Polytope of Type {3,2,6,8}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {3,2,6,8}*1728
if this polytope has a name.
Group : SmallGroup(1728,46549)
Rank : 5
Schlafli Type : {3,2,6,8}
Number of vertices, edges, etc : 3, 3, 18, 72, 24
Order of s0s1s2s3s4 : 24
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 : {3,2,6,4}*864
4-fold quotients : {3,2,6,4}*432
9-fold quotients : {3,2,2,8}*192
18-fold quotients : {3,2,2,4}*96
36-fold quotients : {3,2,2,2}*48
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := (2,3);;
s1 := (1,2);;
s2 := ( 4, 58)( 5, 60)( 6, 59)( 7, 64)( 8, 66)( 9, 65)( 10, 61)( 11, 63)
( 12, 62)( 13, 67)( 14, 69)( 15, 68)( 16, 73)( 17, 75)( 18, 74)( 19, 70)
( 20, 72)( 21, 71)( 22, 49)( 23, 51)( 24, 50)( 25, 55)( 26, 57)( 27, 56)
( 28, 52)( 29, 54)( 30, 53)( 31, 40)( 32, 42)( 33, 41)( 34, 46)( 35, 48)
( 36, 47)( 37, 43)( 38, 45)( 39, 44)( 76,130)( 77,132)( 78,131)( 79,136)
( 80,138)( 81,137)( 82,133)( 83,135)( 84,134)( 85,139)( 86,141)( 87,140)
( 88,145)( 89,147)( 90,146)( 91,142)( 92,144)( 93,143)( 94,121)( 95,123)
( 96,122)( 97,127)( 98,129)( 99,128)(100,124)(101,126)(102,125)(103,112)
(104,114)(105,113)(106,118)(107,120)(108,119)(109,115)(110,117)(111,116);;
s3 := ( 4, 7)( 5, 8)( 6, 9)( 13, 16)( 14, 17)( 15, 18)( 22, 34)( 23, 35)
( 24, 36)( 25, 31)( 26, 32)( 27, 33)( 28, 37)( 29, 38)( 30, 39)( 40, 52)
( 41, 53)( 42, 54)( 43, 49)( 44, 50)( 45, 51)( 46, 55)( 47, 56)( 48, 57)
( 58, 61)( 59, 62)( 60, 63)( 67, 70)( 68, 71)( 69, 72)( 76, 97)( 77, 98)
( 78, 99)( 79, 94)( 80, 95)( 81, 96)( 82,100)( 83,101)( 84,102)( 85,106)
( 86,107)( 87,108)( 88,103)( 89,104)( 90,105)( 91,109)( 92,110)( 93,111)
(112,142)(113,143)(114,144)(115,139)(116,140)(117,141)(118,145)(119,146)
(120,147)(121,133)(122,134)(123,135)(124,130)(125,131)(126,132)(127,136)
(128,137)(129,138);;
s4 := ( 4, 76)( 5, 79)( 6, 82)( 7, 77)( 8, 80)( 9, 83)( 10, 78)( 11, 81)
( 12, 84)( 13, 85)( 14, 88)( 15, 91)( 16, 86)( 17, 89)( 18, 92)( 19, 87)
( 20, 90)( 21, 93)( 22,103)( 23,106)( 24,109)( 25,104)( 26,107)( 27,110)
( 28,105)( 29,108)( 30,111)( 31, 94)( 32, 97)( 33,100)( 34, 95)( 35, 98)
( 36,101)( 37, 96)( 38, 99)( 39,102)( 40,121)( 41,124)( 42,127)( 43,122)
( 44,125)( 45,128)( 46,123)( 47,126)( 48,129)( 49,112)( 50,115)( 51,118)
( 52,113)( 53,116)( 54,119)( 55,114)( 56,117)( 57,120)( 58,130)( 59,133)
( 60,136)( 61,131)( 62,134)( 63,137)( 64,132)( 65,135)( 66,138)( 67,139)
( 68,142)( 69,145)( 70,140)( 71,143)( 72,146)( 73,141)( 74,144)( 75,147);;
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*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3,
s4*s2*s3*s4*s2*s3*s2*s3*s4*s3*s4*s3*s2*s3,
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, 58)( 5, 60)( 6, 59)( 7, 64)( 8, 66)( 9, 65)( 10, 61)
( 11, 63)( 12, 62)( 13, 67)( 14, 69)( 15, 68)( 16, 73)( 17, 75)( 18, 74)
( 19, 70)( 20, 72)( 21, 71)( 22, 49)( 23, 51)( 24, 50)( 25, 55)( 26, 57)
( 27, 56)( 28, 52)( 29, 54)( 30, 53)( 31, 40)( 32, 42)( 33, 41)( 34, 46)
( 35, 48)( 36, 47)( 37, 43)( 38, 45)( 39, 44)( 76,130)( 77,132)( 78,131)
( 79,136)( 80,138)( 81,137)( 82,133)( 83,135)( 84,134)( 85,139)( 86,141)
( 87,140)( 88,145)( 89,147)( 90,146)( 91,142)( 92,144)( 93,143)( 94,121)
( 95,123)( 96,122)( 97,127)( 98,129)( 99,128)(100,124)(101,126)(102,125)
(103,112)(104,114)(105,113)(106,118)(107,120)(108,119)(109,115)(110,117)
(111,116);
s3 := Sym(147)!( 4, 7)( 5, 8)( 6, 9)( 13, 16)( 14, 17)( 15, 18)( 22, 34)
( 23, 35)( 24, 36)( 25, 31)( 26, 32)( 27, 33)( 28, 37)( 29, 38)( 30, 39)
( 40, 52)( 41, 53)( 42, 54)( 43, 49)( 44, 50)( 45, 51)( 46, 55)( 47, 56)
( 48, 57)( 58, 61)( 59, 62)( 60, 63)( 67, 70)( 68, 71)( 69, 72)( 76, 97)
( 77, 98)( 78, 99)( 79, 94)( 80, 95)( 81, 96)( 82,100)( 83,101)( 84,102)
( 85,106)( 86,107)( 87,108)( 88,103)( 89,104)( 90,105)( 91,109)( 92,110)
( 93,111)(112,142)(113,143)(114,144)(115,139)(116,140)(117,141)(118,145)
(119,146)(120,147)(121,133)(122,134)(123,135)(124,130)(125,131)(126,132)
(127,136)(128,137)(129,138);
s4 := Sym(147)!( 4, 76)( 5, 79)( 6, 82)( 7, 77)( 8, 80)( 9, 83)( 10, 78)
( 11, 81)( 12, 84)( 13, 85)( 14, 88)( 15, 91)( 16, 86)( 17, 89)( 18, 92)
( 19, 87)( 20, 90)( 21, 93)( 22,103)( 23,106)( 24,109)( 25,104)( 26,107)
( 27,110)( 28,105)( 29,108)( 30,111)( 31, 94)( 32, 97)( 33,100)( 34, 95)
( 35, 98)( 36,101)( 37, 96)( 38, 99)( 39,102)( 40,121)( 41,124)( 42,127)
( 43,122)( 44,125)( 45,128)( 46,123)( 47,126)( 48,129)( 49,112)( 50,115)
( 51,118)( 52,113)( 53,116)( 54,119)( 55,114)( 56,117)( 57,120)( 58,130)
( 59,133)( 60,136)( 61,131)( 62,134)( 63,137)( 64,132)( 65,135)( 66,138)
( 67,139)( 68,142)( 69,145)( 70,140)( 71,143)( 72,146)( 73,141)( 74,144)
( 75,147);
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*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3,
s4*s2*s3*s4*s2*s3*s2*s3*s4*s3*s4*s3*s2*s3,
s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4 >;
to this polytope