Polytope of Type {12,4,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {12,4,6}*1152a
if this polytope has a name.
Group : SmallGroup(1152,99266)
Rank : 4
Schlafli Type : {12,4,6}
Number of vertices, edges, etc : 24, 48, 24, 6
Order of s₀s₁s₂s₃ : 12
Order of s₀s₁s₂s₃s₂s₁ : 2
Special Properties :
   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 : {12,4,6}*576
   3-fold quotients : {12,4,2}*384a, {4,4,6}*384a
   4-fold quotients : {12,2,6}*288, {6,4,6}*288
   6-fold quotients : {12,4,2}*192a, {4,4,6}*192
   8-fold quotients : {12,2,3}*144, {6,2,6}*144
   9-fold quotients : {4,4,2}*128
   12-fold quotients : {12,2,2}*96, {2,4,6}*96a, {4,2,6}*96, {6,4,2}*96a
   16-fold quotients : {3,2,6}*72, {6,2,3}*72
   18-fold quotients : {4,4,2}*64
   24-fold quotients : {4,2,3}*48, {2,2,6}*48, {6,2,2}*48
   32-fold quotients : {3,2,3}*36
   36-fold quotients : {2,4,2}*32, {4,2,2}*32
   48-fold quotients : {2,2,3}*24, {3,2,2}*24
   72-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

s0 := (  1,109)(  2,116)(  3,114)(  4,115)(  5,113)(  6,111)(  7,112)(  8,110)
(  9,117)( 10,118)( 11,125)( 12,123)( 13,124)( 14,122)( 15,120)( 16,121)
( 17,119)( 18,126)( 19,127)( 20,134)( 21,132)( 22,133)( 23,131)( 24,129)
( 25,130)( 26,128)( 27,135)( 28,136)( 29,143)( 30,141)( 31,142)( 32,140)
( 33,138)( 34,139)( 35,137)( 36,144)( 37, 73)( 38, 80)( 39, 78)( 40, 79)
( 41, 77)( 42, 75)( 43, 76)( 44, 74)( 45, 81)( 46, 82)( 47, 89)( 48, 87)
( 49, 88)( 50, 86)( 51, 84)( 52, 85)( 53, 83)( 54, 90)( 55, 91)( 56, 98)
( 57, 96)( 58, 97)( 59, 95)( 60, 93)( 61, 94)( 62, 92)( 63, 99)( 64,100)
( 65,107)( 66,105)( 67,106)( 68,104)( 69,102)( 70,103)( 71,101)( 72,108);;
s1 := (  1, 40)(  2, 38)(  3, 45)(  4, 37)(  5, 44)(  6, 42)(  7, 43)(  8, 41)
(  9, 39)( 10, 49)( 11, 47)( 12, 54)( 13, 46)( 14, 53)( 15, 51)( 16, 52)
( 17, 50)( 18, 48)( 19, 58)( 20, 56)( 21, 63)( 22, 55)( 23, 62)( 24, 60)
( 25, 61)( 26, 59)( 27, 57)( 28, 67)( 29, 65)( 30, 72)( 31, 64)( 32, 71)
( 33, 69)( 34, 70)( 35, 68)( 36, 66)( 73,130)( 74,128)( 75,135)( 76,127)
( 77,134)( 78,132)( 79,133)( 80,131)( 81,129)( 82,139)( 83,137)( 84,144)
( 85,136)( 86,143)( 87,141)( 88,142)( 89,140)( 90,138)( 91,112)( 92,110)
( 93,117)( 94,109)( 95,116)( 96,114)( 97,115)( 98,113)( 99,111)(100,121)
(101,119)(102,126)(103,118)(104,125)(105,123)(106,124)(107,122)(108,120);;
s2 := (  1, 37)(  2, 42)(  3, 44)(  4, 40)(  5, 45)(  6, 38)(  7, 43)(  8, 39)
(  9, 41)( 10, 46)( 11, 51)( 12, 53)( 13, 49)( 14, 54)( 15, 47)( 16, 52)
( 17, 48)( 18, 50)( 19, 64)( 20, 69)( 21, 71)( 22, 67)( 23, 72)( 24, 65)
( 25, 70)( 26, 66)( 27, 68)( 28, 55)( 29, 60)( 30, 62)( 31, 58)( 32, 63)
( 33, 56)( 34, 61)( 35, 57)( 36, 59)( 73,109)( 74,114)( 75,116)( 76,112)
( 77,117)( 78,110)( 79,115)( 80,111)( 81,113)( 82,118)( 83,123)( 84,125)
( 85,121)( 86,126)( 87,119)( 88,124)( 89,120)( 90,122)( 91,136)( 92,141)
( 93,143)( 94,139)( 95,144)( 96,137)( 97,142)( 98,138)( 99,140)(100,127)
(101,132)(102,134)(103,130)(104,135)(105,128)(106,133)(107,129)(108,131);;
s3 := (  1, 41)(  2, 43)(  3, 39)(  4, 44)(  5, 37)(  6, 42)(  7, 38)(  8, 40)
(  9, 45)( 10, 50)( 11, 52)( 12, 48)( 13, 53)( 14, 46)( 15, 51)( 16, 47)
( 17, 49)( 18, 54)( 19, 59)( 20, 61)( 21, 57)( 22, 62)( 23, 55)( 24, 60)
( 25, 56)( 26, 58)( 27, 63)( 28, 68)( 29, 70)( 30, 66)( 31, 71)( 32, 64)
( 33, 69)( 34, 65)( 35, 67)( 36, 72)( 73,113)( 74,115)( 75,111)( 76,116)
( 77,109)( 78,114)( 79,110)( 80,112)( 81,117)( 82,122)( 83,124)( 84,120)
( 85,125)( 86,118)( 87,123)( 88,119)( 89,121)( 90,126)( 91,131)( 92,133)
( 93,129)( 94,134)( 95,127)( 96,132)( 97,128)( 98,130)( 99,135)(100,140)
(101,142)(102,138)(103,143)(104,136)(105,141)(106,137)(107,139)(108,144);;
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*s1*s2, 
s1*s2*s3*s2*s1*s2*s3*s2, s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
s0*s2*s1*s2*s1*s0*s1*s0*s2*s1*s2*s1*s0*s1, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(144)!(  1,109)(  2,116)(  3,114)(  4,115)(  5,113)(  6,111)(  7,112)
(  8,110)(  9,117)( 10,118)( 11,125)( 12,123)( 13,124)( 14,122)( 15,120)
( 16,121)( 17,119)( 18,126)( 19,127)( 20,134)( 21,132)( 22,133)( 23,131)
( 24,129)( 25,130)( 26,128)( 27,135)( 28,136)( 29,143)( 30,141)( 31,142)
( 32,140)( 33,138)( 34,139)( 35,137)( 36,144)( 37, 73)( 38, 80)( 39, 78)
( 40, 79)( 41, 77)( 42, 75)( 43, 76)( 44, 74)( 45, 81)( 46, 82)( 47, 89)
( 48, 87)( 49, 88)( 50, 86)( 51, 84)( 52, 85)( 53, 83)( 54, 90)( 55, 91)
( 56, 98)( 57, 96)( 58, 97)( 59, 95)( 60, 93)( 61, 94)( 62, 92)( 63, 99)
( 64,100)( 65,107)( 66,105)( 67,106)( 68,104)( 69,102)( 70,103)( 71,101)
( 72,108);
s1 := Sym(144)!(  1, 40)(  2, 38)(  3, 45)(  4, 37)(  5, 44)(  6, 42)(  7, 43)
(  8, 41)(  9, 39)( 10, 49)( 11, 47)( 12, 54)( 13, 46)( 14, 53)( 15, 51)
( 16, 52)( 17, 50)( 18, 48)( 19, 58)( 20, 56)( 21, 63)( 22, 55)( 23, 62)
( 24, 60)( 25, 61)( 26, 59)( 27, 57)( 28, 67)( 29, 65)( 30, 72)( 31, 64)
( 32, 71)( 33, 69)( 34, 70)( 35, 68)( 36, 66)( 73,130)( 74,128)( 75,135)
( 76,127)( 77,134)( 78,132)( 79,133)( 80,131)( 81,129)( 82,139)( 83,137)
( 84,144)( 85,136)( 86,143)( 87,141)( 88,142)( 89,140)( 90,138)( 91,112)
( 92,110)( 93,117)( 94,109)( 95,116)( 96,114)( 97,115)( 98,113)( 99,111)
(100,121)(101,119)(102,126)(103,118)(104,125)(105,123)(106,124)(107,122)
(108,120);
s2 := Sym(144)!(  1, 37)(  2, 42)(  3, 44)(  4, 40)(  5, 45)(  6, 38)(  7, 43)
(  8, 39)(  9, 41)( 10, 46)( 11, 51)( 12, 53)( 13, 49)( 14, 54)( 15, 47)
( 16, 52)( 17, 48)( 18, 50)( 19, 64)( 20, 69)( 21, 71)( 22, 67)( 23, 72)
( 24, 65)( 25, 70)( 26, 66)( 27, 68)( 28, 55)( 29, 60)( 30, 62)( 31, 58)
( 32, 63)( 33, 56)( 34, 61)( 35, 57)( 36, 59)( 73,109)( 74,114)( 75,116)
( 76,112)( 77,117)( 78,110)( 79,115)( 80,111)( 81,113)( 82,118)( 83,123)
( 84,125)( 85,121)( 86,126)( 87,119)( 88,124)( 89,120)( 90,122)( 91,136)
( 92,141)( 93,143)( 94,139)( 95,144)( 96,137)( 97,142)( 98,138)( 99,140)
(100,127)(101,132)(102,134)(103,130)(104,135)(105,128)(106,133)(107,129)
(108,131);
s3 := Sym(144)!(  1, 41)(  2, 43)(  3, 39)(  4, 44)(  5, 37)(  6, 42)(  7, 38)
(  8, 40)(  9, 45)( 10, 50)( 11, 52)( 12, 48)( 13, 53)( 14, 46)( 15, 51)
( 16, 47)( 17, 49)( 18, 54)( 19, 59)( 20, 61)( 21, 57)( 22, 62)( 23, 55)
( 24, 60)( 25, 56)( 26, 58)( 27, 63)( 28, 68)( 29, 70)( 30, 66)( 31, 71)
( 32, 64)( 33, 69)( 34, 65)( 35, 67)( 36, 72)( 73,113)( 74,115)( 75,111)
( 76,116)( 77,109)( 78,114)( 79,110)( 80,112)( 81,117)( 82,122)( 83,124)
( 84,120)( 85,125)( 86,118)( 87,123)( 88,119)( 89,121)( 90,126)( 91,131)
( 92,133)( 93,129)( 94,134)( 95,127)( 96,132)( 97,128)( 98,130)( 99,135)
(100,140)(101,142)(102,138)(103,143)(104,136)(105,141)(106,137)(107,139)
(108,144);
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*s1*s2, s1*s2*s3*s2*s1*s2*s3*s2, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
s0*s2*s1*s2*s1*s0*s1*s0*s2*s1*s2*s1*s0*s1, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 >;

References : None.
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