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