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Polytope of Type {18,30}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {18,30}*1620b
if this polytope has a name.
Group : SmallGroup(1620,138)
Rank : 3
Schlafli Type : {18,30}
Number of vertices, edges, etc : 27, 405, 45
Order of s0s1s2 : 45
Order of s0s1s2s1 : 6
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Non-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 : {6,30}*540
   5-fold quotients : {18,6}*324b
   15-fold quotients : {6,6}*108
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :
s0 := (  2,  3)(  4,  7)(  5,  9)(  6,  8)( 11, 12)( 13, 16)( 14, 18)( 15, 17)
( 20, 21)( 22, 25)( 23, 27)( 24, 26)( 29, 30)( 31, 34)( 32, 36)( 33, 35)
( 38, 39)( 40, 43)( 41, 45)( 42, 44)( 46, 92)( 47, 91)( 48, 93)( 49, 98)
( 50, 97)( 51, 99)( 52, 95)( 53, 94)( 54, 96)( 55,101)( 56,100)( 57,102)
( 58,107)( 59,106)( 60,108)( 61,104)( 62,103)( 63,105)( 64,110)( 65,109)
( 66,111)( 67,116)( 68,115)( 69,117)( 70,113)( 71,112)( 72,114)( 73,119)
( 74,118)( 75,120)( 76,125)( 77,124)( 78,126)( 79,122)( 80,121)( 81,123)
( 82,128)( 83,127)( 84,129)( 85,134)( 86,133)( 87,135)( 88,131)( 89,130)
( 90,132);;
s1 := (  1, 46)(  2, 48)(  3, 47)(  4, 49)(  5, 51)(  6, 50)(  7, 52)(  8, 54)
(  9, 53)( 10, 82)( 11, 84)( 12, 83)( 13, 85)( 14, 87)( 15, 86)( 16, 88)
( 17, 90)( 18, 89)( 19, 73)( 20, 75)( 21, 74)( 22, 76)( 23, 78)( 24, 77)
( 25, 79)( 26, 81)( 27, 80)( 28, 64)( 29, 66)( 30, 65)( 31, 67)( 32, 69)
( 33, 68)( 34, 70)( 35, 72)( 36, 71)( 37, 55)( 38, 57)( 39, 56)( 40, 58)
( 41, 60)( 42, 59)( 43, 61)( 44, 63)( 45, 62)( 91, 92)( 94, 95)( 97, 98)
(100,128)(101,127)(102,129)(103,131)(104,130)(105,132)(106,134)(107,133)
(108,135)(109,119)(110,118)(111,120)(112,122)(113,121)(114,123)(115,125)
(116,124)(117,126);;
s2 := (  1, 10)(  2, 11)(  3, 12)(  4, 17)(  5, 18)(  6, 16)(  7, 15)(  8, 13)
(  9, 14)( 19, 37)( 20, 38)( 21, 39)( 22, 44)( 23, 45)( 24, 43)( 25, 42)
( 26, 40)( 27, 41)( 31, 35)( 32, 36)( 33, 34)( 46, 59)( 47, 60)( 48, 58)
( 49, 57)( 50, 55)( 51, 56)( 52, 61)( 53, 62)( 54, 63)( 64, 86)( 65, 87)
( 66, 85)( 67, 84)( 68, 82)( 69, 83)( 70, 88)( 71, 89)( 72, 90)( 73, 77)
( 74, 78)( 75, 76)( 91,108)( 92,106)( 93,107)( 94,103)( 95,104)( 96,105)
( 97,101)( 98,102)( 99,100)(109,135)(110,133)(111,134)(112,130)(113,131)
(114,132)(115,128)(116,129)(117,127)(118,126)(119,124)(120,125);;
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*s0*s1*s2*s1*s0*s1*s2*s0*s1*s0*s1*s2*s1*s0*s1, 
s2*s0*s1*s2*s1*s2*s0*s1*s0*s1*s2*s1*s2*s1*s2*s1*s0*s1, 
s0*s1*s2*s1*s2*s1*s0*s1*s0*s1*s0*s1*s2*s1*s2*s1*s0*s1*s0*s1, 
s0*s1*s2*s0*s1*s2*s1*s2*s1*s2*s0*s1*s2*s1*s2*s1*s2*s1*s2*s0*s1*s2*s1*s2*s1*s2*s1*s2*s0*s1*s2*s1*s2*s1*s2 ];;
poly := F / rels;;
 
Permutation Representation (Magma) :
s0 := Sym(135)!(  2,  3)(  4,  7)(  5,  9)(  6,  8)( 11, 12)( 13, 16)( 14, 18)
( 15, 17)( 20, 21)( 22, 25)( 23, 27)( 24, 26)( 29, 30)( 31, 34)( 32, 36)
( 33, 35)( 38, 39)( 40, 43)( 41, 45)( 42, 44)( 46, 92)( 47, 91)( 48, 93)
( 49, 98)( 50, 97)( 51, 99)( 52, 95)( 53, 94)( 54, 96)( 55,101)( 56,100)
( 57,102)( 58,107)( 59,106)( 60,108)( 61,104)( 62,103)( 63,105)( 64,110)
( 65,109)( 66,111)( 67,116)( 68,115)( 69,117)( 70,113)( 71,112)( 72,114)
( 73,119)( 74,118)( 75,120)( 76,125)( 77,124)( 78,126)( 79,122)( 80,121)
( 81,123)( 82,128)( 83,127)( 84,129)( 85,134)( 86,133)( 87,135)( 88,131)
( 89,130)( 90,132);
s1 := Sym(135)!(  1, 46)(  2, 48)(  3, 47)(  4, 49)(  5, 51)(  6, 50)(  7, 52)
(  8, 54)(  9, 53)( 10, 82)( 11, 84)( 12, 83)( 13, 85)( 14, 87)( 15, 86)
( 16, 88)( 17, 90)( 18, 89)( 19, 73)( 20, 75)( 21, 74)( 22, 76)( 23, 78)
( 24, 77)( 25, 79)( 26, 81)( 27, 80)( 28, 64)( 29, 66)( 30, 65)( 31, 67)
( 32, 69)( 33, 68)( 34, 70)( 35, 72)( 36, 71)( 37, 55)( 38, 57)( 39, 56)
( 40, 58)( 41, 60)( 42, 59)( 43, 61)( 44, 63)( 45, 62)( 91, 92)( 94, 95)
( 97, 98)(100,128)(101,127)(102,129)(103,131)(104,130)(105,132)(106,134)
(107,133)(108,135)(109,119)(110,118)(111,120)(112,122)(113,121)(114,123)
(115,125)(116,124)(117,126);
s2 := Sym(135)!(  1, 10)(  2, 11)(  3, 12)(  4, 17)(  5, 18)(  6, 16)(  7, 15)
(  8, 13)(  9, 14)( 19, 37)( 20, 38)( 21, 39)( 22, 44)( 23, 45)( 24, 43)
( 25, 42)( 26, 40)( 27, 41)( 31, 35)( 32, 36)( 33, 34)( 46, 59)( 47, 60)
( 48, 58)( 49, 57)( 50, 55)( 51, 56)( 52, 61)( 53, 62)( 54, 63)( 64, 86)
( 65, 87)( 66, 85)( 67, 84)( 68, 82)( 69, 83)( 70, 88)( 71, 89)( 72, 90)
( 73, 77)( 74, 78)( 75, 76)( 91,108)( 92,106)( 93,107)( 94,103)( 95,104)
( 96,105)( 97,101)( 98,102)( 99,100)(109,135)(110,133)(111,134)(112,130)
(113,131)(114,132)(115,128)(116,129)(117,127)(118,126)(119,124)(120,125);
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*s0*s1*s2*s1*s0*s1*s2*s0*s1*s0*s1*s2*s1*s0*s1, 
s2*s0*s1*s2*s1*s2*s0*s1*s0*s1*s2*s1*s2*s1*s2*s1*s0*s1, 
s0*s1*s2*s1*s2*s1*s0*s1*s0*s1*s0*s1*s2*s1*s2*s1*s0*s1*s0*s1, 
s0*s1*s2*s0*s1*s2*s1*s2*s1*s2*s0*s1*s2*s1*s2*s1*s2*s1*s2*s0*s1*s2*s1*s2*s1*s2*s1*s2*s0*s1*s2*s1*s2*s1*s2 >; 
 
References : None.
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