Polytope of Type {4,12,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,12,6}*576e
if this polytope has a name.
Group : SmallGroup(576,8312)
Rank : 4
Schlafli Type : {4,12,6}
Number of vertices, edges, etc : 4, 24, 36, 6
Order of s₀s₁s₂s₃ : 12
Order of s₀s₁s₂s₃s₂s₁ : 2
Special Properties :
   Universal
   Non-Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {4,12,6,2} of size 1152
   {4,12,6,3} of size 1728
Vertex Figure Of :
   {2,4,12,6} of size 1152
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {4,6,6}*288e
   3-fold quotients : {4,12,2}*192b
   4-fold quotients : {4,3,6}*144
   6-fold quotients : {4,6,2}*96c
   12-fold quotients : {4,3,2}*48
Covers (Minimal Covers in Boldface) :
   2-fold covers : {4,24,6}*1152h, {4,24,6}*1152j, {4,12,12}*1152e, {4,12,6}*1152f
   3-fold covers : {4,36,6}*1728d, {4,12,6}*1728d, {4,12,6}*1728l
Permutation Representation (GAP) :

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