Polytope of Type {3}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {3}*6
Also Known As : triangle, 2-simplex, {3}. if this polytope has another name.
Group : SmallGroup(6,1)
Rank : 2
Schlafli Type : {3}
Number of vertices, edges, etc : 3, 3
Order of s0s1 : 3
Special Properties :
Universal
Spherical
Locally Spherical
Orientable
Self-Dual
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{3,2} of size 12
{3,3} of size 24
{3,4} of size 24
{3,6} of size 36
{3,4} of size 48
{3,6} of size 48
{3,5} of size 60
{3,8} of size 96
{3,12} of size 96
{3,6} of size 108
{3,5} of size 120
{3,10} of size 120
{3,10} of size 120
{3,6} of size 144
{3,12} of size 144
{3,6} of size 192
{3,8} of size 192
{3,10} of size 240
{3,12} of size 288
{3,24} of size 288
{3,6} of size 300
{3,10} of size 300
{3,6} of size 324
{3,18} of size 324
{3,7} of size 336
{3,8} of size 336
{3,8} of size 336
{3,10} of size 360
{3,15} of size 360
{3,8} of size 384
{3,12} of size 384
{3,6} of size 432
{3,12} of size 432
{3,20} of size 480
{3,7} of size 504
{3,9} of size 504
{3,6} of size 576
{3,24} of size 576
{3,12} of size 576
{3,6} of size 588
{3,14} of size 588
{3,9} of size 648
{3,12} of size 648
{3,8} of size 672
{3,8} of size 672
{3,14} of size 672
{3,8} of size 720
{3,10} of size 720
{3,10} of size 720
{3,30} of size 720
{3,16} of size 768
{3,24} of size 768
{3,6} of size 768
{3,16} of size 768
{3,12} of size 864
{3,24} of size 864
{3,6} of size 900
{3,30} of size 900
{3,18} of size 972
{3,6} of size 972
{3,18} of size 972
{3,7} of size 1008
{3,9} of size 1008
{3,14} of size 1008
{3,14} of size 1008
{3,18} of size 1008
{3,18} of size 1008
{3,8} of size 1008
{3,21} of size 1008
{3,7} of size 1092
{3,13} of size 1092
{3,12} of size 1152
{3,24} of size 1152
{3,12} of size 1152
{3,24} of size 1152
{3,24} of size 1152
{3,6} of size 1200
{3,20} of size 1200
{3,6} of size 1296
{3,36} of size 1296
{3,12} of size 1296
{3,18} of size 1296
{3,12} of size 1296
{3,18} of size 1296
{3,10} of size 1320
{3,10} of size 1320
{3,11} of size 1320
{3,12} of size 1320
{3,12} of size 1320
{3,16} of size 1344
{3,16} of size 1344
{3,28} of size 1344
{3,20} of size 1440
{3,60} of size 1440
{3,8} of size 1440
{3,10} of size 1440
{3,15} of size 1440
{3,20} of size 1440
{3,6} of size 1452
{3,22} of size 1452
{3,10} of size 1500
{3,30} of size 1500
{3,6} of size 1728
{3,24} of size 1728
{3,12} of size 1728
{3,6} of size 1764
{3,42} of size 1764
Vertex Figure Of :
{2,3} of size 12
{3,3} of size 24
{4,3} of size 24
{6,3} of size 36
{4,3} of size 48
{6,3} of size 48
{5,3} of size 60
{8,3} of size 96
{12,3} of size 96
{6,3} of size 108
{5,3} of size 120
{10,3} of size 120
{10,3} of size 120
{6,3} of size 144
{12,3} of size 144
{6,3} of size 192
{8,3} of size 192
{10,3} of size 240
{12,3} of size 288
{24,3} of size 288
{6,3} of size 300
{10,3} of size 300
{6,3} of size 324
{18,3} of size 324
{7,3} of size 336
{8,3} of size 336
{8,3} of size 336
{10,3} of size 360
{15,3} of size 360
{8,3} of size 384
{12,3} of size 384
{6,3} of size 432
{12,3} of size 432
{20,3} of size 480
{7,3} of size 504
{9,3} of size 504
{6,3} of size 576
{24,3} of size 576
{12,3} of size 576
{6,3} of size 588
{14,3} of size 588
{9,3} of size 648
{12,3} of size 648
{8,3} of size 672
{8,3} of size 672
{14,3} of size 672
{8,3} of size 720
{10,3} of size 720
{10,3} of size 720
{30,3} of size 720
{16,3} of size 768
{24,3} of size 768
{6,3} of size 768
{16,3} of size 768
{12,3} of size 864
{24,3} of size 864
{6,3} of size 900
{30,3} of size 900
{18,3} of size 972
{6,3} of size 972
{18,3} of size 972
{7,3} of size 1008
{9,3} of size 1008
{14,3} of size 1008
{14,3} of size 1008
{18,3} of size 1008
{18,3} of size 1008
{8,3} of size 1008
{21,3} of size 1008
{7,3} of size 1092
{13,3} of size 1092
{12,3} of size 1152
{24,3} of size 1152
{12,3} of size 1152
{24,3} of size 1152
{24,3} of size 1152
{6,3} of size 1200
{20,3} of size 1200
{6,3} of size 1296
{36,3} of size 1296
{12,3} of size 1296
{18,3} of size 1296
{12,3} of size 1296
{18,3} of size 1296
{10,3} of size 1320
{10,3} of size 1320
{11,3} of size 1320
{12,3} of size 1320
{12,3} of size 1320
{16,3} of size 1344
{16,3} of size 1344
{28,3} of size 1344
{20,3} of size 1440
{60,3} of size 1440
{8,3} of size 1440
{10,3} of size 1440
{15,3} of size 1440
{20,3} of size 1440
{6,3} of size 1452
{22,3} of size 1452
{10,3} of size 1500
{30,3} of size 1500
{6,3} of size 1728
{24,3} of size 1728
{12,3} of size 1728
{6,3} of size 1764
{42,3} of size 1764
Quotients (Maximal Quotients in Boldface) :
No Regular Quotients.
Covers (Minimal Covers in Boldface) :
2-fold covers : {6}*12
3-fold covers : {9}*18
4-fold covers : {12}*24
5-fold covers : {15}*30
6-fold covers : {18}*36
7-fold covers : {21}*42
8-fold covers : {24}*48
9-fold covers : {27}*54
10-fold covers : {30}*60
11-fold covers : {33}*66
12-fold covers : {36}*72
13-fold covers : {39}*78
14-fold covers : {42}*84
15-fold covers : {45}*90
16-fold covers : {48}*96
17-fold covers : {51}*102
18-fold covers : {54}*108
19-fold covers : {57}*114
20-fold covers : {60}*120
21-fold covers : {63}*126
22-fold covers : {66}*132
23-fold covers : {69}*138
24-fold covers : {72}*144
25-fold covers : {75}*150
26-fold covers : {78}*156
27-fold covers : {81}*162
28-fold covers : {84}*168
29-fold covers : {87}*174
30-fold covers : {90}*180
31-fold covers : {93}*186
32-fold covers : {96}*192
33-fold covers : {99}*198
34-fold covers : {102}*204
35-fold covers : {105}*210
36-fold covers : {108}*216
37-fold covers : {111}*222
38-fold covers : {114}*228
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40-fold covers : {120}*240
41-fold covers : {123}*246
42-fold covers : {126}*252
43-fold covers : {129}*258
44-fold covers : {132}*264
45-fold covers : {135}*270
46-fold covers : {138}*276
47-fold covers : {141}*282
48-fold covers : {144}*288
49-fold covers : {147}*294
50-fold covers : {150}*300
51-fold covers : {153}*306
52-fold covers : {156}*312
53-fold covers : {159}*318
54-fold covers : {162}*324
55-fold covers : {165}*330
56-fold covers : {168}*336
57-fold covers : {171}*342
58-fold covers : {174}*348
59-fold covers : {177}*354
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61-fold covers : {183}*366
62-fold covers : {186}*372
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66-fold covers : {198}*396
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106-fold covers : {318}*636
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109-fold covers : {327}*654
110-fold covers : {330}*660
111-fold covers : {333}*666
112-fold covers : {336}*672
113-fold covers : {339}*678
114-fold covers : {342}*684
115-fold covers : {345}*690
116-fold covers : {348}*696
117-fold covers : {351}*702
118-fold covers : {354}*708
119-fold covers : {357}*714
120-fold covers : {360}*720
121-fold covers : {363}*726
122-fold covers : {366}*732
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126-fold covers : {378}*756
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130-fold covers : {390}*780
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132-fold covers : {396}*792
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134-fold covers : {402}*804
135-fold covers : {405}*810
136-fold covers : {408}*816
137-fold covers : {411}*822
138-fold covers : {414}*828
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141-fold covers : {423}*846
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152-fold covers : {456}*912
153-fold covers : {459}*918
154-fold covers : {462}*924
155-fold covers : {465}*930
156-fold covers : {468}*936
157-fold covers : {471}*942
158-fold covers : {474}*948
159-fold covers : {477}*954
160-fold covers : {480}*960
161-fold covers : {483}*966
162-fold covers : {486}*972
163-fold covers : {489}*978
164-fold covers : {492}*984
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321-fold covers : {963}*1926
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323-fold covers : {969}*1938
324-fold covers : {972}*1944
325-fold covers : {975}*1950
326-fold covers : {978}*1956
327-fold covers : {981}*1962
328-fold covers : {984}*1968
329-fold covers : {987}*1974
330-fold covers : {990}*1980
331-fold covers : {993}*1986
332-fold covers : {996}*1992
333-fold covers : {999}*1998
Permutation Representation (GAP) :
s0 := (2,3);;
s1 := (1,2);;
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 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(3)!(2,3);
s1 := Sym(3)!(1,2);
poly := sub<Sym(3)|s0,s1>;
Finitely Presented Group Representation (Magma) :
poly<s0,s1> := Group< s0,s1 | s0*s0, s1*s1, s0*s1*s0*s1*s0*s1 >;
References : None.
to this polytope