Polytope of Type {18,30}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {18,30}*1080a
if this polytope has a name.
Group : SmallGroup(1080,286)
Rank : 3
Schlafli Type : {18,30}
Number of vertices, edges, etc : 18, 270, 30
Order of s₀s₁s₂ : 90
Order of s₀s₁s₂s₁ : 6
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
   Flat
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 : {18,10}*360, {6,30}*360a
   5-fold quotients : {18,6}*216b
   9-fold quotients : {6,10}*120
   10-fold quotients : {9,6}*108
   15-fold quotients : {18,2}*72, {6,6}*72c
   27-fold quotients : {2,10}*40
   30-fold quotients : {9,2}*36, {3,6}*36
   45-fold quotients : {6,2}*24
   54-fold quotients : {2,5}*20
   90-fold quotients : {3,2}*12
   135-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

References : None.
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