Polytope of Type {72}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {72}*144
Also Known As : 72-gon, {72}. if this polytope has another name.
Group : SmallGroup(144,8)
Rank : 2
Schlafli Type : {72}
Number of vertices, edges, etc : 72, 72
Order of s0s1 : 72
Special Properties :
Universal
Spherical
Locally Spherical
Orientable
Self-Dual
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{72,2} of size 288
{72,4} of size 576
{72,4} of size 576
{72,4} of size 576
{72,4} of size 576
{72,6} of size 864
{72,6} of size 864
{72,4} of size 1152
{72,8} of size 1152
{72,8} of size 1152
{72,8} of size 1152
{72,8} of size 1152
{72,4} of size 1152
{72,4} of size 1152
{72,4} of size 1152
{72,4} of size 1152
{72,4} of size 1152
{72,10} of size 1440
{72,12} of size 1728
{72,12} of size 1728
{72,12} of size 1728
{72,12} of size 1728
{72,6} of size 1728
{72,6} of size 1728
{72,6} of size 1728
Vertex Figure Of :
{2,72} of size 288
{4,72} of size 576
{4,72} of size 576
{4,72} of size 576
{4,72} of size 576
{6,72} of size 864
{6,72} of size 864
{4,72} of size 1152
{8,72} of size 1152
{8,72} of size 1152
{8,72} of size 1152
{8,72} of size 1152
{4,72} of size 1152
{4,72} of size 1152
{4,72} of size 1152
{4,72} of size 1152
{4,72} of size 1152
{10,72} of size 1440
{12,72} of size 1728
{12,72} of size 1728
{12,72} of size 1728
{12,72} of size 1728
{6,72} of size 1728
{6,72} of size 1728
{6,72} of size 1728
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {36}*72
3-fold quotients : {24}*48
4-fold quotients : {18}*36
6-fold quotients : {12}*24
8-fold quotients : {9}*18
9-fold quotients : {8}*16
12-fold quotients : {6}*12
18-fold quotients : {4}*8
24-fold quotients : {3}*6
36-fold quotients : {2}*4
Covers (Minimal Covers in Boldface) :
2-fold covers : {144}*288
3-fold covers : {216}*432
4-fold covers : {288}*576
5-fold covers : {360}*720
6-fold covers : {432}*864
7-fold covers : {504}*1008
8-fold covers : {576}*1152
9-fold covers : {648}*1296
10-fold covers : {720}*1440
11-fold covers : {792}*1584
12-fold covers : {864}*1728
13-fold covers : {936}*1872
Permutation Representation (GAP) :
s0 := ( 2, 3)( 4, 8)( 5, 7)( 6, 9)(11,12)(13,17)(14,16)(15,18)(19,28)(20,30)
(21,29)(22,35)(23,34)(24,36)(25,32)(26,31)(27,33)(37,55)(38,57)(39,56)(40,62)
(41,61)(42,63)(43,59)(44,58)(45,60)(46,64)(47,66)(48,65)(49,71)(50,70)(51,72)
(52,68)(53,67)(54,69);;
s1 := ( 1,40)( 2,42)( 3,41)( 4,37)( 5,39)( 6,38)( 7,44)( 8,43)( 9,45)(10,49)
(11,51)(12,50)(13,46)(14,48)(15,47)(16,53)(17,52)(18,54)(19,67)(20,69)(21,68)
(22,64)(23,66)(24,65)(25,71)(26,70)(27,72)(28,58)(29,60)(30,59)(31,55)(32,57)
(33,56)(34,62)(35,61)(36,63);;
poly := Group([s0,s1]);;
Finitely Presented Group Representation (GAP) :
F := FreeGroup("s0","s1");;
s0 := F.1;; s1 := F.2;;
rels := [ s0*s0, s1*s1, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(72)!( 2, 3)( 4, 8)( 5, 7)( 6, 9)(11,12)(13,17)(14,16)(15,18)(19,28)
(20,30)(21,29)(22,35)(23,34)(24,36)(25,32)(26,31)(27,33)(37,55)(38,57)(39,56)
(40,62)(41,61)(42,63)(43,59)(44,58)(45,60)(46,64)(47,66)(48,65)(49,71)(50,70)
(51,72)(52,68)(53,67)(54,69);
s1 := Sym(72)!( 1,40)( 2,42)( 3,41)( 4,37)( 5,39)( 6,38)( 7,44)( 8,43)( 9,45)
(10,49)(11,51)(12,50)(13,46)(14,48)(15,47)(16,53)(17,52)(18,54)(19,67)(20,69)
(21,68)(22,64)(23,66)(24,65)(25,71)(26,70)(27,72)(28,58)(29,60)(30,59)(31,55)
(32,57)(33,56)(34,62)(35,61)(36,63);
poly := sub<Sym(72)|s0,s1>;
Finitely Presented Group Representation (Magma) :
poly<s0,s1> := Group< s0,s1 | s0*s0, s1*s1, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 >;
References : None.
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