Polytope of Type {4,3,6}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,3,6}*1728
Also Known As : {{4,3}3,{3,6}12}. if this polytope has another name.
Group : SmallGroup(1728,46099)
Rank : 4
Schlafli Type : {4,3,6}
Number of vertices, edges, etc : 4, 72, 108, 72
Order of s0s1s2s3 : 12
Order of s0s1s2s3s2s1 : 2
Special Properties :
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 : {4,3,6}*576
4-fold quotients : {4,3,6}*432
12-fold quotients : {4,3,6}*144
36-fold quotients : {4,3,2}*48
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := ( 1, 9)( 2, 10)( 3, 11)( 4, 12)( 5, 13)( 6, 14)( 7, 15)( 8, 16)
( 17, 25)( 18, 26)( 19, 27)( 20, 28)( 21, 29)( 22, 30)( 23, 31)( 24, 32)
( 33, 41)( 34, 42)( 35, 43)( 36, 44)( 37, 45)( 38, 46)( 39, 47)( 40, 48)
( 49, 57)( 50, 58)( 51, 59)( 52, 60)( 53, 61)( 54, 62)( 55, 63)( 56, 64)
( 65, 73)( 66, 74)( 67, 75)( 68, 76)( 69, 77)( 70, 78)( 71, 79)( 72, 80)
( 81, 89)( 82, 90)( 83, 91)( 84, 92)( 85, 93)( 86, 94)( 87, 95)( 88, 96)
( 97,105)( 98,106)( 99,107)(100,108)(101,109)(102,110)(103,111)(104,112)
(113,121)(114,122)(115,123)(116,124)(117,125)(118,126)(119,127)(120,128)
(129,137)(130,138)(131,139)(132,140)(133,141)(134,142)(135,143)(136,144);;
s1 := ( 3, 4)( 7, 8)( 9, 13)( 10, 14)( 11, 16)( 12, 15)( 19, 20)( 23, 24)
( 25, 29)( 26, 30)( 27, 32)( 28, 31)( 35, 36)( 39, 40)( 41, 45)( 42, 46)
( 43, 48)( 44, 47)( 49, 97)( 50, 98)( 51,100)( 52, 99)( 53,101)( 54,102)
( 55,104)( 56,103)( 57,109)( 58,110)( 59,112)( 60,111)( 61,105)( 62,106)
( 63,108)( 64,107)( 65,113)( 66,114)( 67,116)( 68,115)( 69,117)( 70,118)
( 71,120)( 72,119)( 73,125)( 74,126)( 75,128)( 76,127)( 77,121)( 78,122)
( 79,124)( 80,123)( 81,129)( 82,130)( 83,132)( 84,131)( 85,133)( 86,134)
( 87,136)( 88,135)( 89,141)( 90,142)( 91,144)( 92,143)( 93,137)( 94,138)
( 95,140)( 96,139);;
s2 := ( 1, 65)( 2, 67)( 3, 66)( 4, 68)( 5, 77)( 6, 79)( 7, 78)( 8, 80)
( 9, 73)( 10, 75)( 11, 74)( 12, 76)( 13, 69)( 14, 71)( 15, 70)( 16, 72)
( 17, 81)( 18, 83)( 19, 82)( 20, 84)( 21, 93)( 22, 95)( 23, 94)( 24, 96)
( 25, 89)( 26, 91)( 27, 90)( 28, 92)( 29, 85)( 30, 87)( 31, 86)( 32, 88)
( 33, 49)( 34, 51)( 35, 50)( 36, 52)( 37, 61)( 38, 63)( 39, 62)( 40, 64)
( 41, 57)( 42, 59)( 43, 58)( 44, 60)( 45, 53)( 46, 55)( 47, 54)( 48, 56)
( 98, 99)(101,109)(102,111)(103,110)(104,112)(106,107)(114,115)(117,125)
(118,127)(119,126)(120,128)(122,123)(130,131)(133,141)(134,143)(135,142)
(136,144)(138,139);;
s3 := ( 1, 2)( 5, 6)( 9, 10)( 13, 14)( 17, 34)( 18, 33)( 19, 35)( 20, 36)
( 21, 38)( 22, 37)( 23, 39)( 24, 40)( 25, 42)( 26, 41)( 27, 43)( 28, 44)
( 29, 46)( 30, 45)( 31, 47)( 32, 48)( 49, 50)( 53, 54)( 57, 58)( 61, 62)
( 65, 82)( 66, 81)( 67, 83)( 68, 84)( 69, 86)( 70, 85)( 71, 87)( 72, 88)
( 73, 90)( 74, 89)( 75, 91)( 76, 92)( 77, 94)( 78, 93)( 79, 95)( 80, 96)
( 97, 98)(101,102)(105,106)(109,110)(113,130)(114,129)(115,131)(116,132)
(117,134)(118,133)(119,135)(120,136)(121,138)(122,137)(123,139)(124,140)
(125,142)(126,141)(127,143)(128,144);;
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*s1*s2*s1*s2,
s0*s1*s0*s1*s0*s1*s0*s1, s2*s0*s1*s2*s0*s1*s2*s0*s1,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3,
s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(144)!( 1, 9)( 2, 10)( 3, 11)( 4, 12)( 5, 13)( 6, 14)( 7, 15)
( 8, 16)( 17, 25)( 18, 26)( 19, 27)( 20, 28)( 21, 29)( 22, 30)( 23, 31)
( 24, 32)( 33, 41)( 34, 42)( 35, 43)( 36, 44)( 37, 45)( 38, 46)( 39, 47)
( 40, 48)( 49, 57)( 50, 58)( 51, 59)( 52, 60)( 53, 61)( 54, 62)( 55, 63)
( 56, 64)( 65, 73)( 66, 74)( 67, 75)( 68, 76)( 69, 77)( 70, 78)( 71, 79)
( 72, 80)( 81, 89)( 82, 90)( 83, 91)( 84, 92)( 85, 93)( 86, 94)( 87, 95)
( 88, 96)( 97,105)( 98,106)( 99,107)(100,108)(101,109)(102,110)(103,111)
(104,112)(113,121)(114,122)(115,123)(116,124)(117,125)(118,126)(119,127)
(120,128)(129,137)(130,138)(131,139)(132,140)(133,141)(134,142)(135,143)
(136,144);
s1 := Sym(144)!( 3, 4)( 7, 8)( 9, 13)( 10, 14)( 11, 16)( 12, 15)( 19, 20)
( 23, 24)( 25, 29)( 26, 30)( 27, 32)( 28, 31)( 35, 36)( 39, 40)( 41, 45)
( 42, 46)( 43, 48)( 44, 47)( 49, 97)( 50, 98)( 51,100)( 52, 99)( 53,101)
( 54,102)( 55,104)( 56,103)( 57,109)( 58,110)( 59,112)( 60,111)( 61,105)
( 62,106)( 63,108)( 64,107)( 65,113)( 66,114)( 67,116)( 68,115)( 69,117)
( 70,118)( 71,120)( 72,119)( 73,125)( 74,126)( 75,128)( 76,127)( 77,121)
( 78,122)( 79,124)( 80,123)( 81,129)( 82,130)( 83,132)( 84,131)( 85,133)
( 86,134)( 87,136)( 88,135)( 89,141)( 90,142)( 91,144)( 92,143)( 93,137)
( 94,138)( 95,140)( 96,139);
s2 := Sym(144)!( 1, 65)( 2, 67)( 3, 66)( 4, 68)( 5, 77)( 6, 79)( 7, 78)
( 8, 80)( 9, 73)( 10, 75)( 11, 74)( 12, 76)( 13, 69)( 14, 71)( 15, 70)
( 16, 72)( 17, 81)( 18, 83)( 19, 82)( 20, 84)( 21, 93)( 22, 95)( 23, 94)
( 24, 96)( 25, 89)( 26, 91)( 27, 90)( 28, 92)( 29, 85)( 30, 87)( 31, 86)
( 32, 88)( 33, 49)( 34, 51)( 35, 50)( 36, 52)( 37, 61)( 38, 63)( 39, 62)
( 40, 64)( 41, 57)( 42, 59)( 43, 58)( 44, 60)( 45, 53)( 46, 55)( 47, 54)
( 48, 56)( 98, 99)(101,109)(102,111)(103,110)(104,112)(106,107)(114,115)
(117,125)(118,127)(119,126)(120,128)(122,123)(130,131)(133,141)(134,143)
(135,142)(136,144)(138,139);
s3 := Sym(144)!( 1, 2)( 5, 6)( 9, 10)( 13, 14)( 17, 34)( 18, 33)( 19, 35)
( 20, 36)( 21, 38)( 22, 37)( 23, 39)( 24, 40)( 25, 42)( 26, 41)( 27, 43)
( 28, 44)( 29, 46)( 30, 45)( 31, 47)( 32, 48)( 49, 50)( 53, 54)( 57, 58)
( 61, 62)( 65, 82)( 66, 81)( 67, 83)( 68, 84)( 69, 86)( 70, 85)( 71, 87)
( 72, 88)( 73, 90)( 74, 89)( 75, 91)( 76, 92)( 77, 94)( 78, 93)( 79, 95)
( 80, 96)( 97, 98)(101,102)(105,106)(109,110)(113,130)(114,129)(115,131)
(116,132)(117,134)(118,133)(119,135)(120,136)(121,138)(122,137)(123,139)
(124,140)(125,142)(126,141)(127,143)(128,144);
poly := sub<Sym(144)|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*s1*s2*s1*s2, s0*s1*s0*s1*s0*s1*s0*s1,
s2*s0*s1*s2*s0*s1*s2*s0*s1, s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3,
s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2 >;
References : None.
to this polytope