234}*1872a, {156,6}*1872a, {12,78}*1872b, {156,6}*1872b, {12,78}*1872c
40-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, {32,30}*1920, {160,6}*1920, {40,6}*1920b, {40,6}*1920d, {20,6}*1920b, {20,12}*1920b, {20,12}*1920c, {4,60}*1920d, {8,30}*1920f, {8,30}*1920g, {4,60}*1920e, {4,30}*1920b
41-fold covers : {164,6}*1968a, {4,246}*1968a
Permutation Representation (GAP) :
```s0 := ( 2, 5)( 6, 9)( 7,10);;
s1 := ( 1, 2)( 3, 7)( 4, 6)( 5, 8)( 9,12)(10,11);;
s2 := ( 1, 3)( 2, 6)( 5, 9)( 8,11);;
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 ];;
poly := F / rels;;

```
Permutation Representation (Magma) :
```s0 := Sym(12)!( 2, 5)( 6, 9)( 7,10);
s1 := Sym(12)!( 1, 2)( 3, 7)( 4, 6)( 5, 8)( 9,12)(10,11);
s2 := Sym(12)!( 1, 3)( 2, 6)( 5, 9)( 8,11);
poly := sub<Sym(12)|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 >;

```
References : None.
to this polytope