Polytope of Type {6,15,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,15,6}*1080
if this polytope has a name.
Group : SmallGroup(1080,539)
Rank : 4
Schlafli Type : {6,15,6}
Number of vertices, edges, etc : 6, 45, 45, 6
Order of s₀s₁s₂s₃ : 30
Order of s₀s₁s₂s₃s₂s₁ : 2
Special Properties :
   Universal
   Orientable
   Flat
   Self-Dual
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 : {2,15,6}*360, {6,15,2}*360
   5-fold quotients : {6,3,6}*216
   9-fold quotients : {2,15,2}*120
   15-fold quotients : {2,3,6}*72, {6,3,2}*72
   27-fold quotients : {2,5,2}*40
   45-fold quotients : {2,3,2}*24
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

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