Polytope of Type {30,18}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {30,18}*1620a
if this polytope has a name.
Group : SmallGroup(1620,132)
Rank : 3
Schlafli Type : {30,18}
Number of vertices, edges, etc : 45, 405, 27
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 : {30,6}*540
   5-fold quotients : {6,18}*324a
   15-fold quotients : {6,6}*108
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

References : None.
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