Polytope of Type {6,18}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,18}*1728c
if this polytope has a name.
Group : SmallGroup(1728,46100)
Rank : 3
Schlafli Type : {6,18}
Number of vertices, edges, etc : 48, 432, 144
Order of s₀s₁s₂ : 18
Order of s₀s₁s₂s₁ : 6
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {6,6}*576d
   16-fold quotients : {6,9}*108
   48-fold quotients : {2,9}*36, {6,3}*36
   144-fold quotients : {2,3}*12
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

References : None.
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