Polytope of Type {15,6,3}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {15,6,3}*1620b
if this polytope has a name.
Group : SmallGroup(1620,136)
Rank : 4
Schlafli Type : {15,6,3}
Number of vertices, edges, etc : 45, 135, 27, 3
Order of s₀s₁s₂s₃ : 45
Order of s₀s₁s₂s₃s₂s₁ : 6
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 : {15,6,3}*540
   5-fold quotients : {3,6,3}*324b
   9-fold quotients : {15,2,3}*180
   15-fold quotients : {3,6,3}*108
   27-fold quotients : {5,2,3}*60
   45-fold quotients : {3,2,3}*36
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Irregular Quotients (of which this is a minimal cover):
   P/N, where N=<s0*s1*s2*s1*s0*s1*s2*s1> of order 3.
      3 facets:
         3 of 3-fold non-regular quotient of {15,6}*540
      25 vertex figures:
         10 of {6,3}*36
         15 of {2,3}*12

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