Polytope of Type {2,4,6,4}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,4,6,4}*768d
if this polytope has a name.
Group : SmallGroup(768,1090234)
Rank : 5
Schlafli Type : {2,4,6,4}
Number of vertices, edges, etc : 2, 8, 24, 24, 4
Order of s0s1s2s3s4 : 6
Order of s0s1s2s3s4s3s2s1 : 2
Special Properties :
Degenerate
Universal
Non-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 : {2,4,3,4}*384b, {2,4,6,4}*384d, {2,4,6,4}*384f
4-fold quotients : {2,2,6,4}*192c, {2,4,3,4}*192
8-fold quotients : {2,2,3,4}*96
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := (1,2);;
s1 := ( 3,107)( 4,108)( 5,109)( 6,110)( 7,111)( 8,112)( 9,113)( 10,114)
( 11, 99)( 12,100)( 13,101)( 14,102)( 15,103)( 16,104)( 17,105)( 18,106)
( 19,123)( 20,124)( 21,125)( 22,126)( 23,127)( 24,128)( 25,129)( 26,130)
( 27,115)( 28,116)( 29,117)( 30,118)( 31,119)( 32,120)( 33,121)( 34,122)
( 35,139)( 36,140)( 37,141)( 38,142)( 39,143)( 40,144)( 41,145)( 42,146)
( 43,131)( 44,132)( 45,133)( 46,134)( 47,135)( 48,136)( 49,137)( 50,138)
( 51,155)( 52,156)( 53,157)( 54,158)( 55,159)( 56,160)( 57,161)( 58,162)
( 59,147)( 60,148)( 61,149)( 62,150)( 63,151)( 64,152)( 65,153)( 66,154)
( 67,171)( 68,172)( 69,173)( 70,174)( 71,175)( 72,176)( 73,177)( 74,178)
( 75,163)( 76,164)( 77,165)( 78,166)( 79,167)( 80,168)( 81,169)( 82,170)
( 83,187)( 84,188)( 85,189)( 86,190)( 87,191)( 88,192)( 89,193)( 90,194)
( 91,179)( 92,180)( 93,181)( 94,182)( 95,183)( 96,184)( 97,185)( 98,186);;
s2 := ( 4, 5)( 7, 11)( 8, 13)( 9, 12)( 10, 14)( 16, 17)( 19, 35)( 20, 37)
( 21, 36)( 22, 38)( 23, 43)( 24, 45)( 25, 44)( 26, 46)( 27, 39)( 28, 41)
( 29, 40)( 30, 42)( 31, 47)( 32, 49)( 33, 48)( 34, 50)( 52, 53)( 55, 59)
( 56, 61)( 57, 60)( 58, 62)( 64, 65)( 67, 83)( 68, 85)( 69, 84)( 70, 86)
( 71, 91)( 72, 93)( 73, 92)( 74, 94)( 75, 87)( 76, 89)( 77, 88)( 78, 90)
( 79, 95)( 80, 97)( 81, 96)( 82, 98)(100,101)(103,107)(104,109)(105,108)
(106,110)(112,113)(115,131)(116,133)(117,132)(118,134)(119,139)(120,141)
(121,140)(122,142)(123,135)(124,137)(125,136)(126,138)(127,143)(128,145)
(129,144)(130,146)(148,149)(151,155)(152,157)(153,156)(154,158)(160,161)
(163,179)(164,181)(165,180)(166,182)(167,187)(168,189)(169,188)(170,190)
(171,183)(172,185)(173,184)(174,186)(175,191)(176,193)(177,192)(178,194);;
s3 := ( 3, 83)( 4, 84)( 5, 86)( 6, 85)( 7, 95)( 8, 96)( 9, 98)( 10, 97)
( 11, 91)( 12, 92)( 13, 94)( 14, 93)( 15, 87)( 16, 88)( 17, 90)( 18, 89)
( 19, 67)( 20, 68)( 21, 70)( 22, 69)( 23, 79)( 24, 80)( 25, 82)( 26, 81)
( 27, 75)( 28, 76)( 29, 78)( 30, 77)( 31, 71)( 32, 72)( 33, 74)( 34, 73)
( 35, 51)( 36, 52)( 37, 54)( 38, 53)( 39, 63)( 40, 64)( 41, 66)( 42, 65)
( 43, 59)( 44, 60)( 45, 62)( 46, 61)( 47, 55)( 48, 56)( 49, 58)( 50, 57)
( 99,179)(100,180)(101,182)(102,181)(103,191)(104,192)(105,194)(106,193)
(107,187)(108,188)(109,190)(110,189)(111,183)(112,184)(113,186)(114,185)
(115,163)(116,164)(117,166)(118,165)(119,175)(120,176)(121,178)(122,177)
(123,171)(124,172)(125,174)(126,173)(127,167)(128,168)(129,170)(130,169)
(131,147)(132,148)(133,150)(134,149)(135,159)(136,160)(137,162)(138,161)
(139,155)(140,156)(141,158)(142,157)(143,151)(144,152)(145,154)(146,153);;
s4 := ( 3, 6)( 4, 5)( 7, 10)( 8, 9)( 11, 14)( 12, 13)( 15, 18)( 16, 17)
( 19, 22)( 20, 21)( 23, 26)( 24, 25)( 27, 30)( 28, 29)( 31, 34)( 32, 33)
( 35, 38)( 36, 37)( 39, 42)( 40, 41)( 43, 46)( 44, 45)( 47, 50)( 48, 49)
( 51, 54)( 52, 53)( 55, 58)( 56, 57)( 59, 62)( 60, 61)( 63, 66)( 64, 65)
( 67, 70)( 68, 69)( 71, 74)( 72, 73)( 75, 78)( 76, 77)( 79, 82)( 80, 81)
( 83, 86)( 84, 85)( 87, 90)( 88, 89)( 91, 94)( 92, 93)( 95, 98)( 96, 97)
( 99,102)(100,101)(103,106)(104,105)(107,110)(108,109)(111,114)(112,113)
(115,118)(116,117)(119,122)(120,121)(123,126)(124,125)(127,130)(128,129)
(131,134)(132,133)(135,138)(136,137)(139,142)(140,141)(143,146)(144,145)
(147,150)(148,149)(151,154)(152,153)(155,158)(156,157)(159,162)(160,161)
(163,166)(164,165)(167,170)(168,169)(171,174)(172,173)(175,178)(176,177)
(179,182)(180,181)(183,186)(184,185)(187,190)(188,189)(191,194)(192,193);;
poly := Group([s0,s1,s2,s3,s4]);;
Finitely Presented Group Representation (GAP) :
F := FreeGroup("s0","s1","s2","s3","s4");;
s0 := F.1;; s1 := F.2;; s2 := F.3;; s3 := F.4;; s4 := F.5;;
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s4*s4, s0*s1*s0*s1,
s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3,
s0*s4*s0*s4, s1*s4*s1*s4, s2*s4*s2*s4,
s1*s2*s1*s2*s1*s2*s1*s2, s3*s4*s3*s4*s3*s4*s3*s4,
s4*s3*s2*s4*s3*s4*s3*s2*s3, s1*s2*s3*s2*s3*s2*s1*s2*s3*s2*s3*s2,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(194)!(1,2);
s1 := Sym(194)!( 3,107)( 4,108)( 5,109)( 6,110)( 7,111)( 8,112)( 9,113)
( 10,114)( 11, 99)( 12,100)( 13,101)( 14,102)( 15,103)( 16,104)( 17,105)
( 18,106)( 19,123)( 20,124)( 21,125)( 22,126)( 23,127)( 24,128)( 25,129)
( 26,130)( 27,115)( 28,116)( 29,117)( 30,118)( 31,119)( 32,120)( 33,121)
( 34,122)( 35,139)( 36,140)( 37,141)( 38,142)( 39,143)( 40,144)( 41,145)
( 42,146)( 43,131)( 44,132)( 45,133)( 46,134)( 47,135)( 48,136)( 49,137)
( 50,138)( 51,155)( 52,156)( 53,157)( 54,158)( 55,159)( 56,160)( 57,161)
( 58,162)( 59,147)( 60,148)( 61,149)( 62,150)( 63,151)( 64,152)( 65,153)
( 66,154)( 67,171)( 68,172)( 69,173)( 70,174)( 71,175)( 72,176)( 73,177)
( 74,178)( 75,163)( 76,164)( 77,165)( 78,166)( 79,167)( 80,168)( 81,169)
( 82,170)( 83,187)( 84,188)( 85,189)( 86,190)( 87,191)( 88,192)( 89,193)
( 90,194)( 91,179)( 92,180)( 93,181)( 94,182)( 95,183)( 96,184)( 97,185)
( 98,186);
s2 := Sym(194)!( 4, 5)( 7, 11)( 8, 13)( 9, 12)( 10, 14)( 16, 17)( 19, 35)
( 20, 37)( 21, 36)( 22, 38)( 23, 43)( 24, 45)( 25, 44)( 26, 46)( 27, 39)
( 28, 41)( 29, 40)( 30, 42)( 31, 47)( 32, 49)( 33, 48)( 34, 50)( 52, 53)
( 55, 59)( 56, 61)( 57, 60)( 58, 62)( 64, 65)( 67, 83)( 68, 85)( 69, 84)
( 70, 86)( 71, 91)( 72, 93)( 73, 92)( 74, 94)( 75, 87)( 76, 89)( 77, 88)
( 78, 90)( 79, 95)( 80, 97)( 81, 96)( 82, 98)(100,101)(103,107)(104,109)
(105,108)(106,110)(112,113)(115,131)(116,133)(117,132)(118,134)(119,139)
(120,141)(121,140)(122,142)(123,135)(124,137)(125,136)(126,138)(127,143)
(128,145)(129,144)(130,146)(148,149)(151,155)(152,157)(153,156)(154,158)
(160,161)(163,179)(164,181)(165,180)(166,182)(167,187)(168,189)(169,188)
(170,190)(171,183)(172,185)(173,184)(174,186)(175,191)(176,193)(177,192)
(178,194);
s3 := Sym(194)!( 3, 83)( 4, 84)( 5, 86)( 6, 85)( 7, 95)( 8, 96)( 9, 98)
( 10, 97)( 11, 91)( 12, 92)( 13, 94)( 14, 93)( 15, 87)( 16, 88)( 17, 90)
( 18, 89)( 19, 67)( 20, 68)( 21, 70)( 22, 69)( 23, 79)( 24, 80)( 25, 82)
( 26, 81)( 27, 75)( 28, 76)( 29, 78)( 30, 77)( 31, 71)( 32, 72)( 33, 74)
( 34, 73)( 35, 51)( 36, 52)( 37, 54)( 38, 53)( 39, 63)( 40, 64)( 41, 66)
( 42, 65)( 43, 59)( 44, 60)( 45, 62)( 46, 61)( 47, 55)( 48, 56)( 49, 58)
( 50, 57)( 99,179)(100,180)(101,182)(102,181)(103,191)(104,192)(105,194)
(106,193)(107,187)(108,188)(109,190)(110,189)(111,183)(112,184)(113,186)
(114,185)(115,163)(116,164)(117,166)(118,165)(119,175)(120,176)(121,178)
(122,177)(123,171)(124,172)(125,174)(126,173)(127,167)(128,168)(129,170)
(130,169)(131,147)(132,148)(133,150)(134,149)(135,159)(136,160)(137,162)
(138,161)(139,155)(140,156)(141,158)(142,157)(143,151)(144,152)(145,154)
(146,153);
s4 := Sym(194)!( 3, 6)( 4, 5)( 7, 10)( 8, 9)( 11, 14)( 12, 13)( 15, 18)
( 16, 17)( 19, 22)( 20, 21)( 23, 26)( 24, 25)( 27, 30)( 28, 29)( 31, 34)
( 32, 33)( 35, 38)( 36, 37)( 39, 42)( 40, 41)( 43, 46)( 44, 45)( 47, 50)
( 48, 49)( 51, 54)( 52, 53)( 55, 58)( 56, 57)( 59, 62)( 60, 61)( 63, 66)
( 64, 65)( 67, 70)( 68, 69)( 71, 74)( 72, 73)( 75, 78)( 76, 77)( 79, 82)
( 80, 81)( 83, 86)( 84, 85)( 87, 90)( 88, 89)( 91, 94)( 92, 93)( 95, 98)
( 96, 97)( 99,102)(100,101)(103,106)(104,105)(107,110)(108,109)(111,114)
(112,113)(115,118)(116,117)(119,122)(120,121)(123,126)(124,125)(127,130)
(128,129)(131,134)(132,133)(135,138)(136,137)(139,142)(140,141)(143,146)
(144,145)(147,150)(148,149)(151,154)(152,153)(155,158)(156,157)(159,162)
(160,161)(163,166)(164,165)(167,170)(168,169)(171,174)(172,173)(175,178)
(176,177)(179,182)(180,181)(183,186)(184,185)(187,190)(188,189)(191,194)
(192,193);
poly := sub<Sym(194)|s0,s1,s2,s3,s4>;
Finitely Presented Group Representation (Magma) :
poly<s0,s1,s2,s3,s4> := Group< s0,s1,s2,s3,s4 | s0*s0, s1*s1, s2*s2,
s3*s3, s4*s4, s0*s1*s0*s1, s0*s2*s0*s2,
s0*s3*s0*s3, s1*s3*s1*s3, s0*s4*s0*s4,
s1*s4*s1*s4, s2*s4*s2*s4, s1*s2*s1*s2*s1*s2*s1*s2,
s3*s4*s3*s4*s3*s4*s3*s4, s4*s3*s2*s4*s3*s4*s3*s2*s3,
s1*s2*s3*s2*s3*s2*s1*s2*s3*s2*s3*s2,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 >;
to this polytope