Polytope of Type {3,6,16}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {3,6,16}*768
Also Known As : {{3,6}4,{6,16|2}}. if this polytope has another name.
Group : SmallGroup(768,1088585)
Rank : 4
Schlafli Type : {3,6,16}
Number of vertices, edges, etc : 4, 12, 64, 16
Order of s0s1s2s3 : 16
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 : {3,6,8}*384
4-fold quotients : {3,6,4}*192
8-fold quotients : {3,6,2}*96
16-fold quotients : {3,3,2}*48
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := ( 3, 4)( 5, 9)( 6, 10)( 7, 12)( 8, 11)( 15, 16)( 17, 21)( 18, 22)
( 19, 24)( 20, 23)( 27, 28)( 29, 33)( 30, 34)( 31, 36)( 32, 35)( 39, 40)
( 41, 45)( 42, 46)( 43, 48)( 44, 47)( 51, 52)( 53, 57)( 54, 58)( 55, 60)
( 56, 59)( 63, 64)( 65, 69)( 66, 70)( 67, 72)( 68, 71)( 75, 76)( 77, 81)
( 78, 82)( 79, 84)( 80, 83)( 87, 88)( 89, 93)( 90, 94)( 91, 96)( 92, 95)
( 99,100)(101,105)(102,106)(103,108)(104,107)(111,112)(113,117)(114,118)
(115,120)(116,119)(123,124)(125,129)(126,130)(127,132)(128,131)(135,136)
(137,141)(138,142)(139,144)(140,143)(147,148)(149,153)(150,154)(151,156)
(152,155)(159,160)(161,165)(162,166)(163,168)(164,167)(171,172)(173,177)
(174,178)(175,180)(176,179)(183,184)(185,189)(186,190)(187,192)(188,191);;
s1 := ( 1, 9)( 2, 11)( 3, 10)( 4, 12)( 6, 7)( 13, 21)( 14, 23)( 15, 22)
( 16, 24)( 18, 19)( 25, 33)( 26, 35)( 27, 34)( 28, 36)( 30, 31)( 37, 45)
( 38, 47)( 39, 46)( 40, 48)( 42, 43)( 49, 57)( 50, 59)( 51, 58)( 52, 60)
( 54, 55)( 61, 69)( 62, 71)( 63, 70)( 64, 72)( 66, 67)( 73, 81)( 74, 83)
( 75, 82)( 76, 84)( 78, 79)( 85, 93)( 86, 95)( 87, 94)( 88, 96)( 90, 91)
( 97,105)( 98,107)( 99,106)(100,108)(102,103)(109,117)(110,119)(111,118)
(112,120)(114,115)(121,129)(122,131)(123,130)(124,132)(126,127)(133,141)
(134,143)(135,142)(136,144)(138,139)(145,153)(146,155)(147,154)(148,156)
(150,151)(157,165)(158,167)(159,166)(160,168)(162,163)(169,177)(170,179)
(171,178)(172,180)(174,175)(181,189)(182,191)(183,190)(184,192)(186,187);;
s2 := ( 1, 2)( 5, 10)( 6, 9)( 7, 11)( 8, 12)( 13, 14)( 17, 22)( 18, 21)
( 19, 23)( 20, 24)( 25, 38)( 26, 37)( 27, 39)( 28, 40)( 29, 46)( 30, 45)
( 31, 47)( 32, 48)( 33, 42)( 34, 41)( 35, 43)( 36, 44)( 49, 74)( 50, 73)
( 51, 75)( 52, 76)( 53, 82)( 54, 81)( 55, 83)( 56, 84)( 57, 78)( 58, 77)
( 59, 79)( 60, 80)( 61, 86)( 62, 85)( 63, 87)( 64, 88)( 65, 94)( 66, 93)
( 67, 95)( 68, 96)( 69, 90)( 70, 89)( 71, 91)( 72, 92)( 97,146)( 98,145)
( 99,147)(100,148)(101,154)(102,153)(103,155)(104,156)(105,150)(106,149)
(107,151)(108,152)(109,158)(110,157)(111,159)(112,160)(113,166)(114,165)
(115,167)(116,168)(117,162)(118,161)(119,163)(120,164)(121,182)(122,181)
(123,183)(124,184)(125,190)(126,189)(127,191)(128,192)(129,186)(130,185)
(131,187)(132,188)(133,170)(134,169)(135,171)(136,172)(137,178)(138,177)
(139,179)(140,180)(141,174)(142,173)(143,175)(144,176);;
s3 := ( 1, 97)( 2, 98)( 3, 99)( 4,100)( 5,101)( 6,102)( 7,103)( 8,104)
( 9,105)( 10,106)( 11,107)( 12,108)( 13,109)( 14,110)( 15,111)( 16,112)
( 17,113)( 18,114)( 19,115)( 20,116)( 21,117)( 22,118)( 23,119)( 24,120)
( 25,133)( 26,134)( 27,135)( 28,136)( 29,137)( 30,138)( 31,139)( 32,140)
( 33,141)( 34,142)( 35,143)( 36,144)( 37,121)( 38,122)( 39,123)( 40,124)
( 41,125)( 42,126)( 43,127)( 44,128)( 45,129)( 46,130)( 47,131)( 48,132)
( 49,169)( 50,170)( 51,171)( 52,172)( 53,173)( 54,174)( 55,175)( 56,176)
( 57,177)( 58,178)( 59,179)( 60,180)( 61,181)( 62,182)( 63,183)( 64,184)
( 65,185)( 66,186)( 67,187)( 68,188)( 69,189)( 70,190)( 71,191)( 72,192)
( 73,145)( 74,146)( 75,147)( 76,148)( 77,149)( 78,150)( 79,151)( 80,152)
( 81,153)( 82,154)( 83,155)( 84,156)( 85,157)( 86,158)( 87,159)( 88,160)
( 89,161)( 90,162)( 91,163)( 92,164)( 93,165)( 94,166)( 95,167)( 96,168);;
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,
s1*s2*s3*s2*s1*s2*s3*s2, s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(192)!( 3, 4)( 5, 9)( 6, 10)( 7, 12)( 8, 11)( 15, 16)( 17, 21)
( 18, 22)( 19, 24)( 20, 23)( 27, 28)( 29, 33)( 30, 34)( 31, 36)( 32, 35)
( 39, 40)( 41, 45)( 42, 46)( 43, 48)( 44, 47)( 51, 52)( 53, 57)( 54, 58)
( 55, 60)( 56, 59)( 63, 64)( 65, 69)( 66, 70)( 67, 72)( 68, 71)( 75, 76)
( 77, 81)( 78, 82)( 79, 84)( 80, 83)( 87, 88)( 89, 93)( 90, 94)( 91, 96)
( 92, 95)( 99,100)(101,105)(102,106)(103,108)(104,107)(111,112)(113,117)
(114,118)(115,120)(116,119)(123,124)(125,129)(126,130)(127,132)(128,131)
(135,136)(137,141)(138,142)(139,144)(140,143)(147,148)(149,153)(150,154)
(151,156)(152,155)(159,160)(161,165)(162,166)(163,168)(164,167)(171,172)
(173,177)(174,178)(175,180)(176,179)(183,184)(185,189)(186,190)(187,192)
(188,191);
s1 := Sym(192)!( 1, 9)( 2, 11)( 3, 10)( 4, 12)( 6, 7)( 13, 21)( 14, 23)
( 15, 22)( 16, 24)( 18, 19)( 25, 33)( 26, 35)( 27, 34)( 28, 36)( 30, 31)
( 37, 45)( 38, 47)( 39, 46)( 40, 48)( 42, 43)( 49, 57)( 50, 59)( 51, 58)
( 52, 60)( 54, 55)( 61, 69)( 62, 71)( 63, 70)( 64, 72)( 66, 67)( 73, 81)
( 74, 83)( 75, 82)( 76, 84)( 78, 79)( 85, 93)( 86, 95)( 87, 94)( 88, 96)
( 90, 91)( 97,105)( 98,107)( 99,106)(100,108)(102,103)(109,117)(110,119)
(111,118)(112,120)(114,115)(121,129)(122,131)(123,130)(124,132)(126,127)
(133,141)(134,143)(135,142)(136,144)(138,139)(145,153)(146,155)(147,154)
(148,156)(150,151)(157,165)(158,167)(159,166)(160,168)(162,163)(169,177)
(170,179)(171,178)(172,180)(174,175)(181,189)(182,191)(183,190)(184,192)
(186,187);
s2 := Sym(192)!( 1, 2)( 5, 10)( 6, 9)( 7, 11)( 8, 12)( 13, 14)( 17, 22)
( 18, 21)( 19, 23)( 20, 24)( 25, 38)( 26, 37)( 27, 39)( 28, 40)( 29, 46)
( 30, 45)( 31, 47)( 32, 48)( 33, 42)( 34, 41)( 35, 43)( 36, 44)( 49, 74)
( 50, 73)( 51, 75)( 52, 76)( 53, 82)( 54, 81)( 55, 83)( 56, 84)( 57, 78)
( 58, 77)( 59, 79)( 60, 80)( 61, 86)( 62, 85)( 63, 87)( 64, 88)( 65, 94)
( 66, 93)( 67, 95)( 68, 96)( 69, 90)( 70, 89)( 71, 91)( 72, 92)( 97,146)
( 98,145)( 99,147)(100,148)(101,154)(102,153)(103,155)(104,156)(105,150)
(106,149)(107,151)(108,152)(109,158)(110,157)(111,159)(112,160)(113,166)
(114,165)(115,167)(116,168)(117,162)(118,161)(119,163)(120,164)(121,182)
(122,181)(123,183)(124,184)(125,190)(126,189)(127,191)(128,192)(129,186)
(130,185)(131,187)(132,188)(133,170)(134,169)(135,171)(136,172)(137,178)
(138,177)(139,179)(140,180)(141,174)(142,173)(143,175)(144,176);
s3 := Sym(192)!( 1, 97)( 2, 98)( 3, 99)( 4,100)( 5,101)( 6,102)( 7,103)
( 8,104)( 9,105)( 10,106)( 11,107)( 12,108)( 13,109)( 14,110)( 15,111)
( 16,112)( 17,113)( 18,114)( 19,115)( 20,116)( 21,117)( 22,118)( 23,119)
( 24,120)( 25,133)( 26,134)( 27,135)( 28,136)( 29,137)( 30,138)( 31,139)
( 32,140)( 33,141)( 34,142)( 35,143)( 36,144)( 37,121)( 38,122)( 39,123)
( 40,124)( 41,125)( 42,126)( 43,127)( 44,128)( 45,129)( 46,130)( 47,131)
( 48,132)( 49,169)( 50,170)( 51,171)( 52,172)( 53,173)( 54,174)( 55,175)
( 56,176)( 57,177)( 58,178)( 59,179)( 60,180)( 61,181)( 62,182)( 63,183)
( 64,184)( 65,185)( 66,186)( 67,187)( 68,188)( 69,189)( 70,190)( 71,191)
( 72,192)( 73,145)( 74,146)( 75,147)( 76,148)( 77,149)( 78,150)( 79,151)
( 80,152)( 81,153)( 82,154)( 83,155)( 84,156)( 85,157)( 86,158)( 87,159)
( 88,160)( 89,161)( 90,162)( 91,163)( 92,164)( 93,165)( 94,166)( 95,167)
( 96,168);
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, s1*s2*s3*s2*s1*s2*s3*s2,
s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 >;
References : None.
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