Polytope of Type {30,6,3}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {30,6,3}*1080b
if this polytope has a name.
Group : SmallGroup(1080,539)
Rank : 4
Schlafli Type : {30,6,3}
Number of vertices, edges, etc : 30, 90, 9, 3
Order of s₀s₁s₂s₃ : 30
Order of s₀s₁s₂s₃s₂s₁ : 2
Special Properties :
   Universal
   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 : {10,6,3}*360, {30,2,3}*360
   5-fold quotients : {6,6,3}*216b
   6-fold quotients : {15,2,3}*180
   9-fold quotients : {10,2,3}*120
   15-fold quotients : {2,6,3}*72, {6,2,3}*72
   18-fold quotients : {5,2,3}*60
   30-fold quotients : {3,2,3}*36
   45-fold quotients : {2,2,3}*24
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

s0 := (  2,  5)(  3,  4)(  6, 11)(  7, 15)(  8, 14)(  9, 13)( 10, 12)( 17, 20)
( 18, 19)( 21, 26)( 22, 30)( 23, 29)( 24, 28)( 25, 27)( 32, 35)( 33, 34)
( 36, 41)( 37, 45)( 38, 44)( 39, 43)( 40, 42)( 47, 50)( 48, 49)( 51, 56)
( 52, 60)( 53, 59)( 54, 58)( 55, 57)( 62, 65)( 63, 64)( 66, 71)( 67, 75)
( 68, 74)( 69, 73)( 70, 72)( 77, 80)( 78, 79)( 81, 86)( 82, 90)( 83, 89)
( 84, 88)( 85, 87)( 92, 95)( 93, 94)( 96,101)( 97,105)( 98,104)( 99,103)
(100,102)(107,110)(108,109)(111,116)(112,120)(113,119)(114,118)(115,117)
(122,125)(123,124)(126,131)(127,135)(128,134)(129,133)(130,132);;
s1 := (  1,  7)(  2,  6)(  3, 10)(  4,  9)(  5,  8)( 11, 12)( 13, 15)( 16, 37)
( 17, 36)( 18, 40)( 19, 39)( 20, 38)( 21, 32)( 22, 31)( 23, 35)( 24, 34)
( 25, 33)( 26, 42)( 27, 41)( 28, 45)( 29, 44)( 30, 43)( 46, 52)( 47, 51)
( 48, 55)( 49, 54)( 50, 53)( 56, 57)( 58, 60)( 61, 82)( 62, 81)( 63, 85)
( 64, 84)( 65, 83)( 66, 77)( 67, 76)( 68, 80)( 69, 79)( 70, 78)( 71, 87)
( 72, 86)( 73, 90)( 74, 89)( 75, 88)( 91, 97)( 92, 96)( 93,100)( 94, 99)
( 95, 98)(101,102)(103,105)(106,127)(107,126)(108,130)(109,129)(110,128)
(111,122)(112,121)(113,125)(114,124)(115,123)(116,132)(117,131)(118,135)
(119,134)(120,133);;
s2 := (  1, 16)(  2, 17)(  3, 18)(  4, 19)(  5, 20)(  6, 21)(  7, 22)(  8, 23)
(  9, 24)( 10, 25)( 11, 26)( 12, 27)( 13, 28)( 14, 29)( 15, 30)( 46,106)
( 47,107)( 48,108)( 49,109)( 50,110)( 51,111)( 52,112)( 53,113)( 54,114)
( 55,115)( 56,116)( 57,117)( 58,118)( 59,119)( 60,120)( 61, 91)( 62, 92)
( 63, 93)( 64, 94)( 65, 95)( 66, 96)( 67, 97)( 68, 98)( 69, 99)( 70,100)
( 71,101)( 72,102)( 73,103)( 74,104)( 75,105)( 76,121)( 77,122)( 78,123)
( 79,124)( 80,125)( 81,126)( 82,127)( 83,128)( 84,129)( 85,130)( 86,131)
( 87,132)( 88,133)( 89,134)( 90,135);;
s3 := (  1, 46)(  2, 47)(  3, 48)(  4, 49)(  5, 50)(  6, 51)(  7, 52)(  8, 53)
(  9, 54)( 10, 55)( 11, 56)( 12, 57)( 13, 58)( 14, 59)( 15, 60)( 16, 76)
( 17, 77)( 18, 78)( 19, 79)( 20, 80)( 21, 81)( 22, 82)( 23, 83)( 24, 84)
( 25, 85)( 26, 86)( 27, 87)( 28, 88)( 29, 89)( 30, 90)( 31, 61)( 32, 62)
( 33, 63)( 34, 64)( 35, 65)( 36, 66)( 37, 67)( 38, 68)( 39, 69)( 40, 70)
( 41, 71)( 42, 72)( 43, 73)( 44, 74)( 45, 75)(106,121)(107,122)(108,123)
(109,124)(110,125)(111,126)(112,127)(113,128)(114,129)(115,130)(116,131)
(117,132)(118,133)(119,134)(120,135);;
poly := Group([s0,s1,s2,s3]);;

Finitely Presented Group Representation (GAP) :

F := FreeGroup("s0","s1","s2","s3");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;  
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s0*s2*s0*s2, 
s0*s3*s0*s3, s1*s3*s1*s3, s2*s3*s2*s3*s2*s3, 
s0*s1*s2*s1*s0*s1*s2*s1, s3*s1*s2*s1*s2*s3*s1*s2*s1*s2, 
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*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,  5)(  3,  4)(  6, 11)(  7, 15)(  8, 14)(  9, 13)( 10, 12)
( 17, 20)( 18, 19)( 21, 26)( 22, 30)( 23, 29)( 24, 28)( 25, 27)( 32, 35)
( 33, 34)( 36, 41)( 37, 45)( 38, 44)( 39, 43)( 40, 42)( 47, 50)( 48, 49)
( 51, 56)( 52, 60)( 53, 59)( 54, 58)( 55, 57)( 62, 65)( 63, 64)( 66, 71)
( 67, 75)( 68, 74)( 69, 73)( 70, 72)( 77, 80)( 78, 79)( 81, 86)( 82, 90)
( 83, 89)( 84, 88)( 85, 87)( 92, 95)( 93, 94)( 96,101)( 97,105)( 98,104)
( 99,103)(100,102)(107,110)(108,109)(111,116)(112,120)(113,119)(114,118)
(115,117)(122,125)(123,124)(126,131)(127,135)(128,134)(129,133)(130,132);
s1 := Sym(135)!(  1,  7)(  2,  6)(  3, 10)(  4,  9)(  5,  8)( 11, 12)( 13, 15)
( 16, 37)( 17, 36)( 18, 40)( 19, 39)( 20, 38)( 21, 32)( 22, 31)( 23, 35)
( 24, 34)( 25, 33)( 26, 42)( 27, 41)( 28, 45)( 29, 44)( 30, 43)( 46, 52)
( 47, 51)( 48, 55)( 49, 54)( 50, 53)( 56, 57)( 58, 60)( 61, 82)( 62, 81)
( 63, 85)( 64, 84)( 65, 83)( 66, 77)( 67, 76)( 68, 80)( 69, 79)( 70, 78)
( 71, 87)( 72, 86)( 73, 90)( 74, 89)( 75, 88)( 91, 97)( 92, 96)( 93,100)
( 94, 99)( 95, 98)(101,102)(103,105)(106,127)(107,126)(108,130)(109,129)
(110,128)(111,122)(112,121)(113,125)(114,124)(115,123)(116,132)(117,131)
(118,135)(119,134)(120,133);
s2 := Sym(135)!(  1, 16)(  2, 17)(  3, 18)(  4, 19)(  5, 20)(  6, 21)(  7, 22)
(  8, 23)(  9, 24)( 10, 25)( 11, 26)( 12, 27)( 13, 28)( 14, 29)( 15, 30)
( 46,106)( 47,107)( 48,108)( 49,109)( 50,110)( 51,111)( 52,112)( 53,113)
( 54,114)( 55,115)( 56,116)( 57,117)( 58,118)( 59,119)( 60,120)( 61, 91)
( 62, 92)( 63, 93)( 64, 94)( 65, 95)( 66, 96)( 67, 97)( 68, 98)( 69, 99)
( 70,100)( 71,101)( 72,102)( 73,103)( 74,104)( 75,105)( 76,121)( 77,122)
( 78,123)( 79,124)( 80,125)( 81,126)( 82,127)( 83,128)( 84,129)( 85,130)
( 86,131)( 87,132)( 88,133)( 89,134)( 90,135);
s3 := Sym(135)!(  1, 46)(  2, 47)(  3, 48)(  4, 49)(  5, 50)(  6, 51)(  7, 52)
(  8, 53)(  9, 54)( 10, 55)( 11, 56)( 12, 57)( 13, 58)( 14, 59)( 15, 60)
( 16, 76)( 17, 77)( 18, 78)( 19, 79)( 20, 80)( 21, 81)( 22, 82)( 23, 83)
( 24, 84)( 25, 85)( 26, 86)( 27, 87)( 28, 88)( 29, 89)( 30, 90)( 31, 61)
( 32, 62)( 33, 63)( 34, 64)( 35, 65)( 36, 66)( 37, 67)( 38, 68)( 39, 69)
( 40, 70)( 41, 71)( 42, 72)( 43, 73)( 44, 74)( 45, 75)(106,121)(107,122)
(108,123)(109,124)(110,125)(111,126)(112,127)(113,128)(114,129)(115,130)
(116,131)(117,132)(118,133)(119,134)(120,135);
poly := sub<Sym(135)|s0,s1,s2,s3>;

Finitely Presented Group Representation (Magma) :

poly<s0,s1,s2,s3> := Group< s0,s1,s2,s3 | s0*s0, s1*s1, s2*s2, 
s3*s3, s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s2*s3*s2*s3*s2*s3, s0*s1*s2*s1*s0*s1*s2*s1, 
s3*s1*s2*s1*s2*s3*s1*s2*s1*s2, 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*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.
to this polytope