Polytope of Type {6}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6}*12
Also Known As : hexagon, {6}. if this polytope has another name.
Group : SmallGroup(12,4)
Rank : 2
Schlafli Type : {6}
Number of vertices, edges, etc : 6, 6
Order of s0s1 : 6
Special Properties :
Universal
Spherical
Locally Spherical
Orientable
Self-Dual
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{6,2} of size 24
{6,3} of size 36
{6,4} of size 48
{6,3} of size 48
{6,4} of size 48
{6,4} of size 48
{6,4} of size 72
{6,6} of size 72
{6,6} of size 72
{6,6} of size 72
{6,8} of size 96
{6,4} of size 96
{6,6} of size 96
{6,9} of size 108
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{6,4} of size 120
{6,5} of size 120
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{6,10} of size 120
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{6,3} of size 144
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{6,4} of size 144
{6,14} of size 168
{6,15} of size 180
{6,16} of size 192
{6,4} of size 192
{6,6} of size 192
{6,3} of size 192
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{6,4} of size 192
{6,12} of size 192
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{6,12} of size 192
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{6,4} of size 216
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{6,20} of size 720
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{6,63} of size 756
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{42,6} of size 1344
{56,6} of size 1344
{56,6} of size 1344
{84,6} of size 1344
{4,6} of size 1344
{6,6} of size 1344
{8,6} of size 1344
{8,6} of size 1344
{8,6} of size 1344
{8,6} of size 1344
{14,6} of size 1344
{14,6} of size 1344
{76,6} of size 1368
{114,6} of size 1368
{114,6} of size 1368
{114,6} of size 1368
{116,6} of size 1392
{87,6} of size 1392
{116,6} of size 1392
{117,6} of size 1404
{39,6} of size 1404
{78,6} of size 1404
{118,6} of size 1416
{120,6} of size 1440
{120,6} of size 1440
{120,6} of size 1440
{12,6} of size 1440
{12,6} of size 1440
{8,6} of size 1440
{8,6} of size 1440
{15,6} of size 1440
{15,6} of size 1440
{24,6} of size 1440
{24,6} of size 1440
{30,6} of size 1440
{30,6} of size 1440
{60,6} of size 1440
{60,6} of size 1440
{40,6} of size 1440
{4,6} of size 1440
{8,6} of size 1440
{5,6} of size 1440
{6,6} of size 1440
{6,6} of size 1440
{6,6} of size 1440
{10,6} of size 1440
{10,6} of size 1440
{8,6} of size 1440
{8,6} of size 1440
{10,6} of size 1440
{10,6} of size 1440
{4,6} of size 1440
{6,6} of size 1440
{6,6} of size 1440
{6,6} of size 1440
{10,6} of size 1440
{12,6} of size 1440
{12,6} of size 1440
{30,6} of size 1440
{30,6} of size 1440
{10,6} of size 1440
{30,6} of size 1440
{30,6} of size 1440
{30,6} of size 1440
{60,6} of size 1440
{30,6} of size 1440
{60,6} of size 1440
{3,6} of size 1452
{22,6} of size 1452
{122,6} of size 1464
{123,6} of size 1476
{124,6} of size 1488
{93,6} of size 1488
{124,6} of size 1488
{10,6} of size 1500
{15,6} of size 1500
{10,6} of size 1500
{15,6} of size 1500
{28,6} of size 1512
{84,6} of size 1512
{84,6} of size 1512
{42,6} of size 1512
{126,6} of size 1512
{126,6} of size 1512
{42,6} of size 1512
{42,6} of size 1512
{84,6} of size 1512
{42,6} of size 1512
{129,6} of size 1548
{65,6} of size 1560
{130,6} of size 1560
{132,6} of size 1584
{132,6} of size 1584
{132,6} of size 1584
{33,6} of size 1584
{132,6} of size 1584
{44,6} of size 1584
{134,6} of size 1608
{45,6} of size 1620
{90,6} of size 1620
{135,6} of size 1620
{30,6} of size 1620
{45,6} of size 1620
{30,6} of size 1620
{45,6} of size 1620
{45,6} of size 1620
{90,6} of size 1620
{15,6} of size 1620
{90,6} of size 1620
{5,6} of size 1620
{10,6} of size 1620
{136,6} of size 1632
{68,6} of size 1632
{102,6} of size 1632
{92,6} of size 1656
{138,6} of size 1656
{138,6} of size 1656
{138,6} of size 1656
{140,6} of size 1680
{35,6} of size 1680
{28,6} of size 1680
{35,6} of size 1680
{42,6} of size 1680
{35,6} of size 1680
{70,6} of size 1680
{70,6} of size 1680
{105,6} of size 1680
{140,6} of size 1680
{141,6} of size 1692
{142,6} of size 1704
{144,6} of size 1728
{144,6} of size 1728
{48,6} of size 1728
{48,6} of size 1728
{48,6} of size 1728
{9,6} of size 1728
{72,6} of size 1728
{3,6} of size 1728
{24,6} of size 1728
{16,6} of size 1728
{48,6} of size 1728
{48,6} of size 1728
{48,6} of size 1728
{36,6} of size 1728
{36,6} of size 1728
{18,6} of size 1728
{72,6} of size 1728
{36,6} of size 1728
{72,6} of size 1728
{12,6} of size 1728
{12,6} of size 1728
{12,6} of size 1728
{24,6} of size 1728
{6,6} of size 1728
{6,6} of size 1728
{24,6} of size 1728
{24,6} of size 1728
{12,6} of size 1728
{24,6} of size 1728
{12,6} of size 1728
{12,6} of size 1728
{16,6} of size 1728
{48,6} of size 1728
{48,6} of size 1728
{18,6} of size 1728
{18,6} of size 1728
{6,6} of size 1728
{6,6} of size 1728
{6,6} of size 1728
{12,6} of size 1728
{24,6} of size 1728
{6,6} of size 1728
{24,6} of size 1728
{12,6} of size 1728
{12,6} of size 1728
{4,6} of size 1728
{12,6} of size 1728
{146,6} of size 1752
{147,6} of size 1764
{3,6} of size 1764
{42,6} of size 1764
{148,6} of size 1776
{111,6} of size 1776
{148,6} of size 1776
{100,6} of size 1800
{150,6} of size 1800
{150,6} of size 1800
{150,6} of size 1800
{15,6} of size 1800
{6,6} of size 1800
{6,6} of size 1800
{30,6} of size 1800
{30,6} of size 1800
{6,6} of size 1800
{6,6} of size 1800
{30,6} of size 1800
{30,6} of size 1800
{152,6} of size 1824
{76,6} of size 1824
{114,6} of size 1824
{153,6} of size 1836
{51,6} of size 1836
{102,6} of size 1836
{154,6} of size 1848
{156,6} of size 1872
{156,6} of size 1872
{156,6} of size 1872
{39,6} of size 1872
{156,6} of size 1872
{52,6} of size 1872
{158,6} of size 1896
{159,6} of size 1908
{160,6} of size 1920
{30,6} of size 1920
{40,6} of size 1920
{40,6} of size 1920
{60,6} of size 1920
{20,6} of size 1920
{30,6} of size 1920
{30,6} of size 1920
{40,6} of size 1920
{40,6} of size 1920
{120,6} of size 1920
{20,6} of size 1920
{120,6} of size 1920
{48,6} of size 1920
{48,6} of size 1920
{80,6} of size 1920
{80,6} of size 1920
{24,6} of size 1920
{8,6} of size 1920
{40,6} of size 1920
{40,6} of size 1920
{48,6} of size 1920
{48,6} of size 1920
{12,6} of size 1920
{24,6} of size 1920
{8,6} of size 1920
{20,6} of size 1920
{4,6} of size 1920
{40,6} of size 1920
{20,6} of size 1920
{40,6} of size 1920
{6,6} of size 1920
{10,6} of size 1920
{5,6} of size 1920
{8,6} of size 1920
{12,6} of size 1920
{5,6} of size 1920
{5,6} of size 1920
{5,6} of size 1920
{10,6} of size 1920
{10,6} of size 1920
{10,6} of size 1920
{10,6} of size 1920
{4,6} of size 1944
{12,6} of size 1944
{12,6} of size 1944
{6,6} 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
{54,6} of size 1944
{54,6} of size 1944
{18,6} of size 1944
{18,6} of size 1944
{18,6} of size 1944
{18,6} of size 1944
{54,6} of size 1944
{54,6} of size 1944
{54,6} of size 1944
{54,6} of size 1944
{162,6} of size 1944
{162,6} of size 1944
{6,6} of size 1944
{6,6} of size 1944
{18,6} of size 1944
{18,6} of size 1944
{36,6} of size 1944
{12,6} of size 1944
{108,6} of size 1944
{108,6} of size 1944
{108,6} of size 1944
{18,6} of size 1944
{18,6} of size 1944
{18,6} of size 1944
{6,6} of size 1944
{6,6} of size 1944
{6,6} of size 1944
{54,6} of size 1944
{6,6} of size 1944
{6,6} of size 1944
{18,6} of size 1944
{18,6} of size 1944
{18,6} of size 1944
{18,6} of size 1944
{6,6} of size 1944
{6,6} of size 1944
{18,6} of size 1944
{18,6} of size 1944
{164,6} of size 1968
{123,6} of size 1968
{164,6} of size 1968
{165,6} of size 1980
{166,6} of size 1992
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {3}*6
3-fold quotients : {2}*4
Covers (Minimal Covers in Boldface) :
2-fold covers : {12}*24
3-fold covers : {18}*36
4-fold covers : {24}*48
5-fold covers : {30}*60
6-fold covers : {36}*72
7-fold covers : {42}*84
8-fold covers : {48}*96
9-fold covers : {54}*108
10-fold covers : {60}*120
11-fold covers : {66}*132
12-fold covers : {72}*144
13-fold covers : {78}*156
14-fold covers : {84}*168
15-fold covers : {90}*180
16-fold covers : {96}*192
17-fold covers : {102}*204
18-fold covers : {108}*216
19-fold covers : {114}*228
20-fold covers : {120}*240
21-fold covers : {126}*252
22-fold covers : {132}*264
23-fold covers : {138}*276
24-fold covers : {144}*288
25-fold covers : {150}*300
26-fold covers : {156}*312
27-fold covers : {162}*324
28-fold covers : {168}*336
29-fold covers : {174}*348
30-fold covers : {180}*360
31-fold covers : {186}*372
32-fold covers : {192}*384
33-fold covers : {198}*396
34-fold covers : {204}*408
35-fold covers : {210}*420
36-fold covers : {216}*432
37-fold covers : {222}*444
38-fold covers : {228}*456
39-fold covers : {234}*468
40-fold covers : {240}*480
41-fold covers : {246}*492
42-fold covers : {252}*504
43-fold covers : {258}*516
44-fold covers : {264}*528
45-fold covers : {270}*540
46-fold covers : {276}*552
47-fold covers : {282}*564
48-fold covers : {288}*576
49-fold covers : {294}*588
50-fold covers : {300}*600
51-fold covers : {306}*612
52-fold covers : {312}*624
53-fold covers : {318}*636
54-fold covers : {324}*648
55-fold covers : {330}*660
56-fold covers : {336}*672
57-fold covers : {342}*684
58-fold covers : {348}*696
59-fold covers : {354}*708
60-fold covers : {360}*720
61-fold covers : {366}*732
62-fold covers : {372}*744
63-fold covers : {378}*756
64-fold covers : {384}*768
65-fold covers : {390}*780
66-fold covers : {396}*792
67-fold covers : {402}*804
68-fold covers : {408}*816
69-fold covers : {414}*828
70-fold covers : {420}*840
71-fold covers : {426}*852
72-fold covers : {432}*864
73-fold covers : {438}*876
74-fold covers : {444}*888
75-fold covers : {450}*900
76-fold covers : {456}*912
77-fold covers : {462}*924
78-fold covers : {468}*936
79-fold covers : {474}*948
80-fold covers : {480}*960
81-fold covers : {486}*972
82-fold covers : {492}*984
83-fold covers : {498}*996
84-fold covers : {504}*1008
85-fold covers : {510}*1020
86-fold covers : {516}*1032
87-fold covers : {522}*1044
88-fold covers : {528}*1056
89-fold covers : {534}*1068
90-fold covers : {540}*1080
91-fold covers : {546}*1092
92-fold covers : {552}*1104
93-fold covers : {558}*1116
94-fold covers : {564}*1128
95-fold covers : {570}*1140
96-fold covers : {576}*1152
97-fold covers : {582}*1164
98-fold covers : {588}*1176
99-fold covers : {594}*1188
100-fold covers : {600}*1200
101-fold covers : {606}*1212
102-fold covers : {612}*1224
103-fold covers : {618}*1236
104-fold covers : {624}*1248
105-fold covers : {630}*1260
106-fold covers : {636}*1272
107-fold covers : {642}*1284
108-fold covers : {648}*1296
109-fold covers : {654}*1308
110-fold covers : {660}*1320
111-fold covers : {666}*1332
112-fold covers : {672}*1344
113-fold covers : {678}*1356
114-fold covers : {684}*1368
115-fold covers : {690}*1380
116-fold covers : {696}*1392
117-fold covers : {702}*1404
118-fold covers : {708}*1416
119-fold covers : {714}*1428
120-fold covers : {720}*1440
121-fold covers : {726}*1452
122-fold covers : {732}*1464
123-fold covers : {738}*1476
124-fold covers : {744}*1488
125-fold covers : {750}*1500
126-fold covers : {756}*1512
127-fold covers : {762}*1524
129-fold covers : {774}*1548
130-fold covers : {780}*1560
131-fold covers : {786}*1572
132-fold covers : {792}*1584
133-fold covers : {798}*1596
134-fold covers : {804}*1608
135-fold covers : {810}*1620
136-fold covers : {816}*1632
137-fold covers : {822}*1644
138-fold covers : {828}*1656
139-fold covers : {834}*1668
140-fold covers : {840}*1680
141-fold covers : {846}*1692
142-fold covers : {852}*1704
143-fold covers : {858}*1716
144-fold covers : {864}*1728
145-fold covers : {870}*1740
146-fold covers : {876}*1752
147-fold covers : {882}*1764
148-fold covers : {888}*1776
149-fold covers : {894}*1788
150-fold covers : {900}*1800
151-fold covers : {906}*1812
152-fold covers : {912}*1824
153-fold covers : {918}*1836
154-fold covers : {924}*1848
155-fold covers : {930}*1860
156-fold covers : {936}*1872
157-fold covers : {942}*1884
158-fold covers : {948}*1896
159-fold covers : {954}*1908
160-fold covers : {960}*1920
161-fold covers : {966}*1932
162-fold covers : {972}*1944
163-fold covers : {978}*1956
164-fold covers : {984}*1968
165-fold covers : {990}*1980
166-fold covers : {996}*1992
Permutation Representation (GAP) :
s0 := (3,4)(5,6);;
s1 := (1,5)(2,3)(4,6);;
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 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(6)!(3,4)(5,6);
s1 := Sym(6)!(1,5)(2,3)(4,6);
poly := sub<Sym(6)|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 >;
References : None.
to this polytope