Polytope of Type {15,6,9}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {15,6,9}*1620
if this polytope has a name.
Group : SmallGroup(1620,132)
Rank : 4
Schlafli Type : {15,6,9}
Number of vertices, edges, etc : 15, 45, 27, 9
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 : {15,2,9}*540, {15,6,3}*540
   5-fold quotients : {3,6,9}*324
   9-fold quotients : {5,2,9}*180, {15,2,3}*180
   15-fold quotients : {3,2,9}*108, {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.
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, 12)(  3, 11)(  4, 13)(  5, 15)(  6, 14)(  7, 16)(  8, 18)
(  9, 17)( 19, 37)( 20, 39)( 21, 38)( 22, 40)( 23, 42)( 24, 41)( 25, 43)
( 26, 45)( 27, 44)( 29, 30)( 32, 33)( 35, 36)( 46, 56)( 47, 55)( 48, 57)
( 49, 59)( 50, 58)( 51, 60)( 52, 62)( 53, 61)( 54, 63)( 64, 83)( 65, 82)
( 66, 84)( 67, 86)( 68, 85)( 69, 87)( 70, 89)( 71, 88)( 72, 90)( 73, 74)
( 76, 77)( 79, 80)( 91,102)( 92,101)( 93,100)( 94,105)( 95,104)( 96,103)
( 97,108)( 98,107)( 99,106)(109,129)(110,128)(111,127)(112,132)(113,131)
(114,130)(115,135)(116,134)(117,133)(118,120)(121,123)(124,126);;
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);;
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, 94)( 47, 96)( 48, 95)( 49, 91)
( 50, 93)( 51, 92)( 52, 97)( 53, 99)( 54, 98)( 55,103)( 56,105)( 57,104)
( 58,100)( 59,102)( 60,101)( 61,106)( 62,108)( 63,107)( 64,112)( 65,114)
( 66,113)( 67,109)( 68,111)( 69,110)( 70,115)( 71,117)( 72,116)( 73,121)
( 74,123)( 75,122)( 76,118)( 77,120)( 78,119)( 79,124)( 80,126)( 81,125)
( 82,130)( 83,132)( 84,131)( 85,127)( 86,129)( 87,128)( 88,133)( 89,135)
( 90,134);;
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*s2*s1*s2*s0*s1*s2*s1, 
s3*s1*s2*s1*s2*s3*s1*s2*s1*s2, s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
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,  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, 12)(  3, 11)(  4, 13)(  5, 15)(  6, 14)(  7, 16)
(  8, 18)(  9, 17)( 19, 37)( 20, 39)( 21, 38)( 22, 40)( 23, 42)( 24, 41)
( 25, 43)( 26, 45)( 27, 44)( 29, 30)( 32, 33)( 35, 36)( 46, 56)( 47, 55)
( 48, 57)( 49, 59)( 50, 58)( 51, 60)( 52, 62)( 53, 61)( 54, 63)( 64, 83)
( 65, 82)( 66, 84)( 67, 86)( 68, 85)( 69, 87)( 70, 89)( 71, 88)( 72, 90)
( 73, 74)( 76, 77)( 79, 80)( 91,102)( 92,101)( 93,100)( 94,105)( 95,104)
( 96,103)( 97,108)( 98,107)( 99,106)(109,129)(110,128)(111,127)(112,132)
(113,131)(114,130)(115,135)(116,134)(117,133)(118,120)(121,123)(124,126);
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);
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, 94)( 47, 96)( 48, 95)
( 49, 91)( 50, 93)( 51, 92)( 52, 97)( 53, 99)( 54, 98)( 55,103)( 56,105)
( 57,104)( 58,100)( 59,102)( 60,101)( 61,106)( 62,108)( 63,107)( 64,112)
( 65,114)( 66,113)( 67,109)( 68,111)( 69,110)( 70,115)( 71,117)( 72,116)
( 73,121)( 74,123)( 75,122)( 76,118)( 77,120)( 78,119)( 79,124)( 80,126)
( 81,125)( 82,130)( 83,132)( 84,131)( 85,127)( 86,129)( 87,128)( 88,133)
( 89,135)( 90,134);
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*s2*s1*s2*s0*s1*s2*s1, s3*s1*s2*s1*s2*s3*s1*s2*s1*s2, 
s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
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.
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