Polytope of Type {4,6,4}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,6,4}*768h
if this polytope has a name.
Group : SmallGroup(768,1090183)
Rank : 4
Schlafli Type : {4,6,4}
Number of vertices, edges, etc : 16, 48, 48, 4
Order of s0s1s2s3 : 12
Order of s0s1s2s3s2s1 : 2
Special Properties :
Universal
Non-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 : {4,6,4}*384d
4-fold quotients : {4,6,4}*192b, {4,3,4}*192b, {4,6,4}*192d, {4,6,4}*192f
8-fold quotients : {2,6,4}*96c, {4,3,4}*96
16-fold quotients : {2,3,4}*48
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := ( 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)
( 97,148)( 98,147)( 99,146)(100,145)(101,152)(102,151)(103,150)(104,149)
(105,156)(106,155)(107,154)(108,153)(109,160)(110,159)(111,158)(112,157)
(113,164)(114,163)(115,162)(116,161)(117,168)(118,167)(119,166)(120,165)
(121,172)(122,171)(123,170)(124,169)(125,176)(126,175)(127,174)(128,173)
(129,180)(130,179)(131,178)(132,177)(133,184)(134,183)(135,182)(136,181)
(137,188)(138,187)(139,186)(140,185)(141,192)(142,191)(143,190)(144,189);;
s1 := ( 1, 97)( 2, 98)( 3,100)( 4, 99)( 5,101)( 6,102)( 7,104)( 8,103)
( 9,109)( 10,110)( 11,112)( 12,111)( 13,105)( 14,106)( 15,108)( 16,107)
( 17,129)( 18,130)( 19,132)( 20,131)( 21,133)( 22,134)( 23,136)( 24,135)
( 25,141)( 26,142)( 27,144)( 28,143)( 29,137)( 30,138)( 31,140)( 32,139)
( 33,113)( 34,114)( 35,116)( 36,115)( 37,117)( 38,118)( 39,120)( 40,119)
( 41,125)( 42,126)( 43,128)( 44,127)( 45,121)( 46,122)( 47,124)( 48,123)
( 49,145)( 50,146)( 51,148)( 52,147)( 53,149)( 54,150)( 55,152)( 56,151)
( 57,157)( 58,158)( 59,160)( 60,159)( 61,153)( 62,154)( 63,156)( 64,155)
( 65,177)( 66,178)( 67,180)( 68,179)( 69,181)( 70,182)( 71,184)( 72,183)
( 73,189)( 74,190)( 75,192)( 76,191)( 77,185)( 78,186)( 79,188)( 80,187)
( 81,161)( 82,162)( 83,164)( 84,163)( 85,165)( 86,166)( 87,168)( 88,167)
( 89,173)( 90,174)( 91,176)( 92,175)( 93,169)( 94,170)( 95,172)( 96,171);;
s2 := ( 1, 33)( 2, 35)( 3, 34)( 4, 36)( 5, 41)( 6, 43)( 7, 42)( 8, 44)
( 9, 37)( 10, 39)( 11, 38)( 12, 40)( 13, 45)( 14, 47)( 15, 46)( 16, 48)
( 18, 19)( 21, 25)( 22, 27)( 23, 26)( 24, 28)( 30, 31)( 49, 81)( 50, 83)
( 51, 82)( 52, 84)( 53, 89)( 54, 91)( 55, 90)( 56, 92)( 57, 85)( 58, 87)
( 59, 86)( 60, 88)( 61, 93)( 62, 95)( 63, 94)( 64, 96)( 66, 67)( 69, 73)
( 70, 75)( 71, 74)( 72, 76)( 78, 79)( 97,129)( 98,131)( 99,130)(100,132)
(101,137)(102,139)(103,138)(104,140)(105,133)(106,135)(107,134)(108,136)
(109,141)(110,143)(111,142)(112,144)(114,115)(117,121)(118,123)(119,122)
(120,124)(126,127)(145,177)(146,179)(147,178)(148,180)(149,185)(150,187)
(151,186)(152,188)(153,181)(154,183)(155,182)(156,184)(157,189)(158,191)
(159,190)(160,192)(162,163)(165,169)(166,171)(167,170)(168,172)(174,175);;
s3 := ( 1, 5)( 2, 6)( 3, 7)( 4, 8)( 9, 13)( 10, 14)( 11, 15)( 12, 16)
( 17, 21)( 18, 22)( 19, 23)( 20, 24)( 25, 29)( 26, 30)( 27, 31)( 28, 32)
( 33, 37)( 34, 38)( 35, 39)( 36, 40)( 41, 45)( 42, 46)( 43, 47)( 44, 48)
( 49, 53)( 50, 54)( 51, 55)( 52, 56)( 57, 61)( 58, 62)( 59, 63)( 60, 64)
( 65, 69)( 66, 70)( 67, 71)( 68, 72)( 73, 77)( 74, 78)( 75, 79)( 76, 80)
( 81, 85)( 82, 86)( 83, 87)( 84, 88)( 89, 93)( 90, 94)( 91, 95)( 92, 96)
( 97,101)( 98,102)( 99,103)(100,104)(105,109)(106,110)(107,111)(108,112)
(113,117)(114,118)(115,119)(116,120)(121,125)(122,126)(123,127)(124,128)
(129,133)(130,134)(131,135)(132,136)(137,141)(138,142)(139,143)(140,144)
(145,149)(146,150)(147,151)(148,152)(153,157)(154,158)(155,159)(156,160)
(161,165)(162,166)(163,167)(164,168)(169,173)(170,174)(171,175)(172,176)
(177,181)(178,182)(179,183)(180,184)(185,189)(186,190)(187,191)(188,192);;
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,
s2*s3*s2*s3*s2*s3*s2*s3, s3*s2*s1*s3*s2*s3*s2*s1*s2,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s2*s0*s1*s2*s1*s2*s0*s1*s2*s0*s1*s2*s1*s2*s0*s1 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(192)!( 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)( 97,148)( 98,147)( 99,146)(100,145)(101,152)(102,151)(103,150)
(104,149)(105,156)(106,155)(107,154)(108,153)(109,160)(110,159)(111,158)
(112,157)(113,164)(114,163)(115,162)(116,161)(117,168)(118,167)(119,166)
(120,165)(121,172)(122,171)(123,170)(124,169)(125,176)(126,175)(127,174)
(128,173)(129,180)(130,179)(131,178)(132,177)(133,184)(134,183)(135,182)
(136,181)(137,188)(138,187)(139,186)(140,185)(141,192)(142,191)(143,190)
(144,189);
s1 := Sym(192)!( 1, 97)( 2, 98)( 3,100)( 4, 99)( 5,101)( 6,102)( 7,104)
( 8,103)( 9,109)( 10,110)( 11,112)( 12,111)( 13,105)( 14,106)( 15,108)
( 16,107)( 17,129)( 18,130)( 19,132)( 20,131)( 21,133)( 22,134)( 23,136)
( 24,135)( 25,141)( 26,142)( 27,144)( 28,143)( 29,137)( 30,138)( 31,140)
( 32,139)( 33,113)( 34,114)( 35,116)( 36,115)( 37,117)( 38,118)( 39,120)
( 40,119)( 41,125)( 42,126)( 43,128)( 44,127)( 45,121)( 46,122)( 47,124)
( 48,123)( 49,145)( 50,146)( 51,148)( 52,147)( 53,149)( 54,150)( 55,152)
( 56,151)( 57,157)( 58,158)( 59,160)( 60,159)( 61,153)( 62,154)( 63,156)
( 64,155)( 65,177)( 66,178)( 67,180)( 68,179)( 69,181)( 70,182)( 71,184)
( 72,183)( 73,189)( 74,190)( 75,192)( 76,191)( 77,185)( 78,186)( 79,188)
( 80,187)( 81,161)( 82,162)( 83,164)( 84,163)( 85,165)( 86,166)( 87,168)
( 88,167)( 89,173)( 90,174)( 91,176)( 92,175)( 93,169)( 94,170)( 95,172)
( 96,171);
s2 := Sym(192)!( 1, 33)( 2, 35)( 3, 34)( 4, 36)( 5, 41)( 6, 43)( 7, 42)
( 8, 44)( 9, 37)( 10, 39)( 11, 38)( 12, 40)( 13, 45)( 14, 47)( 15, 46)
( 16, 48)( 18, 19)( 21, 25)( 22, 27)( 23, 26)( 24, 28)( 30, 31)( 49, 81)
( 50, 83)( 51, 82)( 52, 84)( 53, 89)( 54, 91)( 55, 90)( 56, 92)( 57, 85)
( 58, 87)( 59, 86)( 60, 88)( 61, 93)( 62, 95)( 63, 94)( 64, 96)( 66, 67)
( 69, 73)( 70, 75)( 71, 74)( 72, 76)( 78, 79)( 97,129)( 98,131)( 99,130)
(100,132)(101,137)(102,139)(103,138)(104,140)(105,133)(106,135)(107,134)
(108,136)(109,141)(110,143)(111,142)(112,144)(114,115)(117,121)(118,123)
(119,122)(120,124)(126,127)(145,177)(146,179)(147,178)(148,180)(149,185)
(150,187)(151,186)(152,188)(153,181)(154,183)(155,182)(156,184)(157,189)
(158,191)(159,190)(160,192)(162,163)(165,169)(166,171)(167,170)(168,172)
(174,175);
s3 := Sym(192)!( 1, 5)( 2, 6)( 3, 7)( 4, 8)( 9, 13)( 10, 14)( 11, 15)
( 12, 16)( 17, 21)( 18, 22)( 19, 23)( 20, 24)( 25, 29)( 26, 30)( 27, 31)
( 28, 32)( 33, 37)( 34, 38)( 35, 39)( 36, 40)( 41, 45)( 42, 46)( 43, 47)
( 44, 48)( 49, 53)( 50, 54)( 51, 55)( 52, 56)( 57, 61)( 58, 62)( 59, 63)
( 60, 64)( 65, 69)( 66, 70)( 67, 71)( 68, 72)( 73, 77)( 74, 78)( 75, 79)
( 76, 80)( 81, 85)( 82, 86)( 83, 87)( 84, 88)( 89, 93)( 90, 94)( 91, 95)
( 92, 96)( 97,101)( 98,102)( 99,103)(100,104)(105,109)(106,110)(107,111)
(108,112)(113,117)(114,118)(115,119)(116,120)(121,125)(122,126)(123,127)
(124,128)(129,133)(130,134)(131,135)(132,136)(137,141)(138,142)(139,143)
(140,144)(145,149)(146,150)(147,151)(148,152)(153,157)(154,158)(155,159)
(156,160)(161,165)(162,166)(163,167)(164,168)(169,173)(170,174)(171,175)
(172,176)(177,181)(178,182)(179,183)(180,184)(185,189)(186,190)(187,191)
(188,192);
poly := sub<Sym(192)|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, s2*s3*s2*s3*s2*s3*s2*s3,
s3*s2*s1*s3*s2*s3*s2*s1*s2, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s2*s0*s1*s2*s1*s2*s0*s1*s2*s0*s1*s2*s1*s2*s0*s1 >;
References : None.
to this polytope