Polytope of Type {9,6,10}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {9,6,10}*1080
if this polytope has a name.
Group : SmallGroup(1080,286)
Rank : 4
Schlafli Type : {9,6,10}
Number of vertices, edges, etc : 9, 27, 30, 10
Order of s₀s₁s₂s₃ : 90
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 : {9,2,10}*360, {3,6,10}*360
   5-fold quotients : {9,6,2}*216
   6-fold quotients : {9,2,5}*180
   9-fold quotients : {3,2,10}*120
   15-fold quotients : {9,2,2}*72, {3,6,2}*72
   18-fold quotients : {3,2,5}*60
   45-fold quotients : {3,2,2}*24
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

References : None.
to this polytope