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Polytope of Type {4,6,12}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,6,12}*1728i
if this polytope has a name.
Group : SmallGroup(1728,46119)
Rank : 4
Schlafli Type : {4,6,12}
Number of vertices, edges, etc : 4, 36, 108, 36
Order of s0s1s2s3 : 3
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 : {4,6,12}*576f
   4-fold quotients : {4,6,6}*432
   9-fold quotients : {4,6,4}*192e
   18-fold quotients : {4,3,4}*96
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)( 17, 33)( 18, 34)
( 19, 36)( 20, 35)( 21, 37)( 22, 38)( 23, 40)( 24, 39)( 25, 45)( 26, 46)
( 27, 48)( 28, 47)( 29, 41)( 30, 42)( 31, 44)( 32, 43)( 51, 52)( 55, 56)
( 57, 61)( 58, 62)( 59, 64)( 60, 63)( 65, 81)( 66, 82)( 67, 84)( 68, 83)
( 69, 85)( 70, 86)( 71, 88)( 72, 87)( 73, 93)( 74, 94)( 75, 96)( 76, 95)
( 77, 89)( 78, 90)( 79, 92)( 80, 91)( 99,100)(103,104)(105,109)(106,110)
(107,112)(108,111)(113,129)(114,130)(115,132)(116,131)(117,133)(118,134)
(119,136)(120,135)(121,141)(122,142)(123,144)(124,143)(125,137)(126,138)
(127,140)(128,139);;
s2 := (  2,  4)(  5, 13)(  6, 16)(  7, 15)(  8, 14)( 10, 12)( 18, 20)( 21, 29)
( 22, 32)( 23, 31)( 24, 30)( 26, 28)( 34, 36)( 37, 45)( 38, 48)( 39, 47)
( 40, 46)( 42, 44)( 49,129)( 50,132)( 51,131)( 52,130)( 53,141)( 54,144)
( 55,143)( 56,142)( 57,137)( 58,140)( 59,139)( 60,138)( 61,133)( 62,136)
( 63,135)( 64,134)( 65, 97)( 66,100)( 67, 99)( 68, 98)( 69,109)( 70,112)
( 71,111)( 72,110)( 73,105)( 74,108)( 75,107)( 76,106)( 77,101)( 78,104)
( 79,103)( 80,102)( 81,113)( 82,116)( 83,115)( 84,114)( 85,125)( 86,128)
( 87,127)( 88,126)( 89,121)( 90,124)( 91,123)( 92,122)( 93,117)( 94,120)
( 95,119)( 96,118);;
s3 := (  1, 50)(  2, 49)(  3, 52)(  4, 51)(  5, 54)(  6, 53)(  7, 56)(  8, 55)
(  9, 58)( 10, 57)( 11, 60)( 12, 59)( 13, 62)( 14, 61)( 15, 64)( 16, 63)
( 17, 82)( 18, 81)( 19, 84)( 20, 83)( 21, 86)( 22, 85)( 23, 88)( 24, 87)
( 25, 90)( 26, 89)( 27, 92)( 28, 91)( 29, 94)( 30, 93)( 31, 96)( 32, 95)
( 33, 66)( 34, 65)( 35, 68)( 36, 67)( 37, 70)( 38, 69)( 39, 72)( 40, 71)
( 41, 74)( 42, 73)( 43, 76)( 44, 75)( 45, 78)( 46, 77)( 47, 80)( 48, 79)
( 97, 98)( 99,100)(101,102)(103,104)(105,106)(107,108)(109,110)(111,112)
(113,130)(114,129)(115,132)(116,131)(117,134)(118,133)(119,136)(120,135)
(121,138)(122,137)(123,140)(124,139)(125,142)(126,141)(127,144)(128,143);;
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, s0*s1*s0*s1*s0*s1*s0*s1, 
s0*s1*s2*s1*s0*s1*s2*s0*s1, s1*s2*s3*s1*s2*s3*s1*s2*s3, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s2*s3*s2*s3*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(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)( 17, 33)
( 18, 34)( 19, 36)( 20, 35)( 21, 37)( 22, 38)( 23, 40)( 24, 39)( 25, 45)
( 26, 46)( 27, 48)( 28, 47)( 29, 41)( 30, 42)( 31, 44)( 32, 43)( 51, 52)
( 55, 56)( 57, 61)( 58, 62)( 59, 64)( 60, 63)( 65, 81)( 66, 82)( 67, 84)
( 68, 83)( 69, 85)( 70, 86)( 71, 88)( 72, 87)( 73, 93)( 74, 94)( 75, 96)
( 76, 95)( 77, 89)( 78, 90)( 79, 92)( 80, 91)( 99,100)(103,104)(105,109)
(106,110)(107,112)(108,111)(113,129)(114,130)(115,132)(116,131)(117,133)
(118,134)(119,136)(120,135)(121,141)(122,142)(123,144)(124,143)(125,137)
(126,138)(127,140)(128,139);
s2 := Sym(144)!(  2,  4)(  5, 13)(  6, 16)(  7, 15)(  8, 14)( 10, 12)( 18, 20)
( 21, 29)( 22, 32)( 23, 31)( 24, 30)( 26, 28)( 34, 36)( 37, 45)( 38, 48)
( 39, 47)( 40, 46)( 42, 44)( 49,129)( 50,132)( 51,131)( 52,130)( 53,141)
( 54,144)( 55,143)( 56,142)( 57,137)( 58,140)( 59,139)( 60,138)( 61,133)
( 62,136)( 63,135)( 64,134)( 65, 97)( 66,100)( 67, 99)( 68, 98)( 69,109)
( 70,112)( 71,111)( 72,110)( 73,105)( 74,108)( 75,107)( 76,106)( 77,101)
( 78,104)( 79,103)( 80,102)( 81,113)( 82,116)( 83,115)( 84,114)( 85,125)
( 86,128)( 87,127)( 88,126)( 89,121)( 90,124)( 91,123)( 92,122)( 93,117)
( 94,120)( 95,119)( 96,118);
s3 := Sym(144)!(  1, 50)(  2, 49)(  3, 52)(  4, 51)(  5, 54)(  6, 53)(  7, 56)
(  8, 55)(  9, 58)( 10, 57)( 11, 60)( 12, 59)( 13, 62)( 14, 61)( 15, 64)
( 16, 63)( 17, 82)( 18, 81)( 19, 84)( 20, 83)( 21, 86)( 22, 85)( 23, 88)
( 24, 87)( 25, 90)( 26, 89)( 27, 92)( 28, 91)( 29, 94)( 30, 93)( 31, 96)
( 32, 95)( 33, 66)( 34, 65)( 35, 68)( 36, 67)( 37, 70)( 38, 69)( 39, 72)
( 40, 71)( 41, 74)( 42, 73)( 43, 76)( 44, 75)( 45, 78)( 46, 77)( 47, 80)
( 48, 79)( 97, 98)( 99,100)(101,102)(103,104)(105,106)(107,108)(109,110)
(111,112)(113,130)(114,129)(115,132)(116,131)(117,134)(118,133)(119,136)
(120,135)(121,138)(122,137)(123,140)(124,139)(125,142)(126,141)(127,144)
(128,143);
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, 
s0*s1*s0*s1*s0*s1*s0*s1, s0*s1*s2*s1*s0*s1*s2*s0*s1, 
s1*s2*s3*s1*s2*s3*s1*s2*s3, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 >; 
 
References : None.
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