Polytope of Type {4,6,3,4}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,6,3,4}*576
if this polytope has a name.
Group : SmallGroup(576,8355)
Rank : 5
Schlafli Type : {4,6,3,4}
Number of vertices, edges, etc : 4, 12, 9, 6, 4
Order of s0s1s2s3s4 : 12
Order of s0s1s2s3s4s3s2s1 : 2
Special Properties :
Universal
Non-Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{4,6,3,4,2} of size 1152
Vertex Figure Of :
{2,4,6,3,4} of size 1152
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {2,6,3,4}*288
3-fold quotients : {4,2,3,4}*192
6-fold quotients : {2,2,3,4}*96
Covers (Minimal Covers in Boldface) :
2-fold covers : {8,6,3,4}*1152, {4,6,3,4}*1152, {4,6,6,4}*1152g, {4,6,6,4}*1152h
3-fold covers : {4,6,9,4}*1728, {4,6,3,4}*1728a, {12,6,3,4}*1728b
Permutation Representation (GAP) :
s0 := ( 1, 73)( 2, 74)( 3, 75)( 4, 76)( 5, 77)( 6, 78)( 7, 79)( 8, 80)
( 9, 81)( 10, 82)( 11, 83)( 12, 84)( 13, 85)( 14, 86)( 15, 87)( 16, 88)
( 17, 89)( 18, 90)( 19, 91)( 20, 92)( 21, 93)( 22, 94)( 23, 95)( 24, 96)
( 25, 97)( 26, 98)( 27, 99)( 28,100)( 29,101)( 30,102)( 31,103)( 32,104)
( 33,105)( 34,106)( 35,107)( 36,108)( 37,109)( 38,110)( 39,111)( 40,112)
( 41,113)( 42,114)( 43,115)( 44,116)( 45,117)( 46,118)( 47,119)( 48,120)
( 49,121)( 50,122)( 51,123)( 52,124)( 53,125)( 54,126)( 55,127)( 56,128)
( 57,129)( 58,130)( 59,131)( 60,132)( 61,133)( 62,134)( 63,135)( 64,136)
( 65,137)( 66,138)( 67,139)( 68,140)( 69,141)( 70,142)( 71,143)( 72,144);;
s1 := ( 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)
( 73,109)( 74,110)( 75,111)( 76,112)( 77,117)( 78,118)( 79,119)( 80,120)
( 81,113)( 82,114)( 83,115)( 84,116)( 85,121)( 86,122)( 87,123)( 88,124)
( 89,129)( 90,130)( 91,131)( 92,132)( 93,125)( 94,126)( 95,127)( 96,128)
( 97,133)( 98,134)( 99,135)(100,136)(101,141)(102,142)(103,143)(104,144)
(105,137)(106,138)(107,139)(108,140);;
s2 := ( 1, 5)( 2, 7)( 3, 6)( 4, 8)( 10, 11)( 13, 29)( 14, 31)( 15, 30)
( 16, 32)( 17, 25)( 18, 27)( 19, 26)( 20, 28)( 21, 33)( 22, 35)( 23, 34)
( 24, 36)( 37, 41)( 38, 43)( 39, 42)( 40, 44)( 46, 47)( 49, 65)( 50, 67)
( 51, 66)( 52, 68)( 53, 61)( 54, 63)( 55, 62)( 56, 64)( 57, 69)( 58, 71)
( 59, 70)( 60, 72)( 73, 77)( 74, 79)( 75, 78)( 76, 80)( 82, 83)( 85,101)
( 86,103)( 87,102)( 88,104)( 89, 97)( 90, 99)( 91, 98)( 92,100)( 93,105)
( 94,107)( 95,106)( 96,108)(109,113)(110,115)(111,114)(112,116)(118,119)
(121,137)(122,139)(123,138)(124,140)(125,133)(126,135)(127,134)(128,136)
(129,141)(130,143)(131,142)(132,144);;
s3 := ( 1, 13)( 2, 14)( 3, 16)( 4, 15)( 5, 21)( 6, 22)( 7, 24)( 8, 23)
( 9, 17)( 10, 18)( 11, 20)( 12, 19)( 27, 28)( 29, 33)( 30, 34)( 31, 36)
( 32, 35)( 37, 49)( 38, 50)( 39, 52)( 40, 51)( 41, 57)( 42, 58)( 43, 60)
( 44, 59)( 45, 53)( 46, 54)( 47, 56)( 48, 55)( 63, 64)( 65, 69)( 66, 70)
( 67, 72)( 68, 71)( 73, 85)( 74, 86)( 75, 88)( 76, 87)( 77, 93)( 78, 94)
( 79, 96)( 80, 95)( 81, 89)( 82, 90)( 83, 92)( 84, 91)( 99,100)(101,105)
(102,106)(103,108)(104,107)(109,121)(110,122)(111,124)(112,123)(113,129)
(114,130)(115,132)(116,131)(117,125)(118,126)(119,128)(120,127)(135,136)
(137,141)(138,142)(139,144)(140,143);;
s4 := ( 1, 4)( 2, 3)( 5, 8)( 6, 7)( 9, 12)( 10, 11)( 13, 16)( 14, 15)
( 17, 20)( 18, 19)( 21, 24)( 22, 23)( 25, 28)( 26, 27)( 29, 32)( 30, 31)
( 33, 36)( 34, 35)( 37, 40)( 38, 39)( 41, 44)( 42, 43)( 45, 48)( 46, 47)
( 49, 52)( 50, 51)( 53, 56)( 54, 55)( 57, 60)( 58, 59)( 61, 64)( 62, 63)
( 65, 68)( 66, 67)( 69, 72)( 70, 71)( 73, 76)( 74, 75)( 77, 80)( 78, 79)
( 81, 84)( 82, 83)( 85, 88)( 86, 87)( 89, 92)( 90, 91)( 93, 96)( 94, 95)
( 97,100)( 98, 99)(101,104)(102,103)(105,108)(106,107)(109,112)(110,111)
(113,116)(114,115)(117,120)(118,119)(121,124)(122,123)(125,128)(126,127)
(129,132)(130,131)(133,136)(134,135)(137,140)(138,139)(141,144)(142,143);;
poly := Group([s0,s1,s2,s3,s4]);;
Finitely Presented Group Representation (GAP) :
F := FreeGroup("s0","s1","s2","s3","s4");;
s0 := F.1;; s1 := F.2;; s2 := F.3;; s3 := F.4;; s4 := F.5;;
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s4*s4, s0*s2*s0*s2,
s0*s3*s0*s3, s1*s3*s1*s3, s0*s4*s0*s4,
s1*s4*s1*s4, s2*s4*s2*s4, s2*s3*s2*s3*s2*s3,
s0*s1*s0*s1*s0*s1*s0*s1, s0*s1*s2*s1*s0*s1*s2*s1,
s3*s4*s3*s4*s3*s4*s3*s4, s2*s4*s3*s2*s4*s3*s2*s4*s3,
s3*s1*s2*s1*s2*s3*s1*s2*s1*s2 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(144)!( 1, 73)( 2, 74)( 3, 75)( 4, 76)( 5, 77)( 6, 78)( 7, 79)
( 8, 80)( 9, 81)( 10, 82)( 11, 83)( 12, 84)( 13, 85)( 14, 86)( 15, 87)
( 16, 88)( 17, 89)( 18, 90)( 19, 91)( 20, 92)( 21, 93)( 22, 94)( 23, 95)
( 24, 96)( 25, 97)( 26, 98)( 27, 99)( 28,100)( 29,101)( 30,102)( 31,103)
( 32,104)( 33,105)( 34,106)( 35,107)( 36,108)( 37,109)( 38,110)( 39,111)
( 40,112)( 41,113)( 42,114)( 43,115)( 44,116)( 45,117)( 46,118)( 47,119)
( 48,120)( 49,121)( 50,122)( 51,123)( 52,124)( 53,125)( 54,126)( 55,127)
( 56,128)( 57,129)( 58,130)( 59,131)( 60,132)( 61,133)( 62,134)( 63,135)
( 64,136)( 65,137)( 66,138)( 67,139)( 68,140)( 69,141)( 70,142)( 71,143)
( 72,144);
s1 := 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)( 73,109)( 74,110)( 75,111)( 76,112)( 77,117)( 78,118)( 79,119)
( 80,120)( 81,113)( 82,114)( 83,115)( 84,116)( 85,121)( 86,122)( 87,123)
( 88,124)( 89,129)( 90,130)( 91,131)( 92,132)( 93,125)( 94,126)( 95,127)
( 96,128)( 97,133)( 98,134)( 99,135)(100,136)(101,141)(102,142)(103,143)
(104,144)(105,137)(106,138)(107,139)(108,140);
s2 := Sym(144)!( 1, 5)( 2, 7)( 3, 6)( 4, 8)( 10, 11)( 13, 29)( 14, 31)
( 15, 30)( 16, 32)( 17, 25)( 18, 27)( 19, 26)( 20, 28)( 21, 33)( 22, 35)
( 23, 34)( 24, 36)( 37, 41)( 38, 43)( 39, 42)( 40, 44)( 46, 47)( 49, 65)
( 50, 67)( 51, 66)( 52, 68)( 53, 61)( 54, 63)( 55, 62)( 56, 64)( 57, 69)
( 58, 71)( 59, 70)( 60, 72)( 73, 77)( 74, 79)( 75, 78)( 76, 80)( 82, 83)
( 85,101)( 86,103)( 87,102)( 88,104)( 89, 97)( 90, 99)( 91, 98)( 92,100)
( 93,105)( 94,107)( 95,106)( 96,108)(109,113)(110,115)(111,114)(112,116)
(118,119)(121,137)(122,139)(123,138)(124,140)(125,133)(126,135)(127,134)
(128,136)(129,141)(130,143)(131,142)(132,144);
s3 := Sym(144)!( 1, 13)( 2, 14)( 3, 16)( 4, 15)( 5, 21)( 6, 22)( 7, 24)
( 8, 23)( 9, 17)( 10, 18)( 11, 20)( 12, 19)( 27, 28)( 29, 33)( 30, 34)
( 31, 36)( 32, 35)( 37, 49)( 38, 50)( 39, 52)( 40, 51)( 41, 57)( 42, 58)
( 43, 60)( 44, 59)( 45, 53)( 46, 54)( 47, 56)( 48, 55)( 63, 64)( 65, 69)
( 66, 70)( 67, 72)( 68, 71)( 73, 85)( 74, 86)( 75, 88)( 76, 87)( 77, 93)
( 78, 94)( 79, 96)( 80, 95)( 81, 89)( 82, 90)( 83, 92)( 84, 91)( 99,100)
(101,105)(102,106)(103,108)(104,107)(109,121)(110,122)(111,124)(112,123)
(113,129)(114,130)(115,132)(116,131)(117,125)(118,126)(119,128)(120,127)
(135,136)(137,141)(138,142)(139,144)(140,143);
s4 := Sym(144)!( 1, 4)( 2, 3)( 5, 8)( 6, 7)( 9, 12)( 10, 11)( 13, 16)
( 14, 15)( 17, 20)( 18, 19)( 21, 24)( 22, 23)( 25, 28)( 26, 27)( 29, 32)
( 30, 31)( 33, 36)( 34, 35)( 37, 40)( 38, 39)( 41, 44)( 42, 43)( 45, 48)
( 46, 47)( 49, 52)( 50, 51)( 53, 56)( 54, 55)( 57, 60)( 58, 59)( 61, 64)
( 62, 63)( 65, 68)( 66, 67)( 69, 72)( 70, 71)( 73, 76)( 74, 75)( 77, 80)
( 78, 79)( 81, 84)( 82, 83)( 85, 88)( 86, 87)( 89, 92)( 90, 91)( 93, 96)
( 94, 95)( 97,100)( 98, 99)(101,104)(102,103)(105,108)(106,107)(109,112)
(110,111)(113,116)(114,115)(117,120)(118,119)(121,124)(122,123)(125,128)
(126,127)(129,132)(130,131)(133,136)(134,135)(137,140)(138,139)(141,144)
(142,143);
poly := sub<Sym(144)|s0,s1,s2,s3,s4>;
Finitely Presented Group Representation (Magma) :
poly<s0,s1,s2,s3,s4> := Group< s0,s1,s2,s3,s4 | s0*s0, s1*s1, s2*s2,
s3*s3, s4*s4, s0*s2*s0*s2, s0*s3*s0*s3,
s1*s3*s1*s3, s0*s4*s0*s4, s1*s4*s1*s4,
s2*s4*s2*s4, s2*s3*s2*s3*s2*s3, s0*s1*s0*s1*s0*s1*s0*s1,
s0*s1*s2*s1*s0*s1*s2*s1, s3*s4*s3*s4*s3*s4*s3*s4,
s2*s4*s3*s2*s4*s3*s2*s4*s3, s3*s1*s2*s1*s2*s3*s1*s2*s1*s2 >;
References : None.
to this polytope