Polytope of Type {3,2,8,9}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {3,2,8,9}*1728
if this polytope has a name.
Group : SmallGroup(1728,30284)
Rank : 5
Schlafli Type : {3,2,8,9}
Number of vertices, edges, etc : 3, 3, 16, 72, 18
Order of s0s1s2s3s4 : 36
Order of s0s1s2s3s4s3s2s1 : 2
Special Properties :
Degenerate
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) :
2-fold quotients : {3,2,4,9}*864
3-fold quotients : {3,2,8,3}*576
4-fold quotients : {3,2,4,9}*432
6-fold quotients : {3,2,4,3}*288
8-fold quotients : {3,2,2,9}*216
12-fold quotients : {3,2,4,3}*144
24-fold quotients : {3,2,2,3}*72
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := (2,3);;
s1 := (1,2);;
s2 := ( 4, 78)( 5, 79)( 6, 77)( 7, 76)( 8, 82)( 9, 83)( 10, 81)( 11, 80)
( 12, 86)( 13, 87)( 14, 85)( 15, 84)( 16, 90)( 17, 91)( 18, 89)( 19, 88)
( 20, 94)( 21, 95)( 22, 93)( 23, 92)( 24, 98)( 25, 99)( 26, 97)( 27, 96)
( 28,102)( 29,103)( 30,101)( 31,100)( 32,106)( 33,107)( 34,105)( 35,104)
( 36,110)( 37,111)( 38,109)( 39,108)( 40,114)( 41,115)( 42,113)( 43,112)
( 44,118)( 45,119)( 46,117)( 47,116)( 48,122)( 49,123)( 50,121)( 51,120)
( 52,126)( 53,127)( 54,125)( 55,124)( 56,130)( 57,131)( 58,129)( 59,128)
( 60,134)( 61,135)( 62,133)( 63,132)( 64,138)( 65,139)( 66,137)( 67,136)
( 68,142)( 69,143)( 70,141)( 71,140)( 72,146)( 73,147)( 74,145)( 75,144);;
s3 := ( 6, 8)( 7, 9)( 10, 11)( 12, 20)( 13, 21)( 14, 24)( 15, 25)( 16, 22)
( 17, 23)( 18, 27)( 19, 26)( 28, 60)( 29, 61)( 30, 64)( 31, 65)( 32, 62)
( 33, 63)( 34, 67)( 35, 66)( 36, 52)( 37, 53)( 38, 56)( 39, 57)( 40, 54)
( 41, 55)( 42, 59)( 43, 58)( 44, 68)( 45, 69)( 46, 72)( 47, 73)( 48, 70)
( 49, 71)( 50, 75)( 51, 74)( 76, 77)( 78, 81)( 79, 80)( 84, 93)( 85, 92)
( 86, 97)( 87, 96)( 88, 95)( 89, 94)( 90, 98)( 91, 99)(100,133)(101,132)
(102,137)(103,136)(104,135)(105,134)(106,138)(107,139)(108,125)(109,124)
(110,129)(111,128)(112,127)(113,126)(114,130)(115,131)(116,141)(117,140)
(118,145)(119,144)(120,143)(121,142)(122,146)(123,147);;
s4 := ( 4, 28)( 5, 29)( 6, 31)( 7, 30)( 8, 34)( 9, 35)( 10, 32)( 11, 33)
( 12, 44)( 13, 45)( 14, 47)( 15, 46)( 16, 50)( 17, 51)( 18, 48)( 19, 49)
( 20, 36)( 21, 37)( 22, 39)( 23, 38)( 24, 42)( 25, 43)( 26, 40)( 27, 41)
( 52, 60)( 53, 61)( 54, 63)( 55, 62)( 56, 66)( 57, 67)( 58, 64)( 59, 65)
( 70, 71)( 72, 74)( 73, 75)( 76,101)( 77,100)( 78,102)( 79,103)( 80,107)
( 81,106)( 82,105)( 83,104)( 84,117)( 85,116)( 86,118)( 87,119)( 88,123)
( 89,122)( 90,121)( 91,120)( 92,109)( 93,108)( 94,110)( 95,111)( 96,115)
( 97,114)( 98,113)( 99,112)(124,133)(125,132)(126,134)(127,135)(128,139)
(129,138)(130,137)(131,136)(140,141)(144,147)(145,146);;
poly := Group([s0,s1,s2,s3,s4]);;
Finitely Presented Group Representation (GAP) :
F := FreeGroup("s0","s1","s2","s3","s4");;
s0 := F.1;; s1 := F.2;; s2 := F.3;; s3 := F.4;; s4 := F.5;;
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s4*s4, 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*s1*s0*s1*s0*s1, s2*s3*s4*s3*s4*s3*s2*s3*s4*s3*s4*s3,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3,
s2*s3*s4*s3*s2*s3*s2*s3*s2*s3*s4*s3*s2*s3*s2*s3,
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, 78)( 5, 79)( 6, 77)( 7, 76)( 8, 82)( 9, 83)( 10, 81)
( 11, 80)( 12, 86)( 13, 87)( 14, 85)( 15, 84)( 16, 90)( 17, 91)( 18, 89)
( 19, 88)( 20, 94)( 21, 95)( 22, 93)( 23, 92)( 24, 98)( 25, 99)( 26, 97)
( 27, 96)( 28,102)( 29,103)( 30,101)( 31,100)( 32,106)( 33,107)( 34,105)
( 35,104)( 36,110)( 37,111)( 38,109)( 39,108)( 40,114)( 41,115)( 42,113)
( 43,112)( 44,118)( 45,119)( 46,117)( 47,116)( 48,122)( 49,123)( 50,121)
( 51,120)( 52,126)( 53,127)( 54,125)( 55,124)( 56,130)( 57,131)( 58,129)
( 59,128)( 60,134)( 61,135)( 62,133)( 63,132)( 64,138)( 65,139)( 66,137)
( 67,136)( 68,142)( 69,143)( 70,141)( 71,140)( 72,146)( 73,147)( 74,145)
( 75,144);
s3 := Sym(147)!( 6, 8)( 7, 9)( 10, 11)( 12, 20)( 13, 21)( 14, 24)( 15, 25)
( 16, 22)( 17, 23)( 18, 27)( 19, 26)( 28, 60)( 29, 61)( 30, 64)( 31, 65)
( 32, 62)( 33, 63)( 34, 67)( 35, 66)( 36, 52)( 37, 53)( 38, 56)( 39, 57)
( 40, 54)( 41, 55)( 42, 59)( 43, 58)( 44, 68)( 45, 69)( 46, 72)( 47, 73)
( 48, 70)( 49, 71)( 50, 75)( 51, 74)( 76, 77)( 78, 81)( 79, 80)( 84, 93)
( 85, 92)( 86, 97)( 87, 96)( 88, 95)( 89, 94)( 90, 98)( 91, 99)(100,133)
(101,132)(102,137)(103,136)(104,135)(105,134)(106,138)(107,139)(108,125)
(109,124)(110,129)(111,128)(112,127)(113,126)(114,130)(115,131)(116,141)
(117,140)(118,145)(119,144)(120,143)(121,142)(122,146)(123,147);
s4 := Sym(147)!( 4, 28)( 5, 29)( 6, 31)( 7, 30)( 8, 34)( 9, 35)( 10, 32)
( 11, 33)( 12, 44)( 13, 45)( 14, 47)( 15, 46)( 16, 50)( 17, 51)( 18, 48)
( 19, 49)( 20, 36)( 21, 37)( 22, 39)( 23, 38)( 24, 42)( 25, 43)( 26, 40)
( 27, 41)( 52, 60)( 53, 61)( 54, 63)( 55, 62)( 56, 66)( 57, 67)( 58, 64)
( 59, 65)( 70, 71)( 72, 74)( 73, 75)( 76,101)( 77,100)( 78,102)( 79,103)
( 80,107)( 81,106)( 82,105)( 83,104)( 84,117)( 85,116)( 86,118)( 87,119)
( 88,123)( 89,122)( 90,121)( 91,120)( 92,109)( 93,108)( 94,110)( 95,111)
( 96,115)( 97,114)( 98,113)( 99,112)(124,133)(125,132)(126,134)(127,135)
(128,139)(129,138)(130,137)(131,136)(140,141)(144,147)(145,146);
poly := sub<Sym(147)|s0,s1,s2,s3,s4>;
Finitely Presented Group Representation (Magma) :
poly<s0,s1,s2,s3,s4> := Group< s0,s1,s2,s3,s4 | s0*s0, s1*s1, s2*s2,
s3*s3, s4*s4, 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*s1*s0*s1*s0*s1,
s2*s3*s4*s3*s4*s3*s2*s3*s4*s3*s4*s3,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3,
s2*s3*s4*s3*s2*s3*s2*s3*s2*s3*s4*s3*s2*s3*s2*s3,
s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4 >;
to this polytope