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# Polytope of Type {18,14}

Atlas Canonical Name : {18,14}*504
Also Known As : {18,14|2}. if this polytope has another name.
Group : SmallGroup(504,59)
Rank : 3
Schlafli Type : {18,14}
Number of vertices, edges, etc : 18, 126, 14
Order of s0s1s2 : 126
Order of s0s1s2s1 : 2
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{18,14,2} of size 1008
Vertex Figure Of :
{2,18,14} of size 1008
Quotients (Maximal Quotients in Boldface) :
3-fold quotients : {6,14}*168
7-fold quotients : {18,2}*72
9-fold quotients : {2,14}*56
14-fold quotients : {9,2}*36
18-fold quotients : {2,7}*28
21-fold quotients : {6,2}*24
42-fold quotients : {3,2}*12
63-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
2-fold covers : {36,14}*1008, {18,28}*1008a
3-fold covers : {54,14}*1512, {18,42}*1512a, {18,42}*1512b
Permutation Representation (GAP) :
```s0 := (  2,  3)(  5,  6)(  8,  9)( 11, 12)( 14, 15)( 17, 18)( 20, 21)( 22, 44)
( 23, 43)( 24, 45)( 25, 47)( 26, 46)( 27, 48)( 28, 50)( 29, 49)( 30, 51)
( 31, 53)( 32, 52)( 33, 54)( 34, 56)( 35, 55)( 36, 57)( 37, 59)( 38, 58)
( 39, 60)( 40, 62)( 41, 61)( 42, 63)( 65, 66)( 68, 69)( 71, 72)( 74, 75)
( 77, 78)( 80, 81)( 83, 84)( 85,107)( 86,106)( 87,108)( 88,110)( 89,109)
( 90,111)( 91,113)( 92,112)( 93,114)( 94,116)( 95,115)( 96,117)( 97,119)
( 98,118)( 99,120)(100,122)(101,121)(102,123)(103,125)(104,124)(105,126);;
s1 := (  1, 22)(  2, 24)(  3, 23)(  4, 40)(  5, 42)(  6, 41)(  7, 37)(  8, 39)
(  9, 38)( 10, 34)( 11, 36)( 12, 35)( 13, 31)( 14, 33)( 15, 32)( 16, 28)
( 17, 30)( 18, 29)( 19, 25)( 20, 27)( 21, 26)( 43, 44)( 46, 62)( 47, 61)
( 48, 63)( 49, 59)( 50, 58)( 51, 60)( 52, 56)( 53, 55)( 54, 57)( 64, 85)
( 65, 87)( 66, 86)( 67,103)( 68,105)( 69,104)( 70,100)( 71,102)( 72,101)
( 73, 97)( 74, 99)( 75, 98)( 76, 94)( 77, 96)( 78, 95)( 79, 91)( 80, 93)
( 81, 92)( 82, 88)( 83, 90)( 84, 89)(106,107)(109,125)(110,124)(111,126)
(112,122)(113,121)(114,123)(115,119)(116,118)(117,120);;
s2 := (  1, 67)(  2, 68)(  3, 69)(  4, 64)(  5, 65)(  6, 66)(  7, 82)(  8, 83)
(  9, 84)( 10, 79)( 11, 80)( 12, 81)( 13, 76)( 14, 77)( 15, 78)( 16, 73)
( 17, 74)( 18, 75)( 19, 70)( 20, 71)( 21, 72)( 22, 88)( 23, 89)( 24, 90)
( 25, 85)( 26, 86)( 27, 87)( 28,103)( 29,104)( 30,105)( 31,100)( 32,101)
( 33,102)( 34, 97)( 35, 98)( 36, 99)( 37, 94)( 38, 95)( 39, 96)( 40, 91)
( 41, 92)( 42, 93)( 43,109)( 44,110)( 45,111)( 46,106)( 47,107)( 48,108)
( 49,124)( 50,125)( 51,126)( 52,121)( 53,122)( 54,123)( 55,118)( 56,119)
( 57,120)( 58,115)( 59,116)( 60,117)( 61,112)( 62,113)( 63,114);;
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*s2*s0*s2, s0*s1*s2*s1*s0*s1*s2*s1,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
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(126)!(  2,  3)(  5,  6)(  8,  9)( 11, 12)( 14, 15)( 17, 18)( 20, 21)
( 22, 44)( 23, 43)( 24, 45)( 25, 47)( 26, 46)( 27, 48)( 28, 50)( 29, 49)
( 30, 51)( 31, 53)( 32, 52)( 33, 54)( 34, 56)( 35, 55)( 36, 57)( 37, 59)
( 38, 58)( 39, 60)( 40, 62)( 41, 61)( 42, 63)( 65, 66)( 68, 69)( 71, 72)
( 74, 75)( 77, 78)( 80, 81)( 83, 84)( 85,107)( 86,106)( 87,108)( 88,110)
( 89,109)( 90,111)( 91,113)( 92,112)( 93,114)( 94,116)( 95,115)( 96,117)
( 97,119)( 98,118)( 99,120)(100,122)(101,121)(102,123)(103,125)(104,124)
(105,126);
s1 := Sym(126)!(  1, 22)(  2, 24)(  3, 23)(  4, 40)(  5, 42)(  6, 41)(  7, 37)
(  8, 39)(  9, 38)( 10, 34)( 11, 36)( 12, 35)( 13, 31)( 14, 33)( 15, 32)
( 16, 28)( 17, 30)( 18, 29)( 19, 25)( 20, 27)( 21, 26)( 43, 44)( 46, 62)
( 47, 61)( 48, 63)( 49, 59)( 50, 58)( 51, 60)( 52, 56)( 53, 55)( 54, 57)
( 64, 85)( 65, 87)( 66, 86)( 67,103)( 68,105)( 69,104)( 70,100)( 71,102)
( 72,101)( 73, 97)( 74, 99)( 75, 98)( 76, 94)( 77, 96)( 78, 95)( 79, 91)
( 80, 93)( 81, 92)( 82, 88)( 83, 90)( 84, 89)(106,107)(109,125)(110,124)
(111,126)(112,122)(113,121)(114,123)(115,119)(116,118)(117,120);
s2 := Sym(126)!(  1, 67)(  2, 68)(  3, 69)(  4, 64)(  5, 65)(  6, 66)(  7, 82)
(  8, 83)(  9, 84)( 10, 79)( 11, 80)( 12, 81)( 13, 76)( 14, 77)( 15, 78)
( 16, 73)( 17, 74)( 18, 75)( 19, 70)( 20, 71)( 21, 72)( 22, 88)( 23, 89)
( 24, 90)( 25, 85)( 26, 86)( 27, 87)( 28,103)( 29,104)( 30,105)( 31,100)
( 32,101)( 33,102)( 34, 97)( 35, 98)( 36, 99)( 37, 94)( 38, 95)( 39, 96)
( 40, 91)( 41, 92)( 42, 93)( 43,109)( 44,110)( 45,111)( 46,106)( 47,107)
( 48,108)( 49,124)( 50,125)( 51,126)( 52,121)( 53,122)( 54,123)( 55,118)
( 56,119)( 57,120)( 58,115)( 59,116)( 60,117)( 61,112)( 62,113)( 63,114);
poly := sub<Sym(126)|s0,s1,s2>;

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