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}*768e
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}*384e, {2,4,6,4}*384g
4-fold quotients : {2,2,6,4}*192b, {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, 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,
s2*s3*s4*s2*s3*s4*s2*s3*s4, s1*s2*s3*s2*s3*s2*s1*s2*s3*s2*s3*s2 ];;
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, 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, s2*s3*s4*s2*s3*s4*s2*s3*s4,
s1*s2*s3*s2*s3*s2*s1*s2*s3*s2*s3*s2 >;
to this polytope