Polytope of Type {3,8,4}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {3,8,4}*768d
if this polytope has a name.
Group : SmallGroup(768,1088009)
Rank : 4
Schlafli Type : {3,8,4}
Number of vertices, edges, etc : 12, 48, 64, 4
Order of s0s1s2s3 : 12
Order of s0s1s2s3s2s1 : 4
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) :
4-fold quotients : {3,4,4}*192b
8-fold quotients : {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, 6)( 9, 13)( 10, 14)( 11, 16)( 12, 15)( 17, 33)( 18, 34)
( 19, 36)( 20, 35)( 21, 38)( 22, 37)( 23, 39)( 24, 40)( 25, 45)( 26, 46)
( 27, 48)( 28, 47)( 29, 41)( 30, 42)( 31, 44)( 32, 43)( 51, 52)( 53, 54)
( 57, 61)( 58, 62)( 59, 64)( 60, 63)( 65, 81)( 66, 82)( 67, 84)( 68, 83)
( 69, 86)( 70, 85)( 71, 87)( 72, 88)( 73, 93)( 74, 94)( 75, 96)( 76, 95)
( 77, 89)( 78, 90)( 79, 92)( 80, 91)( 99,100)(101,102)(105,109)(106,110)
(107,112)(108,111)(113,129)(114,130)(115,132)(116,131)(117,134)(118,133)
(119,135)(120,136)(121,141)(122,142)(123,144)(124,143)(125,137)(126,138)
(127,140)(128,139)(147,148)(149,150)(153,157)(154,158)(155,160)(156,159)
(161,177)(162,178)(163,180)(164,179)(165,182)(166,181)(167,183)(168,184)
(169,189)(170,190)(171,192)(172,191)(173,185)(174,186)(175,188)(176,187);;
s1 := ( 1, 33)( 2, 35)( 3, 34)( 4, 36)( 5, 44)( 6, 42)( 7, 43)( 8, 41)
( 9, 40)( 10, 38)( 11, 39)( 12, 37)( 13, 45)( 14, 47)( 15, 46)( 16, 48)
( 18, 19)( 21, 28)( 22, 26)( 23, 27)( 24, 25)( 30, 31)( 49, 81)( 50, 83)
( 51, 82)( 52, 84)( 53, 92)( 54, 90)( 55, 91)( 56, 89)( 57, 88)( 58, 86)
( 59, 87)( 60, 85)( 61, 93)( 62, 95)( 63, 94)( 64, 96)( 66, 67)( 69, 76)
( 70, 74)( 71, 75)( 72, 73)( 78, 79)( 97,129)( 98,131)( 99,130)(100,132)
(101,140)(102,138)(103,139)(104,137)(105,136)(106,134)(107,135)(108,133)
(109,141)(110,143)(111,142)(112,144)(114,115)(117,124)(118,122)(119,123)
(120,121)(126,127)(145,177)(146,179)(147,178)(148,180)(149,188)(150,186)
(151,187)(152,185)(153,184)(154,182)(155,183)(156,181)(157,189)(158,191)
(159,190)(160,192)(162,163)(165,172)(166,170)(167,171)(168,169)(174,175);;
s2 := ( 1, 7)( 2, 8)( 3, 5)( 4, 6)( 9, 13)( 10, 14)( 11, 15)( 12, 16)
( 17, 23)( 18, 24)( 19, 21)( 20, 22)( 25, 29)( 26, 30)( 27, 31)( 28, 32)
( 33, 39)( 34, 40)( 35, 37)( 36, 38)( 41, 45)( 42, 46)( 43, 47)( 44, 48)
( 49, 55)( 50, 56)( 51, 53)( 52, 54)( 57, 61)( 58, 62)( 59, 63)( 60, 64)
( 65, 71)( 66, 72)( 67, 69)( 68, 70)( 73, 77)( 74, 78)( 75, 79)( 76, 80)
( 81, 87)( 82, 88)( 83, 85)( 84, 86)( 89, 93)( 90, 94)( 91, 95)( 92, 96)
( 97,151)( 98,152)( 99,149)(100,150)(101,147)(102,148)(103,145)(104,146)
(105,157)(106,158)(107,159)(108,160)(109,153)(110,154)(111,155)(112,156)
(113,167)(114,168)(115,165)(116,166)(117,163)(118,164)(119,161)(120,162)
(121,173)(122,174)(123,175)(124,176)(125,169)(126,170)(127,171)(128,172)
(129,183)(130,184)(131,181)(132,182)(133,179)(134,180)(135,177)(136,178)
(137,189)(138,190)(139,191)(140,192)(141,185)(142,186)(143,187)(144,188);;
s3 := ( 1, 97)( 2, 98)( 3, 99)( 4,100)( 5,102)( 6,101)( 7,104)( 8,103)
( 9,107)( 10,108)( 11,105)( 12,106)( 13,112)( 14,111)( 15,110)( 16,109)
( 17,113)( 18,114)( 19,115)( 20,116)( 21,118)( 22,117)( 23,120)( 24,119)
( 25,123)( 26,124)( 27,121)( 28,122)( 29,128)( 30,127)( 31,126)( 32,125)
( 33,129)( 34,130)( 35,131)( 36,132)( 37,134)( 38,133)( 39,136)( 40,135)
( 41,139)( 42,140)( 43,137)( 44,138)( 45,144)( 46,143)( 47,142)( 48,141)
( 49,145)( 50,146)( 51,147)( 52,148)( 53,150)( 54,149)( 55,152)( 56,151)
( 57,155)( 58,156)( 59,153)( 60,154)( 61,160)( 62,159)( 63,158)( 64,157)
( 65,161)( 66,162)( 67,163)( 68,164)( 69,166)( 70,165)( 71,168)( 72,167)
( 73,171)( 74,172)( 75,169)( 76,170)( 77,176)( 78,175)( 79,174)( 80,173)
( 81,177)( 82,178)( 83,179)( 84,180)( 85,182)( 86,181)( 87,184)( 88,183)
( 89,187)( 90,188)( 91,185)( 92,186)( 93,192)( 94,191)( 95,190)( 96,189);;
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, s3*s1*s2*s1*s2*s3*s1*s2*s1*s2 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(192)!( 3, 4)( 5, 6)( 9, 13)( 10, 14)( 11, 16)( 12, 15)( 17, 33)
( 18, 34)( 19, 36)( 20, 35)( 21, 38)( 22, 37)( 23, 39)( 24, 40)( 25, 45)
( 26, 46)( 27, 48)( 28, 47)( 29, 41)( 30, 42)( 31, 44)( 32, 43)( 51, 52)
( 53, 54)( 57, 61)( 58, 62)( 59, 64)( 60, 63)( 65, 81)( 66, 82)( 67, 84)
( 68, 83)( 69, 86)( 70, 85)( 71, 87)( 72, 88)( 73, 93)( 74, 94)( 75, 96)
( 76, 95)( 77, 89)( 78, 90)( 79, 92)( 80, 91)( 99,100)(101,102)(105,109)
(106,110)(107,112)(108,111)(113,129)(114,130)(115,132)(116,131)(117,134)
(118,133)(119,135)(120,136)(121,141)(122,142)(123,144)(124,143)(125,137)
(126,138)(127,140)(128,139)(147,148)(149,150)(153,157)(154,158)(155,160)
(156,159)(161,177)(162,178)(163,180)(164,179)(165,182)(166,181)(167,183)
(168,184)(169,189)(170,190)(171,192)(172,191)(173,185)(174,186)(175,188)
(176,187);
s1 := Sym(192)!( 1, 33)( 2, 35)( 3, 34)( 4, 36)( 5, 44)( 6, 42)( 7, 43)
( 8, 41)( 9, 40)( 10, 38)( 11, 39)( 12, 37)( 13, 45)( 14, 47)( 15, 46)
( 16, 48)( 18, 19)( 21, 28)( 22, 26)( 23, 27)( 24, 25)( 30, 31)( 49, 81)
( 50, 83)( 51, 82)( 52, 84)( 53, 92)( 54, 90)( 55, 91)( 56, 89)( 57, 88)
( 58, 86)( 59, 87)( 60, 85)( 61, 93)( 62, 95)( 63, 94)( 64, 96)( 66, 67)
( 69, 76)( 70, 74)( 71, 75)( 72, 73)( 78, 79)( 97,129)( 98,131)( 99,130)
(100,132)(101,140)(102,138)(103,139)(104,137)(105,136)(106,134)(107,135)
(108,133)(109,141)(110,143)(111,142)(112,144)(114,115)(117,124)(118,122)
(119,123)(120,121)(126,127)(145,177)(146,179)(147,178)(148,180)(149,188)
(150,186)(151,187)(152,185)(153,184)(154,182)(155,183)(156,181)(157,189)
(158,191)(159,190)(160,192)(162,163)(165,172)(166,170)(167,171)(168,169)
(174,175);
s2 := Sym(192)!( 1, 7)( 2, 8)( 3, 5)( 4, 6)( 9, 13)( 10, 14)( 11, 15)
( 12, 16)( 17, 23)( 18, 24)( 19, 21)( 20, 22)( 25, 29)( 26, 30)( 27, 31)
( 28, 32)( 33, 39)( 34, 40)( 35, 37)( 36, 38)( 41, 45)( 42, 46)( 43, 47)
( 44, 48)( 49, 55)( 50, 56)( 51, 53)( 52, 54)( 57, 61)( 58, 62)( 59, 63)
( 60, 64)( 65, 71)( 66, 72)( 67, 69)( 68, 70)( 73, 77)( 74, 78)( 75, 79)
( 76, 80)( 81, 87)( 82, 88)( 83, 85)( 84, 86)( 89, 93)( 90, 94)( 91, 95)
( 92, 96)( 97,151)( 98,152)( 99,149)(100,150)(101,147)(102,148)(103,145)
(104,146)(105,157)(106,158)(107,159)(108,160)(109,153)(110,154)(111,155)
(112,156)(113,167)(114,168)(115,165)(116,166)(117,163)(118,164)(119,161)
(120,162)(121,173)(122,174)(123,175)(124,176)(125,169)(126,170)(127,171)
(128,172)(129,183)(130,184)(131,181)(132,182)(133,179)(134,180)(135,177)
(136,178)(137,189)(138,190)(139,191)(140,192)(141,185)(142,186)(143,187)
(144,188);
s3 := Sym(192)!( 1, 97)( 2, 98)( 3, 99)( 4,100)( 5,102)( 6,101)( 7,104)
( 8,103)( 9,107)( 10,108)( 11,105)( 12,106)( 13,112)( 14,111)( 15,110)
( 16,109)( 17,113)( 18,114)( 19,115)( 20,116)( 21,118)( 22,117)( 23,120)
( 24,119)( 25,123)( 26,124)( 27,121)( 28,122)( 29,128)( 30,127)( 31,126)
( 32,125)( 33,129)( 34,130)( 35,131)( 36,132)( 37,134)( 38,133)( 39,136)
( 40,135)( 41,139)( 42,140)( 43,137)( 44,138)( 45,144)( 46,143)( 47,142)
( 48,141)( 49,145)( 50,146)( 51,147)( 52,148)( 53,150)( 54,149)( 55,152)
( 56,151)( 57,155)( 58,156)( 59,153)( 60,154)( 61,160)( 62,159)( 63,158)
( 64,157)( 65,161)( 66,162)( 67,163)( 68,164)( 69,166)( 70,165)( 71,168)
( 72,167)( 73,171)( 74,172)( 75,169)( 76,170)( 77,176)( 78,175)( 79,174)
( 80,173)( 81,177)( 82,178)( 83,179)( 84,180)( 85,182)( 86,181)( 87,184)
( 88,183)( 89,187)( 90,188)( 91,185)( 92,186)( 93,192)( 94,191)( 95,190)
( 96,189);
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,
s3*s1*s2*s1*s2*s3*s1*s2*s1*s2 >;
References : None.
to this polytope