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