Polytope of Type {6,3}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,3}*36
Also Known As : {6,3}(1,1). if this polytope has another name.
Group : SmallGroup(36,10)
Rank : 3
Schlafli Type : {6,3}
Number of vertices, edges, etc : 6, 9, 3
Order of s0s1s2 : 6
Order of s0s1s2s1 : 6
Special Properties :
Toroidal
Locally Spherical
Orientable
Flat
Self-Petrie
Related Polytopes :
Facet
Vertex Figure
Dual
Petrial
Facet Of :
{6,3,2} of size 72
{6,3,4} of size 144
{6,3,6} of size 216
{6,3,4} of size 288
{6,3,8} of size 576
{6,3,6} of size 648
{6,3,6} of size 864
{6,3,12} of size 864
{6,3,8} of size 1152
{6,3,12} of size 1728
{6,3,24} of size 1728
{6,3,10} of size 1800
{6,3,6} of size 1944
{6,3,18} of size 1944
Vertex Figure Of :
{2,6,3} of size 72
{3,6,3} of size 108
{4,6,3} of size 144
{6,6,3} of size 216
{6,6,3} of size 216
{8,6,3} of size 288
{9,6,3} of size 324
{3,6,3} of size 324
{10,6,3} of size 360
{12,6,3} of size 432
{12,6,3} of size 432
{4,6,3} of size 432
{14,6,3} of size 504
{15,6,3} of size 540
{16,6,3} of size 576
{4,6,3} of size 576
{18,6,3} of size 648
{6,6,3} of size 648
{18,6,3} of size 648
{6,6,3} of size 648
{6,6,3} of size 648
{20,6,3} of size 720
{21,6,3} of size 756
{22,6,3} of size 792
{24,6,3} of size 864
{24,6,3} of size 864
{8,6,3} of size 864
{26,6,3} of size 936
{27,6,3} of size 972
{9,6,3} of size 972
{28,6,3} of size 1008
{30,6,3} of size 1080
{30,6,3} of size 1080
{32,6,3} of size 1152
{4,6,3} of size 1152
{33,6,3} of size 1188
{34,6,3} of size 1224
{36,6,3} of size 1296
{12,6,3} of size 1296
{36,6,3} of size 1296
{12,6,3} of size 1296
{12,6,3} of size 1296
{4,6,3} of size 1296
{12,6,3} of size 1296
{38,6,3} of size 1368
{39,6,3} of size 1404
{40,6,3} of size 1440
{42,6,3} of size 1512
{42,6,3} of size 1512
{44,6,3} of size 1584
{45,6,3} of size 1620
{15,6,3} of size 1620
{46,6,3} of size 1656
{48,6,3} of size 1728
{48,6,3} of size 1728
{16,6,3} of size 1728
{50,6,3} of size 1800
{51,6,3} of size 1836
{52,6,3} of size 1872
{54,6,3} of size 1944
{18,6,3} of size 1944
{18,6,3} of size 1944
{18,6,3} of size 1944
{6,6,3} of size 1944
{54,6,3} of size 1944
{6,6,3} of size 1944
{6,6,3} of size 1944
Quotients (Maximal Quotients in Boldface) :
3-fold quotients : {2,3}*12
Covers (Minimal Covers in Boldface) :
2-fold covers : {6,6}*72b
3-fold covers : {6,9}*108, {6,3}*108
4-fold covers : {6,12}*144b, {12,6}*144c, {6,3}*144, {12,3}*144
5-fold covers : {6,15}*180
6-fold covers : {6,18}*216b, {6,6}*216a, {6,6}*216d
7-fold covers : {6,21}*252
8-fold covers : {6,24}*288b, {12,12}*288b, {24,6}*288c, {12,3}*288, {24,3}*288, {6,6}*288a, {12,6}*288b
9-fold covers : {18,9}*324, {6,9}*324a, {6,27}*324, {6,9}*324b, {6,9}*324c, {6,9}*324d, {6,3}*324, {18,3}*324
10-fold covers : {30,6}*360a, {6,30}*360c
11-fold covers : {6,33}*396
12-fold covers : {6,36}*432b, {6,12}*432a, {12,18}*432b, {12,6}*432c, {6,9}*432, {12,9}*432, {6,3}*432, {12,3}*432, {6,12}*432g, {12,6}*432g
13-fold covers : {6,39}*468
14-fold covers : {42,6}*504a, {6,42}*504c
15-fold covers : {6,45}*540, {6,15}*540
16-fold covers : {6,48}*576b, {24,12}*576a, {12,12}*576b, {24,12}*576b, {12,24}*576d, {12,24}*576f, {48,6}*576c, {6,3}*576, {24,3}*576, {12,12}*576g, {6,12}*576a, {12,12}*576i, {12,6}*576c, {24,6}*576b, {6,6}*576a, {24,6}*576d, {6,12}*576d, {12,6}*576e, {12,6}*576f, {12,3}*576, {6,6}*576d
17-fold covers : {6,51}*612
18-fold covers : {18,18}*648b, {6,18}*648a, {6,54}*648b, {6,18}*648c, {6,18}*648d, {6,18}*648e, {6,6}*648c, {18,6}*648h, {6,18}*648i, {18,6}*648i, {6,6}*648e, {6,6}*648f, {6,6}*648g
19-fold covers : {6,57}*684
20-fold covers : {60,6}*720a, {30,12}*720a, {6,60}*720c, {12,30}*720c, {12,15}*720, {6,15}*720e
21-fold covers : {6,63}*756, {6,21}*756
22-fold covers : {66,6}*792a, {6,66}*792c
23-fold covers : {6,69}*828
24-fold covers : {6,72}*864b, {6,24}*864a, {12,36}*864b, {12,12}*864b, {24,18}*864b, {24,6}*864c, {12,9}*864, {24,9}*864, {12,3}*864, {24,3}*864, {6,24}*864f, {24,6}*864f, {12,12}*864h, {6,18}*864, {12,18}*864b, {6,6}*864b, {12,6}*864a, {6,6}*864c, {6,12}*864c, {12,6}*864c
25-fold covers : {6,75}*900, {6,3}*900, {30,3}*900, {30,15}*900
26-fold covers : {78,6}*936a, {6,78}*936c
27-fold covers : {18,9}*972a, {18,3}*972a, {6,9}*972a, {6,9}*972b, {18,9}*972b, {6,9}*972c, {18,9}*972c, {18,9}*972d, {18,9}*972e, {18,27}*972, {6,27}*972a, {6,9}*972d, {18,9}*972f, {18,9}*972g, {18,9}*972h, {18,9}*972i, {6,9}*972e, {18,9}*972j, {6,27}*972b, {6,27}*972c, {6,81}*972, {6,3}*972, {18,3}*972b
28-fold covers : {84,6}*1008a, {42,12}*1008a, {6,84}*1008c, {12,42}*1008c, {12,21}*1008, {6,21}*1008b
29-fold covers : {6,87}*1044
30-fold covers : {30,18}*1080a, {30,6}*1080a, {6,90}*1080b, {6,30}*1080b, {6,30}*1080d, {30,6}*1080d
31-fold covers : {6,93}*1116
32-fold covers : {12,24}*1152a, {24,12}*1152c, {24,24}*1152a, {24,24}*1152f, {24,24}*1152h, {24,24}*1152j, {12,48}*1152a, {48,12}*1152c, {12,48}*1152d, {48,12}*1152f, {12,12}*1152b, {24,12}*1152d, {12,24}*1152f, {96,6}*1152a, {6,96}*1152b, {12,3}*1152a, {24,3}*1152a, {24,6}*1152a, {6,6}*1152b, {24,6}*1152c, {24,12}*1152j, {12,12}*1152d, {24,12}*1152l, {12,12}*1152f, {24,12}*1152m, {6,12}*1152a, {24,12}*1152n, {6,6}*1152c, {12,6}*1152c, {6,6}*1152e, {24,6}*1152f, {12,24}*1152p, {12,24}*1152r, {6,24}*1152g, {6,24}*1152i, {12,24}*1152s, {12,24}*1152t, {12,12}*1152j, {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, {12,3}*1152b, {12,6}*1152g, {24,3}*1152b, {24,3}*1152c, {6,12}*1152h, {6,12}*1152i, {12,6}*1152j, {6,6}*1152k
33-fold covers : {6,99}*1188, {6,33}*1188
34-fold covers : {102,6}*1224a, {6,102}*1224c
35-fold covers : {6,105}*1260
36-fold covers : {18,36}*1296b, {6,36}*1296a, {6,108}*1296b, {6,36}*1296c, {6,36}*1296d, {6,36}*1296e, {18,12}*1296d, {6,12}*1296c, {36,18}*1296c, {12,18}*1296e, {12,54}*1296b, {12,18}*1296f, {12,18}*1296g, {12,18}*1296h, {12,6}*1296d, {36,6}*1296h, {6,27}*1296, {12,27}*1296, {18,9}*1296a, {36,9}*1296, {6,9}*1296a, {6,3}*1296, {36,3}*1296, {6,9}*1296b, {12,3}*1296a, {18,3}*1296a, {12,9}*1296a, {6,9}*1296c, {12,9}*1296b, {12,9}*1296c, {6,9}*1296d, {12,9}*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, {12,6}*1296s, {6,12}*1296u, {12,12}*1296g
37-fold covers : {6,111}*1332
38-fold covers : {114,6}*1368a, {6,114}*1368c
39-fold covers : {6,117}*1404, {6,39}*1404
40-fold covers : {120,6}*1440a, {30,24}*1440a, {60,12}*1440a, {6,120}*1440c, {12,60}*1440c, {24,30}*1440c, {24,15}*1440, {12,15}*1440c, {30,6}*1440g, {60,6}*1440c, {12,30}*1440b, {6,30}*1440h
41-fold covers : {6,123}*1476
42-fold covers : {42,18}*1512a, {42,6}*1512a, {6,126}*1512b, {6,42}*1512b, {6,42}*1512d, {42,6}*1512d
43-fold covers : {6,129}*1548
44-fold covers : {132,6}*1584a, {66,12}*1584a, {6,132}*1584c, {12,66}*1584c, {12,33}*1584, {6,33}*1584
45-fold covers : {18,45}*1620, {6,45}*1620a, {6,135}*1620, {6,45}*1620b, {6,45}*1620c, {6,45}*1620d, {6,15}*1620, {18,15}*1620
46-fold covers : {138,6}*1656a, {6,138}*1656c
47-fold covers : {6,141}*1692
48-fold covers : {6,144}*1728b, {6,48}*1728a, {24,36}*1728a, {24,12}*1728a, {12,36}*1728b, {12,12}*1728b, {24,36}*1728b, {24,12}*1728b, {12,72}*1728b, {12,24}*1728c, {12,72}*1728d, {12,24}*1728e, {48,18}*1728b, {48,6}*1728c, {6,9}*1728, {24,9}*1728, {6,3}*1728, {24,3}*1728, {6,48}*1728f, {48,6}*1728f, {12,24}*1728o, {24,12}*1728o, {12,24}*1728p, {24,12}*1728p, {12,12}*1728h, {6,36}*1728a, {12,18}*1728a, {6,18}*1728a, {6,36}*1728c, {12,18}*1728b, {12,36}*1728f, {12,36}*1728g, {12,12}*1728k, {6,12}*1728a, {12,12}*1728n, {24,18}*1728b, {24,18}*1728d, {12,6}*1728c, {24,6}*1728b, {6,6}*1728a, {24,6}*1728d, {6,12}*1728d, {12,18}*1728d, {12,6}*1728e, {12,6}*1728f, {12,9}*1728, {12,3}*1728, {6,18}*1728c, {6,6}*1728e, {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 : {6,147}*1764, {6,3}*1764, {42,3}*1764, {42,21}*1764
50-fold covers : {150,6}*1800a, {6,150}*1800c, {6,6}*1800a, {30,6}*1800b, {6,6}*1800c, {30,6}*1800c, {30,30}*1800c, {30,30}*1800e, {30,30}*1800h
51-fold covers : {6,153}*1836, {6,51}*1836
52-fold covers : {156,6}*1872a, {78,12}*1872a, {6,156}*1872c, {12,78}*1872c, {12,39}*1872, {6,39}*1872
53-fold covers : {6,159}*1908
54-fold covers : {18,18}*1944b, {6,18}*1944a, {18,6}*1944b, {6,18}*1944d, {18,18}*1944e, {6,18}*1944f, {18,18}*1944g, {18,18}*1944j, {18,18}*1944n, {18,54}*1944b, {6,54}*1944a, {6,18}*1944h, {18,18}*1944p, {18,18}*1944r, {18,18}*1944w, {18,18}*1944aa, {6,18}*1944i, {18,18}*1944ac, {6,54}*1944c, {6,54}*1944e, {6,162}*1944b, {6,6}*1944c, {18,6}*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 : {6,165}*1980
Permutation Representation (GAP) :
s0 := (4,5)(6,7)(8,9);;
s1 := (1,4)(2,8)(3,6)(7,9);;
s2 := (1,2)(4,7)(5,6)(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, s1*s2*s1*s2*s1*s2,
s2*s0*s1*s0*s1*s2*s0*s1*s0*s1 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(9)!(4,5)(6,7)(8,9);
s1 := Sym(9)!(1,4)(2,8)(3,6)(7,9);
s2 := Sym(9)!(1,2)(4,7)(5,6)(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, s1*s2*s1*s2*s1*s2, s2*s0*s1*s0*s1*s2*s0*s1*s0*s1 >;
References : None.
to this polytope