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# Polytope of Type {3,6}

Atlas Canonical Name : {3,6}*36
Also Known As : {3,6}(1,1)if this polytope has another name.
Group : SmallGroup(36,10)
Rank : 3
Schlafli Type : {3,6}
Number of vertices, edges, etc : 3, 9, 6
Order of s0s1s2 : 6
Order of s0s1s2s1 : 6
Special Properties :
Toroidal
Locally Spherical
Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{3,6,2} of size 72
{3,6,3} of size 108
{3,6,4} of size 144
{3,6,6} of size 216
{3,6,6} of size 216
{3,6,8} of size 288
{3,6,9} of size 324
{3,6,3} of size 324
{3,6,10} of size 360
{3,6,12} of size 432
{3,6,12} of size 432
{3,6,4} of size 432
{3,6,14} of size 504
{3,6,15} of size 540
{3,6,16} of size 576
{3,6,4} of size 576
{3,6,18} of size 648
{3,6,6} of size 648
{3,6,18} of size 648
{3,6,6} of size 648
{3,6,6} of size 648
{3,6,20} of size 720
{3,6,21} of size 756
{3,6,22} of size 792
{3,6,24} of size 864
{3,6,24} of size 864
{3,6,8} of size 864
{3,6,26} of size 936
{3,6,27} of size 972
{3,6,9} of size 972
{3,6,28} of size 1008
{3,6,30} of size 1080
{3,6,30} of size 1080
{3,6,32} of size 1152
{3,6,4} of size 1152
{3,6,33} of size 1188
{3,6,34} of size 1224
{3,6,36} of size 1296
{3,6,12} of size 1296
{3,6,36} of size 1296
{3,6,12} of size 1296
{3,6,12} of size 1296
{3,6,4} of size 1296
{3,6,12} of size 1296
{3,6,38} of size 1368
{3,6,39} of size 1404
{3,6,40} of size 1440
{3,6,42} of size 1512
{3,6,42} of size 1512
{3,6,44} of size 1584
{3,6,45} of size 1620
{3,6,15} of size 1620
{3,6,46} of size 1656
{3,6,48} of size 1728
{3,6,48} of size 1728
{3,6,16} of size 1728
{3,6,50} of size 1800
{3,6,51} of size 1836
{3,6,52} of size 1872
{3,6,54} of size 1944
{3,6,18} of size 1944
{3,6,18} of size 1944
{3,6,18} of size 1944
{3,6,6} of size 1944
{3,6,54} of size 1944
{3,6,6} of size 1944
{3,6,6} of size 1944
Vertex Figure Of :
{2,3,6} of size 72
{4,3,6} of size 144
{6,3,6} of size 216
{4,3,6} of size 288
{8,3,6} of size 576
{6,3,6} of size 648
{6,3,6} of size 864
{12,3,6} of size 864
{8,3,6} of size 1152
{12,3,6} of size 1728
{24,3,6} of size 1728
{10,3,6} of size 1800
{6,3,6} of size 1944
{18,3,6} of size 1944
Quotients (Maximal Quotients in Boldface) :
3-fold quotients : {3,2}*12
Covers (Minimal Covers in Boldface) :
2-fold covers : {6,6}*72c
3-fold covers : {9,6}*108, {3,6}*108
4-fold covers : {12,6}*144b, {6,12}*144c, {3,6}*144, {3,12}*144
5-fold covers : {15,6}*180
6-fold covers : {18,6}*216b, {6,6}*216c, {6,6}*216d
7-fold covers : {21,6}*252
8-fold covers : {24,6}*288b, {12,12}*288c, {6,24}*288c, {3,12}*288, {3,24}*288, {6,6}*288b, {6,12}*288b
9-fold covers : {9,18}*324, {9,6}*324a, {27,6}*324, {9,6}*324b, {9,6}*324c, {9,6}*324d, {3,6}*324, {3,18}*324
10-fold covers : {6,30}*360a, {30,6}*360c
11-fold covers : {33,6}*396
12-fold covers : {36,6}*432b, {12,6}*432a, {18,12}*432b, {6,12}*432c, {9,6}*432, {9,12}*432, {3,6}*432, {3,12}*432, {6,12}*432g, {12,6}*432g
13-fold covers : {39,6}*468
14-fold covers : {6,42}*504a, {42,6}*504c
15-fold covers : {45,6}*540, {15,6}*540
16-fold covers : {48,6}*576b, {12,24}*576a, {12,12}*576c, {12,24}*576b, {24,12}*576d, {24,12}*576f, {6,48}*576c, {3,6}*576, {3,24}*576, {12,12}*576e, {12,6}*576a, {12,12}*576h, {6,12}*576c, {6,24}*576b, {6,6}*576b, {6,24}*576d, {12,6}*576d, {6,12}*576e, {6,12}*576f, {3,12}*576, {6,6}*576e
17-fold covers : {51,6}*612
18-fold covers : {18,18}*648c, {18,6}*648a, {54,6}*648b, {18,6}*648c, {18,6}*648d, {18,6}*648e, {6,6}*648d, {6,18}*648h, {6,18}*648i, {18,6}*648i, {6,6}*648e, {6,6}*648f, {6,6}*648g
19-fold covers : {57,6}*684
20-fold covers : {6,60}*720a, {12,30}*720a, {60,6}*720c, {30,12}*720c, {15,12}*720, {15,6}*720e
21-fold covers : {63,6}*756, {21,6}*756
22-fold covers : {6,66}*792a, {66,6}*792c
23-fold covers : {69,6}*828
24-fold covers : {72,6}*864b, {24,6}*864a, {36,12}*864b, {12,12}*864a, {18,24}*864b, {6,24}*864c, {9,12}*864, {9,24}*864, {3,12}*864, {3,24}*864, {6,24}*864f, {24,6}*864f, {12,12}*864h, {18,6}*864, {18,12}*864b, {6,6}*864a, {6,12}*864a, {6,6}*864c, {6,12}*864c, {12,6}*864c
25-fold covers : {75,6}*900, {3,6}*900, {3,30}*900, {15,30}*900
26-fold covers : {6,78}*936a, {78,6}*936c
27-fold covers : {9,18}*972a, {3,18}*972a, {9,6}*972a, {9,6}*972b, {9,18}*972b, {9,6}*972c, {9,18}*972c, {9,18}*972d, {9,18}*972e, {27,18}*972, {27,6}*972a, {9,6}*972d, {9,18}*972f, {9,18}*972g, {9,18}*972h, {9,18}*972i, {9,6}*972e, {9,18}*972j, {27,6}*972b, {27,6}*972c, {81,6}*972, {3,6}*972, {3,18}*972b
28-fold covers : {6,84}*1008a, {12,42}*1008a, {84,6}*1008c, {42,12}*1008c, {21,12}*1008, {21,6}*1008b
29-fold covers : {87,6}*1044
30-fold covers : {18,30}*1080a, {6,30}*1080a, {90,6}*1080b, {30,6}*1080b, {6,30}*1080d, {30,6}*1080d
31-fold covers : {93,6}*1116
32-fold covers : {24,12}*1152a, {12,24}*1152c, {24,24}*1152c, {24,24}*1152d, {24,24}*1152e, {24,24}*1152l, {48,12}*1152a, {12,48}*1152c, {48,12}*1152d, {12,48}*1152f, {12,12}*1152a, {12,24}*1152d, {24,12}*1152f, {6,96}*1152a, {96,6}*1152b, {3,12}*1152a, {3,24}*1152a, {6,24}*1152a, {6,6}*1152a, {6,24}*1152b, {12,24}*1152j, {12,12}*1152e, {12,24}*1152l, {12,12}*1152g, {12,24}*1152m, {12,6}*1152a, {12,24}*1152n, {6,6}*1152d, {6,12}*1152c, {6,6}*1152f, {6,24}*1152f, {24,12}*1152p, {24,12}*1152r, {24,6}*1152g, {24,6}*1152i, {24,12}*1152s, {24,12}*1152t, {12,12}*1152l, {12,12}*1152m, {6,24}*1152j, {6,24}*1152k, {6,12}*1152e, {6,24}*1152l, {12,12}*1152q, {12,12}*1152s, {6,12}*1152f, {6,24}*1152m, {3,12}*1152b, {3,24}*1152b, {6,12}*1152g, {3,24}*1152c, {12,6}*1152h, {12,6}*1152i, {6,12}*1152j, {6,6}*1152i
33-fold covers : {99,6}*1188, {33,6}*1188
34-fold covers : {6,102}*1224a, {102,6}*1224c
35-fold covers : {105,6}*1260
36-fold covers : {36,18}*1296b, {36,6}*1296a, {108,6}*1296b, {36,6}*1296c, {36,6}*1296d, {36,6}*1296e, {12,18}*1296d, {12,6}*1296c, {18,36}*1296c, {18,12}*1296e, {54,12}*1296b, {18,12}*1296f, {18,12}*1296g, {18,12}*1296h, {6,12}*1296d, {6,36}*1296h, {27,6}*1296, {27,12}*1296, {9,18}*1296a, {9,36}*1296, {9,6}*1296a, {3,6}*1296, {3,36}*1296, {9,6}*1296b, {3,12}*1296a, {3,18}*1296a, {9,12}*1296a, {9,6}*1296c, {9,12}*1296b, {9,12}*1296c, {9,6}*1296d, {9,12}*1296d, {6,36}*1296l, {36,6}*1296l, {12,18}*1296l, {18,12}*1296l, {6,12}*1296g, {6,12}*1296h, {12,6}*1296g, {12,6}*1296h, {6,12}*1296i, {12,6}*1296i, {6,12}*1296s, {12,6}*1296u, {12,12}*1296h
37-fold covers : {111,6}*1332
38-fold covers : {6,114}*1368a, {114,6}*1368c
39-fold covers : {117,6}*1404, {39,6}*1404
40-fold covers : {6,120}*1440a, {24,30}*1440a, {12,60}*1440a, {120,6}*1440c, {60,12}*1440c, {30,24}*1440c, {15,24}*1440, {15,12}*1440c, {6,30}*1440g, {6,60}*1440c, {30,12}*1440b, {30,6}*1440h
41-fold covers : {123,6}*1476
42-fold covers : {18,42}*1512a, {6,42}*1512a, {126,6}*1512b, {42,6}*1512b, {6,42}*1512d, {42,6}*1512d
43-fold covers : {129,6}*1548
44-fold covers : {6,132}*1584a, {12,66}*1584a, {132,6}*1584c, {66,12}*1584c, {33,12}*1584, {33,6}*1584
45-fold covers : {45,18}*1620, {45,6}*1620a, {135,6}*1620, {45,6}*1620b, {45,6}*1620c, {45,6}*1620d, {15,6}*1620, {15,18}*1620
46-fold covers : {6,138}*1656a, {138,6}*1656c
47-fold covers : {141,6}*1692
48-fold covers : {144,6}*1728b, {48,6}*1728a, {36,24}*1728a, {12,24}*1728a, {36,12}*1728b, {12,12}*1728a, {36,24}*1728b, {12,24}*1728b, {72,12}*1728b, {24,12}*1728c, {72,12}*1728d, {24,12}*1728e, {18,48}*1728b, {6,48}*1728c, {9,6}*1728, {9,24}*1728, {3,6}*1728, {3,24}*1728, {6,48}*1728f, {48,6}*1728f, {12,24}*1728o, {24,12}*1728o, {12,24}*1728p, {24,12}*1728p, {12,12}*1728h, {36,6}*1728a, {18,12}*1728a, {18,6}*1728a, {36,6}*1728c, {18,12}*1728b, {36,12}*1728f, {36,12}*1728g, {12,12}*1728i, {12,6}*1728a, {12,12}*1728m, {18,24}*1728b, {18,24}*1728d, {6,12}*1728c, {6,24}*1728b, {6,6}*1728b, {6,24}*1728d, {12,6}*1728d, {18,12}*1728d, {6,12}*1728e, {6,12}*1728f, {9,12}*1728, {3,12}*1728, {18,6}*1728b, {6,6}*1728c, {6,12}*1728g, {6,24}*1728f, {12,6}*1728g, {24,6}*1728f, {6,6}*1728f, {6,24}*1728g, {24,6}*1728g, {12,12}*1728v, {12,12}*1728w, {6,12}*1728h, {6,12}*1728i, {12,6}*1728h, {12,6}*1728i, {12,12}*1728x, {12,12}*1728y
49-fold covers : {147,6}*1764, {3,6}*1764, {3,42}*1764, {21,42}*1764
50-fold covers : {6,150}*1800a, {150,6}*1800c, {6,6}*1800b, {6,30}*1800b, {6,6}*1800d, {6,30}*1800d, {30,30}*1800a, {30,30}*1800b, {30,30}*1800i
51-fold covers : {153,6}*1836, {51,6}*1836
52-fold covers : {6,156}*1872a, {12,78}*1872a, {156,6}*1872c, {78,12}*1872c, {39,12}*1872, {39,6}*1872
53-fold covers : {159,6}*1908
54-fold covers : {18,18}*1944a, {18,6}*1944a, {6,18}*1944b, {18,6}*1944d, {18,18}*1944f, {18,6}*1944f, {18,18}*1944h, {18,18}*1944l, {18,18}*1944o, {54,18}*1944b, {54,6}*1944a, {18,6}*1944h, {18,18}*1944q, {18,18}*1944t, {18,18}*1944u, {18,18}*1944y, {18,6}*1944i, {18,18}*1944ab, {54,6}*1944c, {54,6}*1944e, {162,6}*1944b, {6,6}*1944b, {6,18}*1944k, {18,18}*1944ad, {18,18}*1944ae, {18,18}*1944af, {6,18}*1944m, {6,18}*1944n, {18,6}*1944m, {18,6}*1944n, {6,18}*1944o, {18,6}*1944o, {6,6}*1944d, {6,6}*1944e, {6,6}*1944f, {6,54}*1944g, {54,6}*1944g, {6,6}*1944g, {6,6}*1944h, {6,18}*1944p, {6,18}*1944q, {18,6}*1944p, {18,6}*1944q, {6,18}*1944r, {6,18}*1944s, {18,6}*1944r, {18,6}*1944s, {6,6}*1944i, {6,6}*1944j, {6,18}*1944t, {6,18}*1944u, {18,6}*1944t, {18,6}*1944u
55-fold covers : {165,6}*1980
Permutation Representation (GAP) :
s0 := (2,3)(4,5)(6,9)(7,8);;
s1 := (1,6)(2,4)(3,8)(5,7);;
s2 := (4,5)(6,7)(8,9);;
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,
s2*s0*s1*s2*s1*s2*s0*s1*s2*s1 ];;
poly := F / rels;;

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

References : None.
to this polytope