Questions?
See the FAQ
or other info.

# Polytope of Type {6,4}

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

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

```
References : None.
to this polytope