Polytope of Type {3,8,4}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {3,8,4}*768a
if this polytope has a name.
Group : SmallGroup(768,1086320)
Rank : 4
Schlafli Type : {3,8,4}
Number of vertices, edges, etc : 6, 48, 64, 8
Order of s0s1s2s3 : 12
Order of s0s1s2s3s2s1 : 4
Special Properties :
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 : {3,4,4}*384b
4-fold quotients : {3,4,4}*192a, {3,8,2}*192
8-fold quotients : {3,4,2}*96
16-fold quotients : {3,4,2}*48
32-fold quotients : {3,2,2}*24
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := ( 5, 7)( 6, 8)( 9, 10)( 11, 12)( 13, 16)( 14, 15)( 17, 25)( 18, 26)
( 19, 27)( 20, 28)( 21, 31)( 22, 32)( 23, 29)( 24, 30)( 33, 65)( 34, 66)
( 35, 67)( 36, 68)( 37, 71)( 38, 72)( 39, 69)( 40, 70)( 41, 74)( 42, 73)
( 43, 76)( 44, 75)( 45, 80)( 46, 79)( 47, 78)( 48, 77)( 49, 89)( 50, 90)
( 51, 91)( 52, 92)( 53, 95)( 54, 96)( 55, 93)( 56, 94)( 57, 81)( 58, 82)
( 59, 83)( 60, 84)( 61, 87)( 62, 88)( 63, 85)( 64, 86)( 97, 98)( 99,100)
(101,104)(102,103)(109,111)(110,112)(113,122)(114,121)(115,124)(116,123)
(117,128)(118,127)(119,126)(120,125)(129,162)(130,161)(131,164)(132,163)
(133,168)(134,167)(135,166)(136,165)(137,169)(138,170)(139,171)(140,172)
(141,175)(142,176)(143,173)(144,174)(145,186)(146,185)(147,188)(148,187)
(149,192)(150,191)(151,190)(152,189)(153,178)(154,177)(155,180)(156,179)
(157,184)(158,183)(159,182)(160,181);;
s1 := ( 1, 65)( 2, 66)( 3, 69)( 4, 70)( 5, 67)( 6, 68)( 7, 71)( 8, 72)
( 9, 82)( 10, 81)( 11, 86)( 12, 85)( 13, 84)( 14, 83)( 15, 88)( 16, 87)
( 17, 74)( 18, 73)( 19, 78)( 20, 77)( 21, 76)( 22, 75)( 23, 80)( 24, 79)
( 25, 90)( 26, 89)( 27, 94)( 28, 93)( 29, 92)( 30, 91)( 31, 96)( 32, 95)
( 35, 37)( 36, 38)( 41, 50)( 42, 49)( 43, 54)( 44, 53)( 45, 52)( 46, 51)
( 47, 56)( 48, 55)( 57, 58)( 59, 62)( 60, 61)( 63, 64)( 97,162)( 98,161)
( 99,166)(100,165)(101,164)(102,163)(103,168)(104,167)(105,177)(106,178)
(107,181)(108,182)(109,179)(110,180)(111,183)(112,184)(113,169)(114,170)
(115,173)(116,174)(117,171)(118,172)(119,175)(120,176)(121,185)(122,186)
(123,189)(124,190)(125,187)(126,188)(127,191)(128,192)(129,130)(131,134)
(132,133)(135,136)(137,145)(138,146)(139,149)(140,150)(141,147)(142,148)
(143,151)(144,152)(155,157)(156,158);;
s2 := ( 1,106)( 2,105)( 3,108)( 4,107)( 5,110)( 6,109)( 7,112)( 8,111)
( 9, 97)( 10, 98)( 11, 99)( 12,100)( 13,101)( 14,102)( 15,103)( 16,104)
( 17,122)( 18,121)( 19,124)( 20,123)( 21,126)( 22,125)( 23,128)( 24,127)
( 25,113)( 26,114)( 27,115)( 28,116)( 29,117)( 30,118)( 31,119)( 32,120)
( 33,138)( 34,137)( 35,140)( 36,139)( 37,142)( 38,141)( 39,144)( 40,143)
( 41,129)( 42,130)( 43,131)( 44,132)( 45,133)( 46,134)( 47,135)( 48,136)
( 49,154)( 50,153)( 51,156)( 52,155)( 53,158)( 54,157)( 55,160)( 56,159)
( 57,145)( 58,146)( 59,147)( 60,148)( 61,149)( 62,150)( 63,151)( 64,152)
( 65,170)( 66,169)( 67,172)( 68,171)( 69,174)( 70,173)( 71,176)( 72,175)
( 73,161)( 74,162)( 75,163)( 76,164)( 77,165)( 78,166)( 79,167)( 80,168)
( 81,186)( 82,185)( 83,188)( 84,187)( 85,190)( 86,189)( 87,192)( 88,191)
( 89,177)( 90,178)( 91,179)( 92,180)( 93,181)( 94,182)( 95,183)( 96,184);;
s3 := ( 9, 11)( 10, 12)( 13, 15)( 14, 16)( 17, 21)( 18, 22)( 19, 23)( 20, 24)
( 25, 31)( 26, 32)( 27, 29)( 28, 30)( 41, 43)( 42, 44)( 45, 47)( 46, 48)
( 49, 53)( 50, 54)( 51, 55)( 52, 56)( 57, 63)( 58, 64)( 59, 61)( 60, 62)
( 73, 75)( 74, 76)( 77, 79)( 78, 80)( 81, 85)( 82, 86)( 83, 87)( 84, 88)
( 89, 95)( 90, 96)( 91, 93)( 92, 94)(105,107)(106,108)(109,111)(110,112)
(113,117)(114,118)(115,119)(116,120)(121,127)(122,128)(123,125)(124,126)
(137,139)(138,140)(141,143)(142,144)(145,149)(146,150)(147,151)(148,152)
(153,159)(154,160)(155,157)(156,158)(169,171)(170,172)(173,175)(174,176)
(177,181)(178,182)(179,183)(180,184)(185,191)(186,192)(187,189)(188,190);;
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,
s2*s3*s2*s3*s2*s3*s2*s3, s2*s0*s1*s2*s3*s2*s1*s2*s0*s1*s2*s3*s2*s1,
s1*s3*s2*s3*s2*s1*s2*s1*s3*s2*s3*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)!( 5, 7)( 6, 8)( 9, 10)( 11, 12)( 13, 16)( 14, 15)( 17, 25)
( 18, 26)( 19, 27)( 20, 28)( 21, 31)( 22, 32)( 23, 29)( 24, 30)( 33, 65)
( 34, 66)( 35, 67)( 36, 68)( 37, 71)( 38, 72)( 39, 69)( 40, 70)( 41, 74)
( 42, 73)( 43, 76)( 44, 75)( 45, 80)( 46, 79)( 47, 78)( 48, 77)( 49, 89)
( 50, 90)( 51, 91)( 52, 92)( 53, 95)( 54, 96)( 55, 93)( 56, 94)( 57, 81)
( 58, 82)( 59, 83)( 60, 84)( 61, 87)( 62, 88)( 63, 85)( 64, 86)( 97, 98)
( 99,100)(101,104)(102,103)(109,111)(110,112)(113,122)(114,121)(115,124)
(116,123)(117,128)(118,127)(119,126)(120,125)(129,162)(130,161)(131,164)
(132,163)(133,168)(134,167)(135,166)(136,165)(137,169)(138,170)(139,171)
(140,172)(141,175)(142,176)(143,173)(144,174)(145,186)(146,185)(147,188)
(148,187)(149,192)(150,191)(151,190)(152,189)(153,178)(154,177)(155,180)
(156,179)(157,184)(158,183)(159,182)(160,181);
s1 := Sym(192)!( 1, 65)( 2, 66)( 3, 69)( 4, 70)( 5, 67)( 6, 68)( 7, 71)
( 8, 72)( 9, 82)( 10, 81)( 11, 86)( 12, 85)( 13, 84)( 14, 83)( 15, 88)
( 16, 87)( 17, 74)( 18, 73)( 19, 78)( 20, 77)( 21, 76)( 22, 75)( 23, 80)
( 24, 79)( 25, 90)( 26, 89)( 27, 94)( 28, 93)( 29, 92)( 30, 91)( 31, 96)
( 32, 95)( 35, 37)( 36, 38)( 41, 50)( 42, 49)( 43, 54)( 44, 53)( 45, 52)
( 46, 51)( 47, 56)( 48, 55)( 57, 58)( 59, 62)( 60, 61)( 63, 64)( 97,162)
( 98,161)( 99,166)(100,165)(101,164)(102,163)(103,168)(104,167)(105,177)
(106,178)(107,181)(108,182)(109,179)(110,180)(111,183)(112,184)(113,169)
(114,170)(115,173)(116,174)(117,171)(118,172)(119,175)(120,176)(121,185)
(122,186)(123,189)(124,190)(125,187)(126,188)(127,191)(128,192)(129,130)
(131,134)(132,133)(135,136)(137,145)(138,146)(139,149)(140,150)(141,147)
(142,148)(143,151)(144,152)(155,157)(156,158);
s2 := Sym(192)!( 1,106)( 2,105)( 3,108)( 4,107)( 5,110)( 6,109)( 7,112)
( 8,111)( 9, 97)( 10, 98)( 11, 99)( 12,100)( 13,101)( 14,102)( 15,103)
( 16,104)( 17,122)( 18,121)( 19,124)( 20,123)( 21,126)( 22,125)( 23,128)
( 24,127)( 25,113)( 26,114)( 27,115)( 28,116)( 29,117)( 30,118)( 31,119)
( 32,120)( 33,138)( 34,137)( 35,140)( 36,139)( 37,142)( 38,141)( 39,144)
( 40,143)( 41,129)( 42,130)( 43,131)( 44,132)( 45,133)( 46,134)( 47,135)
( 48,136)( 49,154)( 50,153)( 51,156)( 52,155)( 53,158)( 54,157)( 55,160)
( 56,159)( 57,145)( 58,146)( 59,147)( 60,148)( 61,149)( 62,150)( 63,151)
( 64,152)( 65,170)( 66,169)( 67,172)( 68,171)( 69,174)( 70,173)( 71,176)
( 72,175)( 73,161)( 74,162)( 75,163)( 76,164)( 77,165)( 78,166)( 79,167)
( 80,168)( 81,186)( 82,185)( 83,188)( 84,187)( 85,190)( 86,189)( 87,192)
( 88,191)( 89,177)( 90,178)( 91,179)( 92,180)( 93,181)( 94,182)( 95,183)
( 96,184);
s3 := Sym(192)!( 9, 11)( 10, 12)( 13, 15)( 14, 16)( 17, 21)( 18, 22)( 19, 23)
( 20, 24)( 25, 31)( 26, 32)( 27, 29)( 28, 30)( 41, 43)( 42, 44)( 45, 47)
( 46, 48)( 49, 53)( 50, 54)( 51, 55)( 52, 56)( 57, 63)( 58, 64)( 59, 61)
( 60, 62)( 73, 75)( 74, 76)( 77, 79)( 78, 80)( 81, 85)( 82, 86)( 83, 87)
( 84, 88)( 89, 95)( 90, 96)( 91, 93)( 92, 94)(105,107)(106,108)(109,111)
(110,112)(113,117)(114,118)(115,119)(116,120)(121,127)(122,128)(123,125)
(124,126)(137,139)(138,140)(141,143)(142,144)(145,149)(146,150)(147,151)
(148,152)(153,159)(154,160)(155,157)(156,158)(169,171)(170,172)(173,175)
(174,176)(177,181)(178,182)(179,183)(180,184)(185,191)(186,192)(187,189)
(188,190);
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, s2*s3*s2*s3*s2*s3*s2*s3,
s2*s0*s1*s2*s3*s2*s1*s2*s0*s1*s2*s3*s2*s1,
s1*s3*s2*s3*s2*s1*s2*s1*s3*s2*s3*s2*s1*s2,
s2*s0*s1*s2*s1*s2*s0*s1*s2*s0*s1*s2*s1*s2*s0*s1 >;
References : None.
to this polytope