# Polytope of Type {18,16}

Atlas Canonical Name : {18,16}*576
Also Known As : {18,16|2}. if this polytope has another name.
Group : SmallGroup(576,472)
Rank : 3
Schlafli Type : {18,16}
Number of vertices, edges, etc : 18, 144, 16
Order of s0s1s2 : 144
Order of s0s1s2s1 : 2
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{18,16,2} of size 1152
Vertex Figure Of :
{2,18,16} of size 1152
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {18,8}*288
3-fold quotients : {6,16}*192
4-fold quotients : {18,4}*144a
6-fold quotients : {6,8}*96
8-fold quotients : {18,2}*72
9-fold quotients : {2,16}*64
12-fold quotients : {6,4}*48a
16-fold quotients : {9,2}*36
18-fold quotients : {2,8}*32
24-fold quotients : {6,2}*24
36-fold quotients : {2,4}*16
48-fold quotients : {3,2}*12
72-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
2-fold covers : {36,16}*1152a, {18,32}*1152
3-fold covers : {54,16}*1728, {18,48}*1728a, {18,48}*1728b
Permutation Representation (GAP) :
```s0 := (  2,  3)(  4,  8)(  5,  7)(  6,  9)( 11, 12)( 13, 17)( 14, 16)( 15, 18)
( 20, 21)( 22, 26)( 23, 25)( 24, 27)( 29, 30)( 31, 35)( 32, 34)( 33, 36)
( 38, 39)( 40, 44)( 41, 43)( 42, 45)( 47, 48)( 49, 53)( 50, 52)( 51, 54)
( 56, 57)( 58, 62)( 59, 61)( 60, 63)( 65, 66)( 67, 71)( 68, 70)( 69, 72)
( 74, 75)( 76, 80)( 77, 79)( 78, 81)( 83, 84)( 85, 89)( 86, 88)( 87, 90)
( 92, 93)( 94, 98)( 95, 97)( 96, 99)(101,102)(103,107)(104,106)(105,108)
(110,111)(112,116)(113,115)(114,117)(119,120)(121,125)(122,124)(123,126)
(128,129)(130,134)(131,133)(132,135)(137,138)(139,143)(140,142)(141,144);;
s1 := (  1,  4)(  2,  6)(  3,  5)(  7,  8)( 10, 13)( 11, 15)( 12, 14)( 16, 17)
( 19, 31)( 20, 33)( 21, 32)( 22, 28)( 23, 30)( 24, 29)( 25, 35)( 26, 34)
( 27, 36)( 37, 58)( 38, 60)( 39, 59)( 40, 55)( 41, 57)( 42, 56)( 43, 62)
( 44, 61)( 45, 63)( 46, 67)( 47, 69)( 48, 68)( 49, 64)( 50, 66)( 51, 65)
( 52, 71)( 53, 70)( 54, 72)( 73,112)( 74,114)( 75,113)( 76,109)( 77,111)
( 78,110)( 79,116)( 80,115)( 81,117)( 82,121)( 83,123)( 84,122)( 85,118)
( 86,120)( 87,119)( 88,125)( 89,124)( 90,126)( 91,139)( 92,141)( 93,140)
( 94,136)( 95,138)( 96,137)( 97,143)( 98,142)( 99,144)(100,130)(101,132)
(102,131)(103,127)(104,129)(105,128)(106,134)(107,133)(108,135);;
s2 := (  1, 73)(  2, 74)(  3, 75)(  4, 76)(  5, 77)(  6, 78)(  7, 79)(  8, 80)
(  9, 81)( 10, 82)( 11, 83)( 12, 84)( 13, 85)( 14, 86)( 15, 87)( 16, 88)
( 17, 89)( 18, 90)( 19,100)( 20,101)( 21,102)( 22,103)( 23,104)( 24,105)
( 25,106)( 26,107)( 27,108)( 28, 91)( 29, 92)( 30, 93)( 31, 94)( 32, 95)
( 33, 96)( 34, 97)( 35, 98)( 36, 99)( 37,127)( 38,128)( 39,129)( 40,130)
( 41,131)( 42,132)( 43,133)( 44,134)( 45,135)( 46,136)( 47,137)( 48,138)
( 49,139)( 50,140)( 51,141)( 52,142)( 53,143)( 54,144)( 55,109)( 56,110)
( 57,111)( 58,112)( 59,113)( 60,114)( 61,115)( 62,116)( 63,117)( 64,118)
( 65,119)( 66,120)( 67,121)( 68,122)( 69,123)( 70,124)( 71,125)( 72,126);;
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*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(144)!(  2,  3)(  4,  8)(  5,  7)(  6,  9)( 11, 12)( 13, 17)( 14, 16)
( 15, 18)( 20, 21)( 22, 26)( 23, 25)( 24, 27)( 29, 30)( 31, 35)( 32, 34)
( 33, 36)( 38, 39)( 40, 44)( 41, 43)( 42, 45)( 47, 48)( 49, 53)( 50, 52)
( 51, 54)( 56, 57)( 58, 62)( 59, 61)( 60, 63)( 65, 66)( 67, 71)( 68, 70)
( 69, 72)( 74, 75)( 76, 80)( 77, 79)( 78, 81)( 83, 84)( 85, 89)( 86, 88)
( 87, 90)( 92, 93)( 94, 98)( 95, 97)( 96, 99)(101,102)(103,107)(104,106)
(105,108)(110,111)(112,116)(113,115)(114,117)(119,120)(121,125)(122,124)
(123,126)(128,129)(130,134)(131,133)(132,135)(137,138)(139,143)(140,142)
(141,144);
s1 := Sym(144)!(  1,  4)(  2,  6)(  3,  5)(  7,  8)( 10, 13)( 11, 15)( 12, 14)
( 16, 17)( 19, 31)( 20, 33)( 21, 32)( 22, 28)( 23, 30)( 24, 29)( 25, 35)
( 26, 34)( 27, 36)( 37, 58)( 38, 60)( 39, 59)( 40, 55)( 41, 57)( 42, 56)
( 43, 62)( 44, 61)( 45, 63)( 46, 67)( 47, 69)( 48, 68)( 49, 64)( 50, 66)
( 51, 65)( 52, 71)( 53, 70)( 54, 72)( 73,112)( 74,114)( 75,113)( 76,109)
( 77,111)( 78,110)( 79,116)( 80,115)( 81,117)( 82,121)( 83,123)( 84,122)
( 85,118)( 86,120)( 87,119)( 88,125)( 89,124)( 90,126)( 91,139)( 92,141)
( 93,140)( 94,136)( 95,138)( 96,137)( 97,143)( 98,142)( 99,144)(100,130)
(101,132)(102,131)(103,127)(104,129)(105,128)(106,134)(107,133)(108,135);
s2 := Sym(144)!(  1, 73)(  2, 74)(  3, 75)(  4, 76)(  5, 77)(  6, 78)(  7, 79)
(  8, 80)(  9, 81)( 10, 82)( 11, 83)( 12, 84)( 13, 85)( 14, 86)( 15, 87)
( 16, 88)( 17, 89)( 18, 90)( 19,100)( 20,101)( 21,102)( 22,103)( 23,104)
( 24,105)( 25,106)( 26,107)( 27,108)( 28, 91)( 29, 92)( 30, 93)( 31, 94)
( 32, 95)( 33, 96)( 34, 97)( 35, 98)( 36, 99)( 37,127)( 38,128)( 39,129)
( 40,130)( 41,131)( 42,132)( 43,133)( 44,134)( 45,135)( 46,136)( 47,137)
( 48,138)( 49,139)( 50,140)( 51,141)( 52,142)( 53,143)( 54,144)( 55,109)
( 56,110)( 57,111)( 58,112)( 59,113)( 60,114)( 61,115)( 62,116)( 63,117)
( 64,118)( 65,119)( 66,120)( 67,121)( 68,122)( 69,123)( 70,124)( 71,125)
( 72,126);
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, 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*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