Polytope of Type {3,2,4,9,4}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {3,2,4,9,4}*1728
if this polytope has a name.
Group : SmallGroup(1728,46118)
Rank : 6
Schlafli Type : {3,2,4,9,4}
Number of vertices, edges, etc : 3, 3, 4, 18, 18, 4
Order of s0s1s2s3s4s5 : 9
Order of s0s1s2s3s4s5s4s3s2s1 : 2
Special Properties :
Degenerate
Universal
Non-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 : {3,2,4,3,4}*576
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := (2,3);;
s1 := (1,2);;
s2 := ( 4, 5)( 6, 7)( 8, 9)( 10, 11)( 12, 13)( 14, 15)( 16, 17)( 18, 19)
( 20, 21)( 22, 23)( 24, 25)( 26, 27)( 28, 29)( 30, 31)( 32, 33)( 34, 35)
( 36, 37)( 38, 39)( 40, 41)( 42, 43)( 44, 45)( 46, 47)( 48, 49)( 50, 51)
( 52, 53)( 54, 55)( 56, 57)( 58, 59)( 60, 61)( 62, 63)( 64, 65)( 66, 67)
( 68, 69)( 70, 71)( 72, 73)( 74, 75)( 76, 77)( 78, 79)( 80, 81)( 82, 83)
( 84, 85)( 86, 87)( 88, 89)( 90, 91)( 92, 93)( 94, 95)( 96, 97)( 98, 99)
(100,101)(102,103)(104,105)(106,107)(108,109)(110,111)(112,113)(114,115)
(116,117)(118,119)(120,121)(122,123)(124,125)(126,127)(128,129)(130,131)
(132,133)(134,135)(136,137)(138,139)(140,141)(142,143)(144,145)(146,147);;
s3 := ( 5, 6)( 8, 12)( 9, 14)( 10, 13)( 11, 15)( 17, 18)( 20, 36)( 21, 38)
( 22, 37)( 23, 39)( 24, 44)( 25, 46)( 26, 45)( 27, 47)( 28, 40)( 29, 42)
( 30, 41)( 31, 43)( 32, 48)( 33, 50)( 34, 49)( 35, 51)( 52,116)( 53,118)
( 54,117)( 55,119)( 56,124)( 57,126)( 58,125)( 59,127)( 60,120)( 61,122)
( 62,121)( 63,123)( 64,128)( 65,130)( 66,129)( 67,131)( 68,100)( 69,102)
( 70,101)( 71,103)( 72,108)( 73,110)( 74,109)( 75,111)( 76,104)( 77,106)
( 78,105)( 79,107)( 80,112)( 81,114)( 82,113)( 83,115)( 84,132)( 85,134)
( 86,133)( 87,135)( 88,140)( 89,142)( 90,141)( 91,143)( 92,136)( 93,138)
( 94,137)( 95,139)( 96,144)( 97,146)( 98,145)( 99,147);;
s4 := ( 4,100)( 5,101)( 6,103)( 7,102)( 8,112)( 9,113)( 10,115)( 11,114)
( 12,108)( 13,109)( 14,111)( 15,110)( 16,104)( 17,105)( 18,107)( 19,106)
( 20,132)( 21,133)( 22,135)( 23,134)( 24,144)( 25,145)( 26,147)( 27,146)
( 28,140)( 29,141)( 30,143)( 31,142)( 32,136)( 33,137)( 34,139)( 35,138)
( 36,116)( 37,117)( 38,119)( 39,118)( 40,128)( 41,129)( 42,131)( 43,130)
( 44,124)( 45,125)( 46,127)( 47,126)( 48,120)( 49,121)( 50,123)( 51,122)
( 54, 55)( 56, 64)( 57, 65)( 58, 67)( 59, 66)( 62, 63)( 68, 84)( 69, 85)
( 70, 87)( 71, 86)( 72, 96)( 73, 97)( 74, 99)( 75, 98)( 76, 92)( 77, 93)
( 78, 95)( 79, 94)( 80, 88)( 81, 89)( 82, 91)( 83, 90);;
s5 := ( 4, 16)( 5, 17)( 6, 18)( 7, 19)( 8, 12)( 9, 13)( 10, 14)( 11, 15)
( 20, 32)( 21, 33)( 22, 34)( 23, 35)( 24, 28)( 25, 29)( 26, 30)( 27, 31)
( 36, 48)( 37, 49)( 38, 50)( 39, 51)( 40, 44)( 41, 45)( 42, 46)( 43, 47)
( 52, 64)( 53, 65)( 54, 66)( 55, 67)( 56, 60)( 57, 61)( 58, 62)( 59, 63)
( 68, 80)( 69, 81)( 70, 82)( 71, 83)( 72, 76)( 73, 77)( 74, 78)( 75, 79)
( 84, 96)( 85, 97)( 86, 98)( 87, 99)( 88, 92)( 89, 93)( 90, 94)( 91, 95)
(100,112)(101,113)(102,114)(103,115)(104,108)(105,109)(106,110)(107,111)
(116,128)(117,129)(118,130)(119,131)(120,124)(121,125)(122,126)(123,127)
(132,144)(133,145)(134,146)(135,147)(136,140)(137,141)(138,142)(139,143);;
poly := Group([s0,s1,s2,s3,s4,s5]);;
Finitely Presented Group Representation (GAP) :
F := FreeGroup("s0","s1","s2","s3","s4","s5");;
s0 := F.1;; s1 := F.2;; s2 := F.3;; s3 := F.4;; s4 := F.5;; s5 := F.6;;
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s4*s4, s5*s5,
s0*s2*s0*s2, s1*s2*s1*s2, s0*s3*s0*s3,
s1*s3*s1*s3, s0*s4*s0*s4, s1*s4*s1*s4,
s2*s4*s2*s4, s0*s5*s0*s5, s1*s5*s1*s5,
s2*s5*s2*s5, s3*s5*s3*s5, s0*s1*s0*s1*s0*s1,
s2*s3*s2*s3*s2*s3*s2*s3, s4*s5*s4*s5*s4*s5*s4*s5,
s2*s3*s4*s3*s2*s3*s4*s2*s3, s5*s4*s3*s5*s4*s5*s4*s3*s4,
s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(147)!(2,3);
s1 := Sym(147)!(1,2);
s2 := Sym(147)!( 4, 5)( 6, 7)( 8, 9)( 10, 11)( 12, 13)( 14, 15)( 16, 17)
( 18, 19)( 20, 21)( 22, 23)( 24, 25)( 26, 27)( 28, 29)( 30, 31)( 32, 33)
( 34, 35)( 36, 37)( 38, 39)( 40, 41)( 42, 43)( 44, 45)( 46, 47)( 48, 49)
( 50, 51)( 52, 53)( 54, 55)( 56, 57)( 58, 59)( 60, 61)( 62, 63)( 64, 65)
( 66, 67)( 68, 69)( 70, 71)( 72, 73)( 74, 75)( 76, 77)( 78, 79)( 80, 81)
( 82, 83)( 84, 85)( 86, 87)( 88, 89)( 90, 91)( 92, 93)( 94, 95)( 96, 97)
( 98, 99)(100,101)(102,103)(104,105)(106,107)(108,109)(110,111)(112,113)
(114,115)(116,117)(118,119)(120,121)(122,123)(124,125)(126,127)(128,129)
(130,131)(132,133)(134,135)(136,137)(138,139)(140,141)(142,143)(144,145)
(146,147);
s3 := Sym(147)!( 5, 6)( 8, 12)( 9, 14)( 10, 13)( 11, 15)( 17, 18)( 20, 36)
( 21, 38)( 22, 37)( 23, 39)( 24, 44)( 25, 46)( 26, 45)( 27, 47)( 28, 40)
( 29, 42)( 30, 41)( 31, 43)( 32, 48)( 33, 50)( 34, 49)( 35, 51)( 52,116)
( 53,118)( 54,117)( 55,119)( 56,124)( 57,126)( 58,125)( 59,127)( 60,120)
( 61,122)( 62,121)( 63,123)( 64,128)( 65,130)( 66,129)( 67,131)( 68,100)
( 69,102)( 70,101)( 71,103)( 72,108)( 73,110)( 74,109)( 75,111)( 76,104)
( 77,106)( 78,105)( 79,107)( 80,112)( 81,114)( 82,113)( 83,115)( 84,132)
( 85,134)( 86,133)( 87,135)( 88,140)( 89,142)( 90,141)( 91,143)( 92,136)
( 93,138)( 94,137)( 95,139)( 96,144)( 97,146)( 98,145)( 99,147);
s4 := Sym(147)!( 4,100)( 5,101)( 6,103)( 7,102)( 8,112)( 9,113)( 10,115)
( 11,114)( 12,108)( 13,109)( 14,111)( 15,110)( 16,104)( 17,105)( 18,107)
( 19,106)( 20,132)( 21,133)( 22,135)( 23,134)( 24,144)( 25,145)( 26,147)
( 27,146)( 28,140)( 29,141)( 30,143)( 31,142)( 32,136)( 33,137)( 34,139)
( 35,138)( 36,116)( 37,117)( 38,119)( 39,118)( 40,128)( 41,129)( 42,131)
( 43,130)( 44,124)( 45,125)( 46,127)( 47,126)( 48,120)( 49,121)( 50,123)
( 51,122)( 54, 55)( 56, 64)( 57, 65)( 58, 67)( 59, 66)( 62, 63)( 68, 84)
( 69, 85)( 70, 87)( 71, 86)( 72, 96)( 73, 97)( 74, 99)( 75, 98)( 76, 92)
( 77, 93)( 78, 95)( 79, 94)( 80, 88)( 81, 89)( 82, 91)( 83, 90);
s5 := Sym(147)!( 4, 16)( 5, 17)( 6, 18)( 7, 19)( 8, 12)( 9, 13)( 10, 14)
( 11, 15)( 20, 32)( 21, 33)( 22, 34)( 23, 35)( 24, 28)( 25, 29)( 26, 30)
( 27, 31)( 36, 48)( 37, 49)( 38, 50)( 39, 51)( 40, 44)( 41, 45)( 42, 46)
( 43, 47)( 52, 64)( 53, 65)( 54, 66)( 55, 67)( 56, 60)( 57, 61)( 58, 62)
( 59, 63)( 68, 80)( 69, 81)( 70, 82)( 71, 83)( 72, 76)( 73, 77)( 74, 78)
( 75, 79)( 84, 96)( 85, 97)( 86, 98)( 87, 99)( 88, 92)( 89, 93)( 90, 94)
( 91, 95)(100,112)(101,113)(102,114)(103,115)(104,108)(105,109)(106,110)
(107,111)(116,128)(117,129)(118,130)(119,131)(120,124)(121,125)(122,126)
(123,127)(132,144)(133,145)(134,146)(135,147)(136,140)(137,141)(138,142)
(139,143);
poly := sub<Sym(147)|s0,s1,s2,s3,s4,s5>;
Finitely Presented Group Representation (Magma) :
poly<s0,s1,s2,s3,s4,s5> := Group< s0,s1,s2,s3,s4,s5 | s0*s0, s1*s1, s2*s2,
s3*s3, s4*s4, s5*s5, s0*s2*s0*s2, s1*s2*s1*s2,
s0*s3*s0*s3, s1*s3*s1*s3, s0*s4*s0*s4,
s1*s4*s1*s4, s2*s4*s2*s4, s0*s5*s0*s5,
s1*s5*s1*s5, s2*s5*s2*s5, s3*s5*s3*s5,
s0*s1*s0*s1*s0*s1, s2*s3*s2*s3*s2*s3*s2*s3,
s4*s5*s4*s5*s4*s5*s4*s5, s2*s3*s4*s3*s2*s3*s4*s2*s3,
s5*s4*s3*s5*s4*s5*s4*s3*s4, s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4 >;
to this polytope