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# Polytope of Type {18,4}

Atlas Canonical Name : {18,4}*288
if this polytope has a name.
Group : SmallGroup(288,835)
Rank : 3
Schlafli Type : {18,4}
Number of vertices, edges, etc : 36, 72, 8
Order of s0s1s2 : 18
Order of s0s1s2s1 : 4
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Self-Petrie
Related Polytopes :
Facet
Vertex Figure
Dual
Petrial
Skewing Operation
Facet Of :
{18,4,2} of size 576
{18,4,4} of size 1152
{18,4,6} of size 1728
Vertex Figure Of :
{2,18,4} of size 576
{4,18,4} of size 1152
{4,18,4} of size 1152
{4,18,4} of size 1152
{6,18,4} of size 1728
{6,18,4} of size 1728
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {9,4}*144, {18,4}*144b, {18,4}*144c
3-fold quotients : {6,4}*96
4-fold quotients : {9,4}*72, {18,2}*72
6-fold quotients : {3,4}*48, {6,4}*48b, {6,4}*48c
8-fold quotients : {9,2}*36
12-fold quotients : {3,4}*24, {6,2}*24
24-fold quotients : {3,2}*12
36-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
2-fold covers : {36,4}*576b, {18,4}*576b, {36,4}*576c, {18,8}*576b, {18,8}*576c
3-fold covers : {54,4}*864, {18,12}*864a, {18,12}*864b
4-fold covers : {36,4}*1152d, {36,8}*1152e, {36,8}*1152f, {18,4}*1152a, {18,8}*1152d, {18,8}*1152e, {18,8}*1152f, {36,8}*1152g, {36,8}*1152h, {72,4}*1152c, {72,4}*1152d, {18,8}*1152g, {36,4}*1152e, {72,4}*1152e, {18,4}*1152b, {72,4}*1152f
5-fold covers : {18,20}*1440, {90,4}*1440
6-fold covers : {108,4}*1728b, {54,4}*1728b, {108,4}*1728c, {54,8}*1728b, {54,8}*1728c, {36,12}*1728e, {36,12}*1728f, {18,12}*1728c, {36,12}*1728g, {18,24}*1728b, {18,24}*1728c, {18,24}*1728d, {18,24}*1728e, {18,12}*1728d, {36,12}*1728h
Permutation Representation (GAP) :
```s0 := (  2,  3)(  5,  9)(  6, 11)(  7, 10)(  8, 12)( 13, 29)( 14, 31)( 15, 30)
( 16, 32)( 17, 25)( 18, 27)( 19, 26)( 20, 28)( 21, 33)( 22, 35)( 23, 34)
( 24, 36)( 38, 39)( 41, 45)( 42, 47)( 43, 46)( 44, 48)( 49, 65)( 50, 67)
( 51, 66)( 52, 68)( 53, 61)( 54, 63)( 55, 62)( 56, 64)( 57, 69)( 58, 71)
( 59, 70)( 60, 72)( 74, 75)( 77, 81)( 78, 83)( 79, 82)( 80, 84)( 85,101)
( 86,103)( 87,102)( 88,104)( 89, 97)( 90, 99)( 91, 98)( 92,100)( 93,105)
( 94,107)( 95,106)( 96,108)(110,111)(113,117)(114,119)(115,118)(116,120)
(121,137)(122,139)(123,138)(124,140)(125,133)(126,135)(127,134)(128,136)
(129,141)(130,143)(131,142)(132,144);;
s1 := (  1, 49)(  2, 50)(  3, 52)(  4, 51)(  5, 57)(  6, 58)(  7, 60)(  8, 59)
(  9, 53)( 10, 54)( 11, 56)( 12, 55)( 13, 37)( 14, 38)( 15, 40)( 16, 39)
( 17, 45)( 18, 46)( 19, 48)( 20, 47)( 21, 41)( 22, 42)( 23, 44)( 24, 43)
( 25, 65)( 26, 66)( 27, 68)( 28, 67)( 29, 61)( 30, 62)( 31, 64)( 32, 63)
( 33, 69)( 34, 70)( 35, 72)( 36, 71)( 73,121)( 74,122)( 75,124)( 76,123)
( 77,129)( 78,130)( 79,132)( 80,131)( 81,125)( 82,126)( 83,128)( 84,127)
( 85,109)( 86,110)( 87,112)( 88,111)( 89,117)( 90,118)( 91,120)( 92,119)
( 93,113)( 94,114)( 95,116)( 96,115)( 97,137)( 98,138)( 99,140)(100,139)
(101,133)(102,134)(103,136)(104,135)(105,141)(106,142)(107,144)(108,143);;
s2 := (  1, 76)(  2, 75)(  3, 74)(  4, 73)(  5, 80)(  6, 79)(  7, 78)(  8, 77)
(  9, 84)( 10, 83)( 11, 82)( 12, 81)( 13, 88)( 14, 87)( 15, 86)( 16, 85)
( 17, 92)( 18, 91)( 19, 90)( 20, 89)( 21, 96)( 22, 95)( 23, 94)( 24, 93)
( 25,100)( 26, 99)( 27, 98)( 28, 97)( 29,104)( 30,103)( 31,102)( 32,101)
( 33,108)( 34,107)( 35,106)( 36,105)( 37,112)( 38,111)( 39,110)( 40,109)
( 41,116)( 42,115)( 43,114)( 44,113)( 45,120)( 46,119)( 47,118)( 48,117)
( 49,124)( 50,123)( 51,122)( 52,121)( 53,128)( 54,127)( 55,126)( 56,125)
( 57,132)( 58,131)( 59,130)( 60,129)( 61,136)( 62,135)( 63,134)( 64,133)
( 65,140)( 66,139)( 67,138)( 68,137)( 69,144)( 70,143)( 71,142)( 72,141);;
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, s1*s2*s1*s2*s1*s2*s1*s2,
s0*s1*s2*s1*s0*s1*s0*s1*s2*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*s0*s1 ];;
poly := F / rels;;

```
Permutation Representation (Magma) :
```s0 := Sym(144)!(  2,  3)(  5,  9)(  6, 11)(  7, 10)(  8, 12)( 13, 29)( 14, 31)
( 15, 30)( 16, 32)( 17, 25)( 18, 27)( 19, 26)( 20, 28)( 21, 33)( 22, 35)
( 23, 34)( 24, 36)( 38, 39)( 41, 45)( 42, 47)( 43, 46)( 44, 48)( 49, 65)
( 50, 67)( 51, 66)( 52, 68)( 53, 61)( 54, 63)( 55, 62)( 56, 64)( 57, 69)
( 58, 71)( 59, 70)( 60, 72)( 74, 75)( 77, 81)( 78, 83)( 79, 82)( 80, 84)
( 85,101)( 86,103)( 87,102)( 88,104)( 89, 97)( 90, 99)( 91, 98)( 92,100)
( 93,105)( 94,107)( 95,106)( 96,108)(110,111)(113,117)(114,119)(115,118)
(116,120)(121,137)(122,139)(123,138)(124,140)(125,133)(126,135)(127,134)
(128,136)(129,141)(130,143)(131,142)(132,144);
s1 := Sym(144)!(  1, 49)(  2, 50)(  3, 52)(  4, 51)(  5, 57)(  6, 58)(  7, 60)
(  8, 59)(  9, 53)( 10, 54)( 11, 56)( 12, 55)( 13, 37)( 14, 38)( 15, 40)
( 16, 39)( 17, 45)( 18, 46)( 19, 48)( 20, 47)( 21, 41)( 22, 42)( 23, 44)
( 24, 43)( 25, 65)( 26, 66)( 27, 68)( 28, 67)( 29, 61)( 30, 62)( 31, 64)
( 32, 63)( 33, 69)( 34, 70)( 35, 72)( 36, 71)( 73,121)( 74,122)( 75,124)
( 76,123)( 77,129)( 78,130)( 79,132)( 80,131)( 81,125)( 82,126)( 83,128)
( 84,127)( 85,109)( 86,110)( 87,112)( 88,111)( 89,117)( 90,118)( 91,120)
( 92,119)( 93,113)( 94,114)( 95,116)( 96,115)( 97,137)( 98,138)( 99,140)
(100,139)(101,133)(102,134)(103,136)(104,135)(105,141)(106,142)(107,144)
(108,143);
s2 := Sym(144)!(  1, 76)(  2, 75)(  3, 74)(  4, 73)(  5, 80)(  6, 79)(  7, 78)
(  8, 77)(  9, 84)( 10, 83)( 11, 82)( 12, 81)( 13, 88)( 14, 87)( 15, 86)
( 16, 85)( 17, 92)( 18, 91)( 19, 90)( 20, 89)( 21, 96)( 22, 95)( 23, 94)
( 24, 93)( 25,100)( 26, 99)( 27, 98)( 28, 97)( 29,104)( 30,103)( 31,102)
( 32,101)( 33,108)( 34,107)( 35,106)( 36,105)( 37,112)( 38,111)( 39,110)
( 40,109)( 41,116)( 42,115)( 43,114)( 44,113)( 45,120)( 46,119)( 47,118)
( 48,117)( 49,124)( 50,123)( 51,122)( 52,121)( 53,128)( 54,127)( 55,126)
( 56,125)( 57,132)( 58,131)( 59,130)( 60,129)( 61,136)( 62,135)( 63,134)
( 64,133)( 65,140)( 66,139)( 67,138)( 68,137)( 69,144)( 70,143)( 71,142)
( 72,141);
poly := sub<Sym(144)|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, s1*s2*s1*s2*s1*s2*s1*s2,
s0*s1*s2*s1*s0*s1*s0*s1*s2*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*s0*s1 >;

```
References : None.
to this polytope