Polytope of Type {30,18}

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

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

Permutation Representation (Magma) :

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

References : None.
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