Polytope of Type {30,18}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {30,18}*1620c
if this polytope has a name.
Group : SmallGroup(1620,140)
Rank : 3
Schlafli Type : {30,18}
Number of vertices, edges, etc : 45, 405, 27
Order of s₀s₁s₂ : 15
Order of s₀s₁s₂s₁ : 6
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Non-Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {30,6}*540
   5-fold quotients : {6,18}*324c
   15-fold quotients : {6,6}*108
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

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