Polytope of Type {6,6}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,6}*72b
if this polytope has a name.
Group : SmallGroup(72,46)
Rank : 3
Schlafli Type : {6,6}
Number of vertices, edges, etc : 6, 18, 6
Order of s0s1s2 : 6
Order of s0s1s2s1 : 6
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Flat
Self-Petrie
Related Polytopes :
Facet
Vertex Figure
Dual
Petrial
Facet Of :
{6,6,2} of size 144
{6,6,4} of size 288
{6,6,4} of size 288
{6,6,4} of size 288
{6,6,6} of size 432
{6,6,6} of size 432
{6,6,8} of size 576
{6,6,4} of size 576
{6,6,6} of size 648
{6,6,10} of size 720
{6,6,12} of size 864
{6,6,12} of size 864
{6,6,12} of size 864
{6,6,14} of size 1008
{6,6,16} of size 1152
{6,6,4} of size 1152
{6,6,8} of size 1152
{6,6,4} of size 1152
{6,6,8} of size 1152
{6,6,8} of size 1152
{6,6,18} of size 1296
{6,6,6} of size 1296
{6,6,6} of size 1296
{6,6,6} of size 1296
{6,6,20} of size 1440
{6,6,20} of size 1440
{6,6,22} of size 1584
{6,6,24} of size 1728
{6,6,24} of size 1728
{6,6,6} of size 1728
{6,6,12} of size 1728
{6,6,12} of size 1728
{6,6,10} of size 1800
{6,6,26} of size 1872
{6,6,18} of size 1944
{6,6,6} of size 1944
{6,6,6} of size 1944
{6,6,18} of size 1944
{6,6,18} of size 1944
Vertex Figure Of :
{2,6,6} of size 144
{3,6,6} of size 216
{4,6,6} of size 288
{6,6,6} of size 432
{6,6,6} of size 432
{8,6,6} of size 576
{9,6,6} of size 648
{3,6,6} of size 648
{10,6,6} of size 720
{12,6,6} of size 864
{12,6,6} of size 864
{4,6,6} of size 864
{14,6,6} of size 1008
{15,6,6} of size 1080
{16,6,6} of size 1152
{4,6,6} of size 1152
{18,6,6} of size 1296
{6,6,6} of size 1296
{18,6,6} of size 1296
{6,6,6} of size 1296
{6,6,6} of size 1296
{20,6,6} of size 1440
{21,6,6} of size 1512
{22,6,6} of size 1584
{24,6,6} of size 1728
{24,6,6} of size 1728
{8,6,6} of size 1728
{26,6,6} of size 1872
{27,6,6} of size 1944
{9,6,6} of size 1944
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {6,3}*36
3-fold quotients : {2,6}*24
6-fold quotients : {2,3}*12
9-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
2-fold covers : {6,12}*144b, {12,6}*144c
3-fold covers : {6,18}*216b, {6,6}*216a, {6,6}*216d
4-fold covers : {6,24}*288b, {12,12}*288b, {24,6}*288c, {6,6}*288a, {12,6}*288b
5-fold covers : {30,6}*360a, {6,30}*360c
6-fold covers : {6,36}*432b, {6,12}*432a, {12,18}*432b, {12,6}*432c, {6,12}*432g, {12,6}*432g
7-fold covers : {42,6}*504a, {6,42}*504c
8-fold covers : {6,48}*576b, {24,12}*576a, {12,12}*576b, {24,12}*576b, {12,24}*576d, {12,24}*576f, {48,6}*576c, {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
9-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
10-fold covers : {60,6}*720a, {30,12}*720a, {6,60}*720c, {12,30}*720c
11-fold covers : {66,6}*792a, {6,66}*792c
12-fold covers : {6,72}*864b, {6,24}*864a, {12,36}*864b, {12,12}*864b, {24,18}*864b, {24,6}*864c, {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
13-fold covers : {78,6}*936a, {6,78}*936c
14-fold covers : {84,6}*1008a, {42,12}*1008a, {6,84}*1008c, {12,42}*1008c
15-fold covers : {30,18}*1080a, {30,6}*1080a, {6,90}*1080b, {6,30}*1080b, {6,30}*1080d, {30,6}*1080d
16-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, {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, {12,6}*1152j, {6,6}*1152k
17-fold covers : {102,6}*1224a, {6,102}*1224c
18-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, {6,36}*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
19-fold covers : {114,6}*1368a, {6,114}*1368c
20-fold covers : {120,6}*1440a, {30,24}*1440a, {60,12}*1440a, {6,120}*1440c, {12,60}*1440c, {24,30}*1440c, {30,6}*1440g, {60,6}*1440c, {12,30}*1440b, {6,30}*1440h
21-fold covers : {42,18}*1512a, {42,6}*1512a, {6,126}*1512b, {6,42}*1512b, {6,42}*1512d, {42,6}*1512d
22-fold covers : {132,6}*1584a, {66,12}*1584a, {6,132}*1584c, {12,66}*1584c
23-fold covers : {138,6}*1656a, {6,138}*1656c
24-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, {6,48}*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, {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
25-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
26-fold covers : {156,6}*1872a, {78,12}*1872a, {6,156}*1872c, {12,78}*1872c
27-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
Permutation Representation (GAP) :
s0 := ( 5, 6)( 9,10)(11,12)(13,14)(15,16)(17,18);;
s1 := ( 1, 5)( 2, 9)( 3,13)( 4,11)( 7,17)( 8,15)(12,14)(16,18);;
s2 := ( 1, 7)( 2, 3)( 4, 8)( 5,16)( 6,15)( 9,12)(10,11)(13,18)(14,17);;
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, s2*s0*s1*s0*s1*s2*s0*s1*s0*s1,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(18)!( 5, 6)( 9,10)(11,12)(13,14)(15,16)(17,18);
s1 := Sym(18)!( 1, 5)( 2, 9)( 3,13)( 4,11)( 7,17)( 8,15)(12,14)(16,18);
s2 := Sym(18)!( 1, 7)( 2, 3)( 4, 8)( 5,16)( 6,15)( 9,12)(10,11)(13,18)(14,17);
poly := sub<Sym(18)|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, s2*s0*s1*s0*s1*s2*s0*s1*s0*s1,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 >;
References : None.
to this polytope