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# Polytope of Type {2,9}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,9}*36
if this polytope has a name.
Group : SmallGroup(36,4)
Rank : 3
Schlafli Type : {2,9}
Number of vertices, edges, etc : 2, 9, 9
Order of s0s1s2 : 18
Order of s0s1s2s1 : 2
Special Properties :
Degenerate
Universal
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{2,9,2} of size 72
{2,9,4} of size 144
{2,9,6} of size 216
{2,9,4} of size 288
{2,9,8} of size 576
{2,9,18} of size 648
{2,9,6} of size 648
{2,9,6} of size 648
{2,9,6} of size 648
{2,9,6} of size 648
{2,9,6} of size 864
{2,9,12} of size 864
{2,9,3} of size 1008
{2,9,7} of size 1008
{2,9,7} of size 1008
{2,9,7} of size 1008
{2,9,9} of size 1008
{2,9,9} of size 1008
{2,9,8} of size 1152
{2,9,3} of size 1296
{2,9,4} of size 1296
{2,9,9} of size 1296
{2,9,9} of size 1296
{2,9,12} of size 1296
{2,9,12} of size 1296
{2,9,12} of size 1728
{2,9,24} of size 1728
{2,9,10} of size 1800
{2,9,18} of size 1944
{2,9,6} of size 1944
{2,9,6} of size 1944
{2,9,18} of size 1944
{2,9,6} of size 1944
{2,9,18} of size 1944
{2,9,18} of size 1944
{2,9,18} of size 1944
{2,9,6} of size 1944
{2,9,18} of size 1944
{2,9,18} of size 1944
{2,9,18} of size 1944
{2,9,18} of size 1944
{2,9,6} of size 1944
{2,9,18} of size 1944
Vertex Figure Of :
{2,2,9} of size 72
{3,2,9} of size 108
{4,2,9} of size 144
{5,2,9} of size 180
{6,2,9} of size 216
{7,2,9} of size 252
{8,2,9} of size 288
{9,2,9} of size 324
{10,2,9} of size 360
{11,2,9} of size 396
{12,2,9} of size 432
{13,2,9} of size 468
{14,2,9} of size 504
{15,2,9} of size 540
{16,2,9} of size 576
{17,2,9} of size 612
{18,2,9} of size 648
{19,2,9} of size 684
{20,2,9} of size 720
{21,2,9} of size 756
{22,2,9} of size 792
{23,2,9} of size 828
{24,2,9} of size 864
{25,2,9} of size 900
{26,2,9} of size 936
{27,2,9} of size 972
{28,2,9} of size 1008
{29,2,9} of size 1044
{30,2,9} of size 1080
{31,2,9} of size 1116
{32,2,9} of size 1152
{33,2,9} of size 1188
{34,2,9} of size 1224
{35,2,9} of size 1260
{36,2,9} of size 1296
{37,2,9} of size 1332
{38,2,9} of size 1368
{39,2,9} of size 1404
{40,2,9} of size 1440
{41,2,9} of size 1476
{42,2,9} of size 1512
{43,2,9} of size 1548
{44,2,9} of size 1584
{45,2,9} of size 1620
{46,2,9} of size 1656
{47,2,9} of size 1692
{48,2,9} of size 1728
{49,2,9} of size 1764
{50,2,9} of size 1800
{51,2,9} of size 1836
{52,2,9} of size 1872
{53,2,9} of size 1908
{54,2,9} of size 1944
{55,2,9} of size 1980
Quotients (Maximal Quotients in Boldface) :
3-fold quotients : {2,3}*12
Covers (Minimal Covers in Boldface) :
2-fold covers : {2,18}*72
3-fold covers : {2,27}*108, {6,9}*108
4-fold covers : {2,36}*144, {4,18}*144a, {4,9}*144
5-fold covers : {2,45}*180
6-fold covers : {2,54}*216, {6,18}*216a, {6,18}*216b
7-fold covers : {2,63}*252
8-fold covers : {4,36}*288a, {2,72}*288, {8,18}*288, {8,9}*288, {4,18}*288
9-fold covers : {2,81}*324, {18,9}*324, {6,9}*324a, {6,27}*324
10-fold covers : {10,18}*360, {2,90}*360
11-fold covers : {2,99}*396
12-fold covers : {2,108}*432, {4,54}*432a, {4,27}*432, {6,36}*432a, {6,36}*432b, {12,18}*432a, {12,18}*432b, {6,9}*432, {12,9}*432
13-fold covers : {2,117}*468
14-fold covers : {14,18}*504, {2,126}*504
15-fold covers : {2,135}*540, {6,45}*540
16-fold covers : {4,72}*576a, {4,36}*576a, {4,72}*576b, {8,36}*576a, {8,36}*576b, {2,144}*576, {16,18}*576, {8,9}*576, {4,36}*576b, {4,18}*576b, {4,36}*576c, {8,18}*576b, {8,18}*576c
17-fold covers : {2,153}*612
18-fold covers : {2,162}*648, {18,18}*648a, {18,18}*648b, {6,18}*648a, {6,18}*648b, {6,54}*648a, {6,54}*648b, {6,18}*648i
19-fold covers : {2,171}*684
20-fold covers : {10,36}*720, {20,18}*720a, {2,180}*720, {4,90}*720a, {4,45}*720
21-fold covers : {2,189}*756, {6,63}*756
22-fold covers : {22,18}*792, {2,198}*792
23-fold covers : {2,207}*828
24-fold covers : {4,108}*864a, {2,216}*864, {8,54}*864, {8,27}*864, {6,72}*864a, {6,72}*864b, {24,18}*864a, {12,36}*864a, {12,36}*864b, {24,18}*864b, {4,54}*864, {12,9}*864, {24,9}*864, {6,18}*864, {6,36}*864, {12,18}*864a, {12,18}*864b
25-fold covers : {2,225}*900, {10,9}*900, {10,45}*900
26-fold covers : {26,18}*936, {2,234}*936
27-fold covers : {2,243}*972, {18,9}*972a, {18,27}*972, {6,27}*972a, {6,9}*972d, {18,9}*972h, {18,9}*972i, {6,9}*972e, {6,27}*972b, {6,27}*972c, {6,81}*972
28-fold covers : {14,36}*1008, {28,18}*1008a, {2,252}*1008, {4,126}*1008a, {4,63}*1008
29-fold covers : {2,261}*1044
30-fold covers : {10,54}*1080, {2,270}*1080, {30,18}*1080a, {6,90}*1080a, {6,90}*1080b, {30,18}*1080b
31-fold covers : {2,279}*1116
32-fold covers : {8,36}*1152a, {4,72}*1152a, {8,72}*1152a, {8,72}*1152b, {8,72}*1152c, {8,72}*1152d, {16,36}*1152a, {4,144}*1152a, {16,36}*1152b, {4,144}*1152b, {4,36}*1152a, {4,72}*1152b, {8,36}*1152b, {32,18}*1152, {2,288}*1152, {8,9}*1152, {8,18}*1152a, {4,36}*1152d, {8,36}*1152e, {8,36}*1152f, {4,18}*1152a, {8,18}*1152d, {8,18}*1152e, {8,18}*1152f, {8,36}*1152g, {8,36}*1152h, {4,72}*1152c, {4,72}*1152d, {8,18}*1152g, {4,36}*1152e, {4,72}*1152e, {4,18}*1152b, {4,72}*1152f
33-fold covers : {2,297}*1188, {6,99}*1188
34-fold covers : {34,18}*1224, {2,306}*1224
35-fold covers : {2,315}*1260
36-fold covers : {2,324}*1296, {4,162}*1296a, {4,81}*1296, {18,36}*1296a, {18,36}*1296b, {36,18}*1296a, {12,18}*1296a, {6,36}*1296a, {6,36}*1296b, {12,54}*1296a, {6,108}*1296a, {6,108}*1296b, {36,18}*1296c, {12,18}*1296e, {12,54}*1296b, {6,27}*1296, {12,27}*1296, {18,9}*1296a, {36,9}*1296, {6,9}*1296b, {12,9}*1296c, {6,36}*1296l, {12,18}*1296l, {4,18}*1296b, {4,36}*1296, {6,36}*1296m
37-fold covers : {2,333}*1332
38-fold covers : {38,18}*1368, {2,342}*1368
39-fold covers : {2,351}*1404, {6,117}*1404
40-fold covers : {10,72}*1440, {40,18}*1440, {20,36}*1440, {4,180}*1440a, {2,360}*1440, {8,90}*1440, {8,45}*1440, {20,18}*1440, {4,90}*1440
41-fold covers : {2,369}*1476
42-fold covers : {14,54}*1512, {2,378}*1512, {42,18}*1512a, {6,126}*1512a, {6,126}*1512b, {42,18}*1512b
43-fold covers : {2,387}*1548
44-fold covers : {22,36}*1584, {44,18}*1584a, {2,396}*1584, {4,198}*1584a, {4,99}*1584
45-fold covers : {2,405}*1620, {18,45}*1620, {6,45}*1620a, {6,135}*1620
46-fold covers : {46,18}*1656, {2,414}*1656
47-fold covers : {2,423}*1692
48-fold covers : {4,216}*1728a, {4,108}*1728a, {4,216}*1728b, {8,108}*1728a, {8,108}*1728b, {2,432}*1728, {16,54}*1728, {8,27}*1728, {6,144}*1728a, {6,144}*1728b, {48,18}*1728a, {24,36}*1728a, {12,36}*1728a, {12,36}*1728b, {24,36}*1728b, {12,72}*1728a, {12,72}*1728b, {24,36}*1728c, {12,72}*1728c, {12,72}*1728d, {24,36}*1728d, {48,18}*1728b, {4,108}*1728b, {4,54}*1728b, {4,108}*1728c, {8,54}*1728b, {8,54}*1728c, {6,9}*1728, {24,9}*1728, {12,36}*1728c, {6,36}*1728a, {6,36}*1728b, {12,18}*1728a, {6,18}*1728a, {6,72}*1728b, {6,36}*1728c, {6,72}*1728c, {12,18}*1728b, {12,36}*1728d, {12,36}*1728e, {12,36}*1728f, {12,18}*1728c, {12,36}*1728g, {24,18}*1728b, {24,18}*1728c, {24,18}*1728d, {24,18}*1728e, {12,18}*1728d, {12,36}*1728h, {12,9}*1728, {6,18}*1728c
49-fold covers : {2,441}*1764, {14,9}*1764, {14,63}*1764
50-fold covers : {50,18}*1800, {2,450}*1800, {10,18}*1800a, {10,18}*1800b, {10,90}*1800a, {10,90}*1800b, {10,90}*1800c
51-fold covers : {2,459}*1836, {6,153}*1836
52-fold covers : {26,36}*1872, {52,18}*1872a, {2,468}*1872, {4,234}*1872a, {4,117}*1872
53-fold covers : {2,477}*1908
54-fold covers : {2,486}*1944, {18,18}*1944b, {18,18}*1944c, {18,54}*1944a, {18,54}*1944b, {54,18}*1944a, {6,54}*1944a, {6,54}*1944b, {6,18}*1944g, {6,18}*1944h, {18,18}*1944v, {18,18}*1944w, {18,18}*1944z, {18,18}*1944aa, {6,18}*1944i, {6,18}*1944j, {6,54}*1944c, {6,54}*1944d, {6,54}*1944e, {6,54}*1944f, {6,162}*1944a, {6,162}*1944b, {18,18}*1944ad, {18,18}*1944ae, {6,18}*1944m, {6,18}*1944n, {6,18}*1944o, {6,54}*1944g
55-fold covers : {2,495}*1980
Permutation Representation (GAP) :
```s0 := (1,2);;
s1 := ( 4, 5)( 6, 7)( 8, 9)(10,11);;
s2 := ( 3, 4)( 5, 6)( 7, 8)( 9,10);;
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*s1*s0*s1, s0*s2*s0*s2,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 ];;
poly := F / rels;;

```
Permutation Representation (Magma) :
```s0 := Sym(11)!(1,2);
s1 := Sym(11)!( 4, 5)( 6, 7)( 8, 9)(10,11);
s2 := Sym(11)!( 3, 4)( 5, 6)( 7, 8)( 9,10);
poly := sub<Sym(11)|s0,s1,s2>;

```
Finitely Presented Group Representation (Magma) :
```poly<s0,s1,s2> := Group< s0,s1,s2 | s0*s0, s1*s1, s2*s2,
s0*s1*s0*s1, s0*s2*s0*s2, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 >;

```

to this polytope