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