Polytope of Type {3,8,8}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {3,8,8}*768
if this polytope has a name.
Group : SmallGroup(768,1086649)
Rank : 4
Schlafli Type : {3,8,8}
Number of vertices, edges, etc : 6, 24, 64, 8
Order of s0s1s2s3 : 24
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 : {3,8,4}*384, {3,4,8}*384
4-fold quotients : {3,4,4}*192b, {3,8,2}*192
8-fold quotients : {3,2,8}*96, {3,4,2}*96
16-fold quotients : {3,2,4}*48, {3,4,2}*48
32-fold quotients : {3,2,2}*24
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := ( 3, 4)( 5, 7)( 6, 8)( 9, 17)( 10, 18)( 11, 20)( 12, 19)( 13, 23)
( 14, 24)( 15, 21)( 16, 22)( 27, 28)( 29, 31)( 30, 32)( 33, 41)( 34, 42)
( 35, 44)( 36, 43)( 37, 47)( 38, 48)( 39, 45)( 40, 46)( 51, 52)( 53, 55)
( 54, 56)( 57, 65)( 58, 66)( 59, 68)( 60, 67)( 61, 71)( 62, 72)( 63, 69)
( 64, 70)( 75, 76)( 77, 79)( 78, 80)( 81, 89)( 82, 90)( 83, 92)( 84, 91)
( 85, 95)( 86, 96)( 87, 93)( 88, 94)( 97, 98)(101,104)(102,103)(105,114)
(106,113)(107,115)(108,116)(109,120)(110,119)(111,118)(112,117)(121,122)
(125,128)(126,127)(129,138)(130,137)(131,139)(132,140)(133,144)(134,143)
(135,142)(136,141)(145,146)(149,152)(150,151)(153,162)(154,161)(155,163)
(156,164)(157,168)(158,167)(159,166)(160,165)(169,170)(173,176)(174,175)
(177,186)(178,185)(179,187)(180,188)(181,192)(182,191)(183,190)(184,189);;
s1 := ( 1, 17)( 2, 18)( 3, 22)( 4, 21)( 5, 20)( 6, 19)( 7, 24)( 8, 23)
( 11, 14)( 12, 13)( 15, 16)( 25, 41)( 26, 42)( 27, 46)( 28, 45)( 29, 44)
( 30, 43)( 31, 48)( 32, 47)( 35, 38)( 36, 37)( 39, 40)( 49, 65)( 50, 66)
( 51, 70)( 52, 69)( 53, 68)( 54, 67)( 55, 72)( 56, 71)( 59, 62)( 60, 61)
( 63, 64)( 73, 89)( 74, 90)( 75, 94)( 76, 93)( 77, 92)( 78, 91)( 79, 96)
( 80, 95)( 83, 86)( 84, 85)( 87, 88)( 97,114)( 98,113)( 99,117)(100,118)
(101,115)(102,116)(103,119)(104,120)(105,106)(107,109)(108,110)(121,138)
(122,137)(123,141)(124,142)(125,139)(126,140)(127,143)(128,144)(129,130)
(131,133)(132,134)(145,162)(146,161)(147,165)(148,166)(149,163)(150,164)
(151,167)(152,168)(153,154)(155,157)(156,158)(169,186)(170,185)(171,189)
(172,190)(173,187)(174,188)(175,191)(176,192)(177,178)(179,181)(180,182);;
s2 := ( 1,100)( 2, 99)( 3, 97)( 4, 98)( 5,104)( 6,103)( 7,101)( 8,102)
( 9,108)( 10,107)( 11,105)( 12,106)( 13,112)( 14,111)( 15,109)( 16,110)
( 17,116)( 18,115)( 19,113)( 20,114)( 21,120)( 22,119)( 23,117)( 24,118)
( 25,124)( 26,123)( 27,121)( 28,122)( 29,128)( 30,127)( 31,125)( 32,126)
( 33,132)( 34,131)( 35,129)( 36,130)( 37,136)( 38,135)( 39,133)( 40,134)
( 41,140)( 42,139)( 43,137)( 44,138)( 45,144)( 46,143)( 47,141)( 48,142)
( 49,172)( 50,171)( 51,169)( 52,170)( 53,176)( 54,175)( 55,173)( 56,174)
( 57,180)( 58,179)( 59,177)( 60,178)( 61,184)( 62,183)( 63,181)( 64,182)
( 65,188)( 66,187)( 67,185)( 68,186)( 69,192)( 70,191)( 71,189)( 72,190)
( 73,148)( 74,147)( 75,145)( 76,146)( 77,152)( 78,151)( 79,149)( 80,150)
( 81,156)( 82,155)( 83,153)( 84,154)( 85,160)( 86,159)( 87,157)( 88,158)
( 89,164)( 90,163)( 91,161)( 92,162)( 93,168)( 94,167)( 95,165)( 96,166);;
s3 := ( 49, 73)( 50, 74)( 51, 75)( 52, 76)( 53, 77)( 54, 78)( 55, 79)( 56, 80)
( 57, 81)( 58, 82)( 59, 83)( 60, 84)( 61, 85)( 62, 86)( 63, 87)( 64, 88)
( 65, 89)( 66, 90)( 67, 91)( 68, 92)( 69, 93)( 70, 94)( 71, 95)( 72, 96)
( 97,145)( 98,146)( 99,147)(100,148)(101,149)(102,150)(103,151)(104,152)
(105,153)(106,154)(107,155)(108,156)(109,157)(110,158)(111,159)(112,160)
(113,161)(114,162)(115,163)(116,164)(117,165)(118,166)(119,167)(120,168)
(121,169)(122,170)(123,171)(124,172)(125,173)(126,174)(127,175)(128,176)
(129,177)(130,178)(131,179)(132,180)(133,181)(134,182)(135,183)(136,184)
(137,185)(138,186)(139,187)(140,188)(141,189)(142,190)(143,191)(144,192);;
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,
s1*s2*s3*s2*s1*s2*s3*s2, s2*s0*s1*s2*s1*s2*s0*s1*s2*s0*s1*s2*s1*s2*s0*s1,
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, 7)( 6, 8)( 9, 17)( 10, 18)( 11, 20)( 12, 19)
( 13, 23)( 14, 24)( 15, 21)( 16, 22)( 27, 28)( 29, 31)( 30, 32)( 33, 41)
( 34, 42)( 35, 44)( 36, 43)( 37, 47)( 38, 48)( 39, 45)( 40, 46)( 51, 52)
( 53, 55)( 54, 56)( 57, 65)( 58, 66)( 59, 68)( 60, 67)( 61, 71)( 62, 72)
( 63, 69)( 64, 70)( 75, 76)( 77, 79)( 78, 80)( 81, 89)( 82, 90)( 83, 92)
( 84, 91)( 85, 95)( 86, 96)( 87, 93)( 88, 94)( 97, 98)(101,104)(102,103)
(105,114)(106,113)(107,115)(108,116)(109,120)(110,119)(111,118)(112,117)
(121,122)(125,128)(126,127)(129,138)(130,137)(131,139)(132,140)(133,144)
(134,143)(135,142)(136,141)(145,146)(149,152)(150,151)(153,162)(154,161)
(155,163)(156,164)(157,168)(158,167)(159,166)(160,165)(169,170)(173,176)
(174,175)(177,186)(178,185)(179,187)(180,188)(181,192)(182,191)(183,190)
(184,189);
s1 := Sym(192)!( 1, 17)( 2, 18)( 3, 22)( 4, 21)( 5, 20)( 6, 19)( 7, 24)
( 8, 23)( 11, 14)( 12, 13)( 15, 16)( 25, 41)( 26, 42)( 27, 46)( 28, 45)
( 29, 44)( 30, 43)( 31, 48)( 32, 47)( 35, 38)( 36, 37)( 39, 40)( 49, 65)
( 50, 66)( 51, 70)( 52, 69)( 53, 68)( 54, 67)( 55, 72)( 56, 71)( 59, 62)
( 60, 61)( 63, 64)( 73, 89)( 74, 90)( 75, 94)( 76, 93)( 77, 92)( 78, 91)
( 79, 96)( 80, 95)( 83, 86)( 84, 85)( 87, 88)( 97,114)( 98,113)( 99,117)
(100,118)(101,115)(102,116)(103,119)(104,120)(105,106)(107,109)(108,110)
(121,138)(122,137)(123,141)(124,142)(125,139)(126,140)(127,143)(128,144)
(129,130)(131,133)(132,134)(145,162)(146,161)(147,165)(148,166)(149,163)
(150,164)(151,167)(152,168)(153,154)(155,157)(156,158)(169,186)(170,185)
(171,189)(172,190)(173,187)(174,188)(175,191)(176,192)(177,178)(179,181)
(180,182);
s2 := Sym(192)!( 1,100)( 2, 99)( 3, 97)( 4, 98)( 5,104)( 6,103)( 7,101)
( 8,102)( 9,108)( 10,107)( 11,105)( 12,106)( 13,112)( 14,111)( 15,109)
( 16,110)( 17,116)( 18,115)( 19,113)( 20,114)( 21,120)( 22,119)( 23,117)
( 24,118)( 25,124)( 26,123)( 27,121)( 28,122)( 29,128)( 30,127)( 31,125)
( 32,126)( 33,132)( 34,131)( 35,129)( 36,130)( 37,136)( 38,135)( 39,133)
( 40,134)( 41,140)( 42,139)( 43,137)( 44,138)( 45,144)( 46,143)( 47,141)
( 48,142)( 49,172)( 50,171)( 51,169)( 52,170)( 53,176)( 54,175)( 55,173)
( 56,174)( 57,180)( 58,179)( 59,177)( 60,178)( 61,184)( 62,183)( 63,181)
( 64,182)( 65,188)( 66,187)( 67,185)( 68,186)( 69,192)( 70,191)( 71,189)
( 72,190)( 73,148)( 74,147)( 75,145)( 76,146)( 77,152)( 78,151)( 79,149)
( 80,150)( 81,156)( 82,155)( 83,153)( 84,154)( 85,160)( 86,159)( 87,157)
( 88,158)( 89,164)( 90,163)( 91,161)( 92,162)( 93,168)( 94,167)( 95,165)
( 96,166);
s3 := Sym(192)!( 49, 73)( 50, 74)( 51, 75)( 52, 76)( 53, 77)( 54, 78)( 55, 79)
( 56, 80)( 57, 81)( 58, 82)( 59, 83)( 60, 84)( 61, 85)( 62, 86)( 63, 87)
( 64, 88)( 65, 89)( 66, 90)( 67, 91)( 68, 92)( 69, 93)( 70, 94)( 71, 95)
( 72, 96)( 97,145)( 98,146)( 99,147)(100,148)(101,149)(102,150)(103,151)
(104,152)(105,153)(106,154)(107,155)(108,156)(109,157)(110,158)(111,159)
(112,160)(113,161)(114,162)(115,163)(116,164)(117,165)(118,166)(119,167)
(120,168)(121,169)(122,170)(123,171)(124,172)(125,173)(126,174)(127,175)
(128,176)(129,177)(130,178)(131,179)(132,180)(133,181)(134,182)(135,183)
(136,184)(137,185)(138,186)(139,187)(140,188)(141,189)(142,190)(143,191)
(144,192);
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, s1*s2*s3*s2*s1*s2*s3*s2,
s2*s0*s1*s2*s1*s2*s0*s1*s2*s0*s1*s2*s1*s2*s0*s1,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 >;
References : None.
to this polytope