Polytope of Type {2,6,8,4}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,6,8,4}*768b
if this polytope has a name.
Group : SmallGroup(768,1036167)
Rank : 5
Schlafli Type : {2,6,8,4}
Number of vertices, edges, etc : 2, 6, 24, 16, 4
Order of s0s1s2s3s4 : 24
Order of s0s1s2s3s4s3s2s1 : 2
Special Properties :
Degenerate
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 : {2,6,4,4}*384
3-fold quotients : {2,2,8,4}*256b
4-fold quotients : {2,6,2,4}*192, {2,6,4,2}*192a
6-fold quotients : {2,2,4,4}*128
8-fold quotients : {2,3,2,4}*96, {2,6,2,2}*96
12-fold quotients : {2,2,2,4}*64, {2,2,4,2}*64
16-fold quotients : {2,3,2,2}*48
24-fold quotients : {2,2,2,2}*32
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := (1,2);;
s1 := ( 3, 27)( 4, 29)( 5, 28)( 6, 30)( 7, 32)( 8, 31)( 9, 33)( 10, 35)
( 11, 34)( 12, 36)( 13, 38)( 14, 37)( 15, 39)( 16, 41)( 17, 40)( 18, 42)
( 19, 44)( 20, 43)( 21, 45)( 22, 47)( 23, 46)( 24, 48)( 25, 50)( 26, 49)
( 51, 75)( 52, 77)( 53, 76)( 54, 78)( 55, 80)( 56, 79)( 57, 81)( 58, 83)
( 59, 82)( 60, 84)( 61, 86)( 62, 85)( 63, 87)( 64, 89)( 65, 88)( 66, 90)
( 67, 92)( 68, 91)( 69, 93)( 70, 95)( 71, 94)( 72, 96)( 73, 98)( 74, 97)
( 99,123)(100,125)(101,124)(102,126)(103,128)(104,127)(105,129)(106,131)
(107,130)(108,132)(109,134)(110,133)(111,135)(112,137)(113,136)(114,138)
(115,140)(116,139)(117,141)(118,143)(119,142)(120,144)(121,146)(122,145)
(147,171)(148,173)(149,172)(150,174)(151,176)(152,175)(153,177)(154,179)
(155,178)(156,180)(157,182)(158,181)(159,183)(160,185)(161,184)(162,186)
(163,188)(164,187)(165,189)(166,191)(167,190)(168,192)(169,194)(170,193);;
s2 := ( 3,124)( 4,123)( 5,125)( 6,127)( 7,126)( 8,128)( 9,133)( 10,132)
( 11,134)( 12,130)( 13,129)( 14,131)( 15,139)( 16,138)( 17,140)( 18,136)
( 19,135)( 20,137)( 21,142)( 22,141)( 23,143)( 24,145)( 25,144)( 26,146)
( 27,100)( 28, 99)( 29,101)( 30,103)( 31,102)( 32,104)( 33,109)( 34,108)
( 35,110)( 36,106)( 37,105)( 38,107)( 39,115)( 40,114)( 41,116)( 42,112)
( 43,111)( 44,113)( 45,118)( 46,117)( 47,119)( 48,121)( 49,120)( 50,122)
( 51,172)( 52,171)( 53,173)( 54,175)( 55,174)( 56,176)( 57,181)( 58,180)
( 59,182)( 60,178)( 61,177)( 62,179)( 63,187)( 64,186)( 65,188)( 66,184)
( 67,183)( 68,185)( 69,190)( 70,189)( 71,191)( 72,193)( 73,192)( 74,194)
( 75,148)( 76,147)( 77,149)( 78,151)( 79,150)( 80,152)( 81,157)( 82,156)
( 83,158)( 84,154)( 85,153)( 86,155)( 87,163)( 88,162)( 89,164)( 90,160)
( 91,159)( 92,161)( 93,166)( 94,165)( 95,167)( 96,169)( 97,168)( 98,170);;
s3 := ( 15, 18)( 16, 19)( 17, 20)( 21, 24)( 22, 25)( 23, 26)( 39, 42)( 40, 43)
( 41, 44)( 45, 48)( 46, 49)( 47, 50)( 51, 57)( 52, 58)( 53, 59)( 54, 60)
( 55, 61)( 56, 62)( 63, 72)( 64, 73)( 65, 74)( 66, 69)( 67, 70)( 68, 71)
( 75, 81)( 76, 82)( 77, 83)( 78, 84)( 79, 85)( 80, 86)( 87, 96)( 88, 97)
( 89, 98)( 90, 93)( 91, 94)( 92, 95)( 99,111)(100,112)(101,113)(102,114)
(103,115)(104,116)(105,117)(106,118)(107,119)(108,120)(109,121)(110,122)
(123,135)(124,136)(125,137)(126,138)(127,139)(128,140)(129,141)(130,142)
(131,143)(132,144)(133,145)(134,146)(147,168)(148,169)(149,170)(150,165)
(151,166)(152,167)(153,162)(154,163)(155,164)(156,159)(157,160)(158,161)
(171,192)(172,193)(173,194)(174,189)(175,190)(176,191)(177,186)(178,187)
(179,188)(180,183)(181,184)(182,185);;
s4 := ( 3, 51)( 4, 52)( 5, 53)( 6, 54)( 7, 55)( 8, 56)( 9, 57)( 10, 58)
( 11, 59)( 12, 60)( 13, 61)( 14, 62)( 15, 66)( 16, 67)( 17, 68)( 18, 63)
( 19, 64)( 20, 65)( 21, 72)( 22, 73)( 23, 74)( 24, 69)( 25, 70)( 26, 71)
( 27, 75)( 28, 76)( 29, 77)( 30, 78)( 31, 79)( 32, 80)( 33, 81)( 34, 82)
( 35, 83)( 36, 84)( 37, 85)( 38, 86)( 39, 90)( 40, 91)( 41, 92)( 42, 87)
( 43, 88)( 44, 89)( 45, 96)( 46, 97)( 47, 98)( 48, 93)( 49, 94)( 50, 95)
( 99,147)(100,148)(101,149)(102,150)(103,151)(104,152)(105,153)(106,154)
(107,155)(108,156)(109,157)(110,158)(111,162)(112,163)(113,164)(114,159)
(115,160)(116,161)(117,168)(118,169)(119,170)(120,165)(121,166)(122,167)
(123,171)(124,172)(125,173)(126,174)(127,175)(128,176)(129,177)(130,178)
(131,179)(132,180)(133,181)(134,182)(135,186)(136,187)(137,188)(138,183)
(139,184)(140,185)(141,192)(142,193)(143,194)(144,189)(145,190)(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*s3*s2*s1*s2*s3*s2, s3*s4*s3*s4*s3*s4*s3*s4,
s4*s2*s3*s2*s3*s4*s2*s3*s2*s3, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(194)!(1,2);
s1 := Sym(194)!( 3, 27)( 4, 29)( 5, 28)( 6, 30)( 7, 32)( 8, 31)( 9, 33)
( 10, 35)( 11, 34)( 12, 36)( 13, 38)( 14, 37)( 15, 39)( 16, 41)( 17, 40)
( 18, 42)( 19, 44)( 20, 43)( 21, 45)( 22, 47)( 23, 46)( 24, 48)( 25, 50)
( 26, 49)( 51, 75)( 52, 77)( 53, 76)( 54, 78)( 55, 80)( 56, 79)( 57, 81)
( 58, 83)( 59, 82)( 60, 84)( 61, 86)( 62, 85)( 63, 87)( 64, 89)( 65, 88)
( 66, 90)( 67, 92)( 68, 91)( 69, 93)( 70, 95)( 71, 94)( 72, 96)( 73, 98)
( 74, 97)( 99,123)(100,125)(101,124)(102,126)(103,128)(104,127)(105,129)
(106,131)(107,130)(108,132)(109,134)(110,133)(111,135)(112,137)(113,136)
(114,138)(115,140)(116,139)(117,141)(118,143)(119,142)(120,144)(121,146)
(122,145)(147,171)(148,173)(149,172)(150,174)(151,176)(152,175)(153,177)
(154,179)(155,178)(156,180)(157,182)(158,181)(159,183)(160,185)(161,184)
(162,186)(163,188)(164,187)(165,189)(166,191)(167,190)(168,192)(169,194)
(170,193);
s2 := Sym(194)!( 3,124)( 4,123)( 5,125)( 6,127)( 7,126)( 8,128)( 9,133)
( 10,132)( 11,134)( 12,130)( 13,129)( 14,131)( 15,139)( 16,138)( 17,140)
( 18,136)( 19,135)( 20,137)( 21,142)( 22,141)( 23,143)( 24,145)( 25,144)
( 26,146)( 27,100)( 28, 99)( 29,101)( 30,103)( 31,102)( 32,104)( 33,109)
( 34,108)( 35,110)( 36,106)( 37,105)( 38,107)( 39,115)( 40,114)( 41,116)
( 42,112)( 43,111)( 44,113)( 45,118)( 46,117)( 47,119)( 48,121)( 49,120)
( 50,122)( 51,172)( 52,171)( 53,173)( 54,175)( 55,174)( 56,176)( 57,181)
( 58,180)( 59,182)( 60,178)( 61,177)( 62,179)( 63,187)( 64,186)( 65,188)
( 66,184)( 67,183)( 68,185)( 69,190)( 70,189)( 71,191)( 72,193)( 73,192)
( 74,194)( 75,148)( 76,147)( 77,149)( 78,151)( 79,150)( 80,152)( 81,157)
( 82,156)( 83,158)( 84,154)( 85,153)( 86,155)( 87,163)( 88,162)( 89,164)
( 90,160)( 91,159)( 92,161)( 93,166)( 94,165)( 95,167)( 96,169)( 97,168)
( 98,170);
s3 := Sym(194)!( 15, 18)( 16, 19)( 17, 20)( 21, 24)( 22, 25)( 23, 26)( 39, 42)
( 40, 43)( 41, 44)( 45, 48)( 46, 49)( 47, 50)( 51, 57)( 52, 58)( 53, 59)
( 54, 60)( 55, 61)( 56, 62)( 63, 72)( 64, 73)( 65, 74)( 66, 69)( 67, 70)
( 68, 71)( 75, 81)( 76, 82)( 77, 83)( 78, 84)( 79, 85)( 80, 86)( 87, 96)
( 88, 97)( 89, 98)( 90, 93)( 91, 94)( 92, 95)( 99,111)(100,112)(101,113)
(102,114)(103,115)(104,116)(105,117)(106,118)(107,119)(108,120)(109,121)
(110,122)(123,135)(124,136)(125,137)(126,138)(127,139)(128,140)(129,141)
(130,142)(131,143)(132,144)(133,145)(134,146)(147,168)(148,169)(149,170)
(150,165)(151,166)(152,167)(153,162)(154,163)(155,164)(156,159)(157,160)
(158,161)(171,192)(172,193)(173,194)(174,189)(175,190)(176,191)(177,186)
(178,187)(179,188)(180,183)(181,184)(182,185);
s4 := Sym(194)!( 3, 51)( 4, 52)( 5, 53)( 6, 54)( 7, 55)( 8, 56)( 9, 57)
( 10, 58)( 11, 59)( 12, 60)( 13, 61)( 14, 62)( 15, 66)( 16, 67)( 17, 68)
( 18, 63)( 19, 64)( 20, 65)( 21, 72)( 22, 73)( 23, 74)( 24, 69)( 25, 70)
( 26, 71)( 27, 75)( 28, 76)( 29, 77)( 30, 78)( 31, 79)( 32, 80)( 33, 81)
( 34, 82)( 35, 83)( 36, 84)( 37, 85)( 38, 86)( 39, 90)( 40, 91)( 41, 92)
( 42, 87)( 43, 88)( 44, 89)( 45, 96)( 46, 97)( 47, 98)( 48, 93)( 49, 94)
( 50, 95)( 99,147)(100,148)(101,149)(102,150)(103,151)(104,152)(105,153)
(106,154)(107,155)(108,156)(109,157)(110,158)(111,162)(112,163)(113,164)
(114,159)(115,160)(116,161)(117,168)(118,169)(119,170)(120,165)(121,166)
(122,167)(123,171)(124,172)(125,173)(126,174)(127,175)(128,176)(129,177)
(130,178)(131,179)(132,180)(133,181)(134,182)(135,186)(136,187)(137,188)
(138,183)(139,184)(140,185)(141,192)(142,193)(143,194)(144,189)(145,190)
(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*s3*s2*s1*s2*s3*s2,
s3*s4*s3*s4*s3*s4*s3*s4, s4*s2*s3*s2*s3*s4*s2*s3*s2*s3,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 >;
to this polytope