Polytope of Type {4,6,12}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,6,12}*1152b
if this polytope has a name.
Group : SmallGroup(1152,157559)
Rank : 4
Schlafli Type : {4,6,12}
Number of vertices, edges, etc : 4, 24, 72, 24
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 : {2,6,12}*576a
   3-fold quotients : {4,6,4}*384a
   4-fold quotients : {4,6,6}*288a, {2,6,12}*288d
   6-fold quotients : {4,6,4}*192b, {2,6,4}*192
   8-fold quotients : {2,6,6}*144a
   12-fold quotients : {4,2,6}*96, {4,6,2}*96a, {2,3,4}*96, {2,6,4}*96b, {2,6,4}*96c
   24-fold quotients : {4,2,3}*48, {2,3,4}*48, {2,2,6}*48, {2,6,2}*48
   36-fold quotients : {4,2,2}*32
   48-fold quotients : {2,2,3}*24, {2,3,2}*24
   72-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

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

References : None.
to this polytope