Polytope of Type {18}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {18}*36
Also Known As : 18-gon, {18}. if this polytope has another name.
Group : SmallGroup(36,4)
Rank : 2
Schlafli Type : {18}
Number of vertices, edges, etc : 18, 18
Order of s0s1 : 18
Special Properties :
Universal
Spherical
Locally Spherical
Orientable
Self-Dual
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{18,2} of size 72
{18,4} of size 144
{18,4} of size 144
{18,4} of size 144
{18,6} of size 216
{18,6} of size 216
{18,8} of size 288
{18,4} of size 288
{18,9} of size 324
{18,6} of size 324
{18,6} of size 324
{18,3} of size 324
{18,6} of size 324
{18,10} of size 360
{18,12} of size 432
{18,12} of size 432
{18,12} of size 432
{18,14} of size 504
{18,16} of size 576
{18,4} of size 576
{18,8} of size 576
{18,4} of size 576
{18,8} of size 576
{18,8} of size 576
{18,4} of size 648
{18,18} of size 648
{18,18} of size 648
{18,18} of size 648
{18,6} of size 648
{18,6} of size 648
{18,6} of size 648
{18,6} of size 648
{18,6} of size 648
{18,6} of size 648
{18,6} of size 648
{18,6} of size 648
{18,6} of size 648
{18,20} of size 720
{18,20} of size 720
{18,22} of size 792
{18,24} of size 864
{18,24} of size 864
{18,6} of size 864
{18,12} of size 864
{18,12} of size 864
{18,10} of size 900
{18,26} of size 936
{18,9} of size 972
{18,18} of size 972
{18,3} of size 972
{18,18} of size 972
{18,6} of size 972
{18,9} of size 972
{18,6} of size 972
{18,9} of size 972
{18,9} of size 972
{18,18} of size 972
{18,18} of size 972
{18,9} of size 972
{18,18} of size 972
{18,27} of size 972
{18,6} of size 972
{18,9} of size 972
{18,9} of size 972
{18,9} of size 972
{18,18} of size 972
{18,18} of size 972
{18,9} of size 972
{18,18} of size 972
{18,18} of size 972
{18,6} of size 972
{18,9} of size 972
{18,3} of size 972
{18,6} of size 972
{18,28} of size 1008
{18,28} of size 1008
{18,3} of size 1008
{18,3} of size 1008
{18,7} of size 1008
{18,7} of size 1008
{18,7} of size 1008
{18,7} of size 1008
{18,7} of size 1008
{18,7} of size 1008
{18,9} of size 1008
{18,9} of size 1008
{18,9} of size 1008
{18,9} of size 1008
{18,30} of size 1080
{18,30} of size 1080
{18,32} of size 1152
{18,8} of size 1152
{18,8} of size 1152
{18,8} of size 1152
{18,4} of size 1152
{18,8} of size 1152
{18,8} of size 1152
{18,8} of size 1152
{18,8} of size 1152
{18,4} of size 1152
{18,34} of size 1224
{18,36} of size 1296
{18,36} of size 1296
{18,12} of size 1296
{18,12} of size 1296
{18,12} of size 1296
{18,12} of size 1296
{18,36} of size 1296
{18,12} of size 1296
{18,12} of size 1296
{18,12} of size 1296
{18,12} of size 1296
{18,9} of size 1296
{18,36} of size 1296
{18,3} of size 1296
{18,12} of size 1296
{18,12} of size 1296
{18,12} of size 1296
{18,4} of size 1296
{18,12} of size 1296
{18,4} of size 1296
{18,3} of size 1296
{18,4} of size 1296
{18,4} of size 1296
{18,9} of size 1296
{18,9} of size 1296
{18,12} of size 1296
{18,12} of size 1296
{18,12} of size 1296
{18,12} of size 1296
{18,38} of size 1368
{18,40} of size 1440
{18,20} of size 1440
{18,42} of size 1512
{18,42} of size 1512
{18,44} of size 1584
{18,44} of size 1584
{18,45} of size 1620
{18,30} of size 1620
{18,30} of size 1620
{18,15} of size 1620
{18,30} of size 1620
{18,46} of size 1656
{18,48} of size 1728
{18,48} of size 1728
{18,24} of size 1728
{18,12} of size 1728
{18,6} of size 1728
{18,12} of size 1728
{18,12} of size 1728
{18,24} of size 1728
{18,24} of size 1728
{18,24} of size 1728
{18,24} of size 1728
{18,12} of size 1728
{18,6} of size 1728
{18,6} of size 1728
{18,14} of size 1764
{18,50} of size 1800
{18,10} of size 1800
{18,10} of size 1800
{18,52} of size 1872
{18,52} of size 1872
{18,4} of size 1944
{18,12} of size 1944
{18,12} of size 1944
{18,4} of size 1944
{18,12} of size 1944
{18,12} of size 1944
{18,4} of size 1944
{18,12} of size 1944
{18,12} of size 1944
{18,18} of size 1944
{18,18} of size 1944
{18,18} of size 1944
{18,6} of size 1944
{18,6} of size 1944
{18,18} of size 1944
{18,6} of size 1944
{18,6} of size 1944
{18,18} of size 1944
{18,18} of size 1944
{18,6} of size 1944
{18,6} of size 1944
{18,18} of size 1944
{18,18} of size 1944
{18,18} of size 1944
{18,18} of size 1944
{18,18} of size 1944
{18,18} of size 1944
{18,18} of size 1944
{18,18} of size 1944
{18,18} of size 1944
{18,54} of size 1944
{18,54} of size 1944
{18,6} of size 1944
{18,6} of size 1944
{18,18} of size 1944
{18,18} of size 1944
{18,18} of size 1944
{18,18} of size 1944
{18,18} of size 1944
{18,18} of size 1944
{18,18} of size 1944
{18,18} of size 1944
{18,18} of size 1944
{18,18} of size 1944
{18,18} of size 1944
{18,18} of size 1944
{18,6} of size 1944
{18,6} of size 1944
{18,18} of size 1944
{18,18} of size 1944
{18,6} of size 1944
{18,6} of size 1944
{18,12} of size 1944
{18,12} of size 1944
{18,12} of size 1944
{18,18} of size 1944
{18,18} of size 1944
{18,18} of size 1944
{18,6} of size 1944
{18,6} of size 1944
{18,6} of size 1944
{18,6} of size 1944
{18,6} of size 1944
{18,6} of size 1944
{18,6} of size 1944
{18,6} of size 1944
{18,6} of size 1944
Vertex Figure Of :
{2,18} of size 72
{4,18} of size 144
{4,18} of size 144
{4,18} of size 144
{6,18} of size 216
{6,18} of size 216
{8,18} of size 288
{4,18} of size 288
{9,18} of size 324
{6,18} of size 324
{6,18} of size 324
{3,18} of size 324
{6,18} of size 324
{10,18} of size 360
{12,18} of size 432
{12,18} of size 432
{12,18} of size 432
{14,18} of size 504
{16,18} of size 576
{4,18} of size 576
{8,18} of size 576
{4,18} of size 576
{8,18} of size 576
{8,18} of size 576
{4,18} of size 648
{18,18} of size 648
{18,18} of size 648
{18,18} of size 648
{6,18} of size 648
{6,18} of size 648
{6,18} of size 648
{6,18} of size 648
{6,18} of size 648
{6,18} of size 648
{6,18} of size 648
{6,18} of size 648
{6,18} of size 648
{20,18} of size 720
{20,18} of size 720
{22,18} of size 792
{24,18} of size 864
{24,18} of size 864
{6,18} of size 864
{12,18} of size 864
{12,18} of size 864
{10,18} of size 900
{26,18} of size 936
{9,18} of size 972
{18,18} of size 972
{3,18} of size 972
{18,18} of size 972
{6,18} of size 972
{9,18} of size 972
{6,18} of size 972
{9,18} of size 972
{9,18} of size 972
{18,18} of size 972
{18,18} of size 972
{9,18} of size 972
{18,18} of size 972
{27,18} of size 972
{6,18} of size 972
{9,18} of size 972
{9,18} of size 972
{9,18} of size 972
{18,18} of size 972
{18,18} of size 972
{9,18} of size 972
{18,18} of size 972
{18,18} of size 972
{6,18} of size 972
{9,18} of size 972
{3,18} of size 972
{6,18} of size 972
{28,18} of size 1008
{28,18} of size 1008
{3,18} of size 1008
{3,18} of size 1008
{7,18} of size 1008
{7,18} of size 1008
{7,18} of size 1008
{7,18} of size 1008
{7,18} of size 1008
{7,18} of size 1008
{9,18} of size 1008
{9,18} of size 1008
{9,18} of size 1008
{9,18} of size 1008
{30,18} of size 1080
{30,18} of size 1080
{32,18} of size 1152
{8,18} of size 1152
{8,18} of size 1152
{8,18} of size 1152
{4,18} of size 1152
{8,18} of size 1152
{8,18} of size 1152
{8,18} of size 1152
{8,18} of size 1152
{4,18} of size 1152
{34,18} of size 1224
{36,18} of size 1296
{36,18} of size 1296
{12,18} of size 1296
{12,18} of size 1296
{12,18} of size 1296
{12,18} of size 1296
{36,18} of size 1296
{12,18} of size 1296
{12,18} of size 1296
{12,18} of size 1296
{12,18} of size 1296
{9,18} of size 1296
{36,18} of size 1296
{3,18} of size 1296
{12,18} of size 1296
{12,18} of size 1296
{12,18} of size 1296
{4,18} of size 1296
{12,18} of size 1296
{4,18} of size 1296
{3,18} of size 1296
{4,18} of size 1296
{4,18} of size 1296
{9,18} of size 1296
{9,18} of size 1296
{12,18} of size 1296
{12,18} of size 1296
{12,18} of size 1296
{12,18} of size 1296
{38,18} of size 1368
{40,18} of size 1440
{20,18} of size 1440
{42,18} of size 1512
{42,18} of size 1512
{44,18} of size 1584
{44,18} of size 1584
{45,18} of size 1620
{30,18} of size 1620
{30,18} of size 1620
{15,18} of size 1620
{30,18} of size 1620
{46,18} of size 1656
{48,18} of size 1728
{48,18} of size 1728
{24,18} of size 1728
{12,18} of size 1728
{6,18} of size 1728
{12,18} of size 1728
{12,18} of size 1728
{24,18} of size 1728
{24,18} of size 1728
{24,18} of size 1728
{24,18} of size 1728
{12,18} of size 1728
{6,18} of size 1728
{6,18} of size 1728
{14,18} of size 1764
{50,18} of size 1800
{10,18} of size 1800
{10,18} of size 1800
{52,18} of size 1872
{52,18} of size 1872
{4,18} of size 1944
{12,18} of size 1944
{12,18} of size 1944
{4,18} of size 1944
{12,18} of size 1944
{12,18} of size 1944
{4,18} of size 1944
{12,18} of size 1944
{12,18} of size 1944
{18,18} of size 1944
{18,18} of size 1944
{18,18} of size 1944
{6,18} of size 1944
{6,18} of size 1944
{18,18} of size 1944
{6,18} of size 1944
{6,18} of size 1944
{18,18} of size 1944
{18,18} of size 1944
{6,18} of size 1944
{6,18} of size 1944
{18,18} of size 1944
{18,18} of size 1944
{18,18} of size 1944
{18,18} of size 1944
{18,18} of size 1944
{18,18} of size 1944
{18,18} of size 1944
{18,18} of size 1944
{18,18} of size 1944
{54,18} of size 1944
{54,18} of size 1944
{6,18} of size 1944
{6,18} of size 1944
{18,18} of size 1944
{18,18} of size 1944
{18,18} of size 1944
{18,18} of size 1944
{18,18} of size 1944
{18,18} of size 1944
{18,18} of size 1944
{18,18} of size 1944
{18,18} of size 1944
{18,18} of size 1944
{18,18} of size 1944
{18,18} of size 1944
{6,18} of size 1944
{6,18} of size 1944
{18,18} of size 1944
{18,18} of size 1944
{6,18} of size 1944
{6,18} of size 1944
{12,18} of size 1944
{12,18} of size 1944
{12,18} of size 1944
{18,18} of size 1944
{18,18} of size 1944
{18,18} of size 1944
{6,18} of size 1944
{6,18} of size 1944
{6,18} of size 1944
{6,18} of size 1944
{6,18} of size 1944
{6,18} of size 1944
{6,18} of size 1944
{6,18} of size 1944
{6,18} of size 1944
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {9}*18
3-fold quotients : {6}*12
6-fold quotients : {3}*6
9-fold quotients : {2}*4
Covers (Minimal Covers in Boldface) :
2-fold covers : {36}*72
3-fold covers : {54}*108
4-fold covers : {72}*144
5-fold covers : {90}*180
6-fold covers : {108}*216
7-fold covers : {126}*252
8-fold covers : {144}*288
9-fold covers : {162}*324
10-fold covers : {180}*360
11-fold covers : {198}*396
12-fold covers : {216}*432
13-fold covers : {234}*468
14-fold covers : {252}*504
15-fold covers : {270}*540
16-fold covers : {288}*576
17-fold covers : {306}*612
18-fold covers : {324}*648
19-fold covers : {342}*684
20-fold covers : {360}*720
21-fold covers : {378}*756
22-fold covers : {396}*792
23-fold covers : {414}*828
24-fold covers : {432}*864
25-fold covers : {450}*900
26-fold covers : {468}*936
27-fold covers : {486}*972
28-fold covers : {504}*1008
29-fold covers : {522}*1044
30-fold covers : {540}*1080
31-fold covers : {558}*1116
32-fold covers : {576}*1152
33-fold covers : {594}*1188
34-fold covers : {612}*1224
35-fold covers : {630}*1260
36-fold covers : {648}*1296
37-fold covers : {666}*1332
38-fold covers : {684}*1368
39-fold covers : {702}*1404
40-fold covers : {720}*1440
41-fold covers : {738}*1476
42-fold covers : {756}*1512
43-fold covers : {774}*1548
44-fold covers : {792}*1584
45-fold covers : {810}*1620
46-fold covers : {828}*1656
47-fold covers : {846}*1692
48-fold covers : {864}*1728
49-fold covers : {882}*1764
50-fold covers : {900}*1800
51-fold covers : {918}*1836
52-fold covers : {936}*1872
53-fold covers : {954}*1908
54-fold covers : {972}*1944
55-fold covers : {990}*1980
Permutation Representation (GAP) :
s0 := ( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18);;
s1 := ( 1, 5)( 2, 3)( 4, 9)( 6, 7)( 8,13)(10,11)(12,17)(14,15)(16,18);;
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 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(18)!( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18);
s1 := Sym(18)!( 1, 5)( 2, 3)( 4, 9)( 6, 7)( 8,13)(10,11)(12,17)(14,15)(16,18);
poly := sub<Sym(18)|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 >;
References : None.
to this polytope