Polytope of Type {4,12,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,12,6}*1728k
if this polytope has a name.
Group : SmallGroup(1728,46099)
Rank : 4
Schlafli Type : {4,12,6}
Number of vertices, edges, etc : 4, 72, 108, 18
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 : {4,12,6}*576h
   4-fold quotients : {4,6,6}*432
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 := (  1,  3)(  2,  4)(  5,  7)(  6,  8)(  9, 15)( 10, 16)( 11, 13)( 12, 14)
( 17, 35)( 18, 36)( 19, 33)( 20, 34)( 21, 39)( 22, 40)( 23, 37)( 24, 38)
( 25, 47)( 26, 48)( 27, 45)( 28, 46)( 29, 43)( 30, 44)( 31, 41)( 32, 42)
( 49, 99)( 50,100)( 51, 97)( 52, 98)( 53,103)( 54,104)( 55,101)( 56,102)
( 57,111)( 58,112)( 59,109)( 60,110)( 61,107)( 62,108)( 63,105)( 64,106)
( 65,131)( 66,132)( 67,129)( 68,130)( 69,135)( 70,136)( 71,133)( 72,134)
( 73,143)( 74,144)( 75,141)( 76,142)( 77,139)( 78,140)( 79,137)( 80,138)
( 81,115)( 82,116)( 83,113)( 84,114)( 85,119)( 86,120)( 87,117)( 88,118)
( 89,127)( 90,128)( 91,125)( 92,126)( 93,123)( 94,124)( 95,121)( 96,122);;
s2 := (  1, 49)(  2, 50)(  3, 52)(  4, 51)(  5, 61)(  6, 62)(  7, 64)(  8, 63)
(  9, 57)( 10, 58)( 11, 60)( 12, 59)( 13, 53)( 14, 54)( 15, 56)( 16, 55)
( 17, 65)( 18, 66)( 19, 68)( 20, 67)( 21, 77)( 22, 78)( 23, 80)( 24, 79)
( 25, 73)( 26, 74)( 27, 76)( 28, 75)( 29, 69)( 30, 70)( 31, 72)( 32, 71)
( 33, 81)( 34, 82)( 35, 84)( 36, 83)( 37, 93)( 38, 94)( 39, 96)( 40, 95)
( 41, 89)( 42, 90)( 43, 92)( 44, 91)( 45, 85)( 46, 86)( 47, 88)( 48, 87)
( 99,100)(101,109)(102,110)(103,112)(104,111)(107,108)(115,116)(117,125)
(118,126)(119,128)(120,127)(123,124)(131,132)(133,141)(134,142)(135,144)
(136,143)(139,140);;
s3 := (  2,  4)(  6,  8)( 10, 12)( 14, 16)( 17, 33)( 18, 36)( 19, 35)( 20, 34)
( 21, 37)( 22, 40)( 23, 39)( 24, 38)( 25, 41)( 26, 44)( 27, 43)( 28, 42)
( 29, 45)( 30, 48)( 31, 47)( 32, 46)( 49, 65)( 50, 68)( 51, 67)( 52, 66)
( 53, 69)( 54, 72)( 55, 71)( 56, 70)( 57, 73)( 58, 76)( 59, 75)( 60, 74)
( 61, 77)( 62, 80)( 63, 79)( 64, 78)( 82, 84)( 86, 88)( 90, 92)( 94, 96)
( 97,129)( 98,132)( 99,131)(100,130)(101,133)(102,136)(103,135)(104,134)
(105,137)(106,140)(107,139)(108,138)(109,141)(110,144)(111,143)(112,142)
(114,116)(118,120)(122,124)(126,128);;
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, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*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)!(  1,  3)(  2,  4)(  5,  7)(  6,  8)(  9, 15)( 10, 16)( 11, 13)
( 12, 14)( 17, 35)( 18, 36)( 19, 33)( 20, 34)( 21, 39)( 22, 40)( 23, 37)
( 24, 38)( 25, 47)( 26, 48)( 27, 45)( 28, 46)( 29, 43)( 30, 44)( 31, 41)
( 32, 42)( 49, 99)( 50,100)( 51, 97)( 52, 98)( 53,103)( 54,104)( 55,101)
( 56,102)( 57,111)( 58,112)( 59,109)( 60,110)( 61,107)( 62,108)( 63,105)
( 64,106)( 65,131)( 66,132)( 67,129)( 68,130)( 69,135)( 70,136)( 71,133)
( 72,134)( 73,143)( 74,144)( 75,141)( 76,142)( 77,139)( 78,140)( 79,137)
( 80,138)( 81,115)( 82,116)( 83,113)( 84,114)( 85,119)( 86,120)( 87,117)
( 88,118)( 89,127)( 90,128)( 91,125)( 92,126)( 93,123)( 94,124)( 95,121)
( 96,122);
s2 := Sym(144)!(  1, 49)(  2, 50)(  3, 52)(  4, 51)(  5, 61)(  6, 62)(  7, 64)
(  8, 63)(  9, 57)( 10, 58)( 11, 60)( 12, 59)( 13, 53)( 14, 54)( 15, 56)
( 16, 55)( 17, 65)( 18, 66)( 19, 68)( 20, 67)( 21, 77)( 22, 78)( 23, 80)
( 24, 79)( 25, 73)( 26, 74)( 27, 76)( 28, 75)( 29, 69)( 30, 70)( 31, 72)
( 32, 71)( 33, 81)( 34, 82)( 35, 84)( 36, 83)( 37, 93)( 38, 94)( 39, 96)
( 40, 95)( 41, 89)( 42, 90)( 43, 92)( 44, 91)( 45, 85)( 46, 86)( 47, 88)
( 48, 87)( 99,100)(101,109)(102,110)(103,112)(104,111)(107,108)(115,116)
(117,125)(118,126)(119,128)(120,127)(123,124)(131,132)(133,141)(134,142)
(135,144)(136,143)(139,140);
s3 := Sym(144)!(  2,  4)(  6,  8)( 10, 12)( 14, 16)( 17, 33)( 18, 36)( 19, 35)
( 20, 34)( 21, 37)( 22, 40)( 23, 39)( 24, 38)( 25, 41)( 26, 44)( 27, 43)
( 28, 42)( 29, 45)( 30, 48)( 31, 47)( 32, 46)( 49, 65)( 50, 68)( 51, 67)
( 52, 66)( 53, 69)( 54, 72)( 55, 71)( 56, 70)( 57, 73)( 58, 76)( 59, 75)
( 60, 74)( 61, 77)( 62, 80)( 63, 79)( 64, 78)( 82, 84)( 86, 88)( 90, 92)
( 94, 96)( 97,129)( 98,132)( 99,131)(100,130)(101,133)(102,136)(103,135)
(104,134)(105,137)(106,140)(107,139)(108,138)(109,141)(110,144)(111,143)
(112,142)(114,116)(118,120)(122,124)(126,128);
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, s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 >;

References : None.
to this polytope