Polytope of Type {16,18}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {16,18}*576
Also Known As : {16,18|2}. if this polytope has another name.
Group : SmallGroup(576,472)
Rank : 3
Schlafli Type : {16,18}
Number of vertices, edges, etc : 16, 144, 18
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 :
   {16,18,2} of size 1152
Vertex Figure Of :
   {2,16,18} of size 1152
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {8,18}*288
   3-fold quotients : {16,6}*192
   4-fold quotients : {4,18}*144a
   6-fold quotients : {8,6}*96
   8-fold quotients : {2,18}*72
   9-fold quotients : {16,2}*64
   12-fold quotients : {4,6}*48a
   16-fold quotients : {2,9}*36
   18-fold quotients : {8,2}*32
   24-fold quotients : {2,6}*24
   36-fold quotients : {4,2}*16
   48-fold quotients : {2,3}*12
   72-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   2-fold covers : {16,36}*1152a, {32,18}*1152
   3-fold covers : {16,54}*1728, {48,18}*1728a, {48,18}*1728b
Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

References : None.
to this polytope