Polytope of Type {6,4,12}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,4,12}*1152d
if this polytope has a name.
Group : SmallGroup(1152,157559)
Rank : 4
Schlafli Type : {6,4,12}
Number of vertices, edges, etc : 6, 24, 48, 24
Order of s₀s₁s₂s₃ : 6
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 : {6,4,6}*576a
   3-fold quotients : {2,4,12}*384c
   4-fold quotients : {6,4,3}*288
   6-fold quotients : {2,4,6}*192
   8-fold quotients : {6,2,6}*144
   12-fold quotients : {2,4,3}*96, {2,4,6}*96b, {2,4,6}*96c
   16-fold quotients : {3,2,6}*72, {6,2,3}*72
   24-fold quotients : {2,4,3}*48, {2,2,6}*48, {6,2,2}*48
   32-fold quotients : {3,2,3}*36
   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 := (  5,  9)(  6, 10)(  7, 11)(  8, 12)( 17, 21)( 18, 22)( 19, 23)( 20, 24)
( 29, 33)( 30, 34)( 31, 35)( 32, 36)( 41, 45)( 42, 46)( 43, 47)( 44, 48)
( 53, 57)( 54, 58)( 55, 59)( 56, 60)( 65, 69)( 66, 70)( 67, 71)( 68, 72)
( 77, 81)( 78, 82)( 79, 83)( 80, 84)( 89, 93)( 90, 94)( 91, 95)( 92, 96)
(101,105)(102,106)(103,107)(104,108)(113,117)(114,118)(115,119)(116,120)
(125,129)(126,130)(127,131)(128,132)(137,141)(138,142)(139,143)(140,144);;
s1 := (  1,  7)(  2,  8)(  3,  5)(  4,  6)(  9, 11)( 10, 12)( 13, 19)( 14, 20)
( 15, 17)( 16, 18)( 21, 23)( 22, 24)( 25, 31)( 26, 32)( 27, 29)( 28, 30)
( 33, 35)( 34, 36)( 37, 43)( 38, 44)( 39, 41)( 40, 42)( 45, 47)( 46, 48)
( 49, 55)( 50, 56)( 51, 53)( 52, 54)( 57, 59)( 58, 60)( 61, 67)( 62, 68)
( 63, 65)( 64, 66)( 69, 71)( 70, 72)( 73,115)( 74,116)( 75,113)( 76,114)
( 77,111)( 78,112)( 79,109)( 80,110)( 81,119)( 82,120)( 83,117)( 84,118)
( 85,127)( 86,128)( 87,125)( 88,126)( 89,123)( 90,124)( 91,121)( 92,122)
( 93,131)( 94,132)( 95,129)( 96,130)( 97,139)( 98,140)( 99,137)(100,138)
(101,135)(102,136)(103,133)(104,134)(105,143)(106,144)(107,141)(108,142);;
s2 := (  1, 73)(  2, 74)(  3, 76)(  4, 75)(  5, 77)(  6, 78)(  7, 80)(  8, 79)
(  9, 81)( 10, 82)( 11, 84)( 12, 83)( 13, 97)( 14, 98)( 15,100)( 16, 99)
( 17,101)( 18,102)( 19,104)( 20,103)( 21,105)( 22,106)( 23,108)( 24,107)
( 25, 85)( 26, 86)( 27, 88)( 28, 87)( 29, 89)( 30, 90)( 31, 92)( 32, 91)
( 33, 93)( 34, 94)( 35, 96)( 36, 95)( 37,109)( 38,110)( 39,112)( 40,111)
( 41,113)( 42,114)( 43,116)( 44,115)( 45,117)( 46,118)( 47,120)( 48,119)
( 49,133)( 50,134)( 51,136)( 52,135)( 53,137)( 54,138)( 55,140)( 56,139)
( 57,141)( 58,142)( 59,144)( 60,143)( 61,121)( 62,122)( 63,124)( 64,123)
( 65,125)( 66,126)( 67,128)( 68,127)( 69,129)( 70,130)( 71,132)( 72,131);;
s3 := (  1, 13)(  2, 16)(  3, 15)(  4, 14)(  5, 17)(  6, 20)(  7, 19)(  8, 18)
(  9, 21)( 10, 24)( 11, 23)( 12, 22)( 26, 28)( 30, 32)( 34, 36)( 37, 49)
( 38, 52)( 39, 51)( 40, 50)( 41, 53)( 42, 56)( 43, 55)( 44, 54)( 45, 57)
( 46, 60)( 47, 59)( 48, 58)( 62, 64)( 66, 68)( 70, 72)( 73,121)( 74,124)
( 75,123)( 76,122)( 77,125)( 78,128)( 79,127)( 80,126)( 81,129)( 82,132)
( 83,131)( 84,130)( 85,109)( 86,112)( 87,111)( 88,110)( 89,113)( 90,116)
( 91,115)( 92,114)( 93,117)( 94,120)( 95,119)( 96,118)( 97,133)( 98,136)
( 99,135)(100,134)(101,137)(102,140)(103,139)(104,138)(105,141)(106,144)
(107,143)(108,142);;
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*s2*s1*s0*s1*s2*s1, 
s1*s2*s1*s2*s1*s2*s1*s2, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s3*s1*s2*s3*s2*s3*s2*s3*s1*s2*s3*s2*s3*s2, 
s1*s2*s3*s2*s1*s2*s3*s2*s1*s2*s3*s2*s1*s2*s3*s2 ];;
poly := F / rels;;

Permutation Representation (Magma) :

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

References : None.
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