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}*768a
if this polytope has a name.
Group : SmallGroup(768,1035859)
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, {2,6,8,2}*384
3-fold quotients : {2,2,8,4}*256a
4-fold quotients : {2,6,2,4}*192, {2,6,4,2}*192a
6-fold quotients : {2,2,4,4}*128, {2,2,8,2}*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)( 9,130)( 10,129)
( 11,131)( 12,133)( 13,132)( 14,134)( 15,139)( 16,138)( 17,140)( 18,136)
( 19,135)( 20,137)( 21,145)( 22,144)( 23,146)( 24,142)( 25,141)( 26,143)
( 27,100)( 28, 99)( 29,101)( 30,103)( 31,102)( 32,104)( 33,106)( 34,105)
( 35,107)( 36,109)( 37,108)( 38,110)( 39,115)( 40,114)( 41,116)( 42,112)
( 43,111)( 44,113)( 45,121)( 46,120)( 47,122)( 48,118)( 49,117)( 50,119)
( 51,172)( 52,171)( 53,173)( 54,175)( 55,174)( 56,176)( 57,178)( 58,177)
( 59,179)( 60,181)( 61,180)( 62,182)( 63,187)( 64,186)( 65,188)( 66,184)
( 67,183)( 68,185)( 69,193)( 70,192)( 71,194)( 72,190)( 73,189)( 74,191)
( 75,148)( 76,147)( 77,149)( 78,151)( 79,150)( 80,152)( 81,154)( 82,153)
( 83,155)( 84,157)( 85,156)( 86,158)( 87,163)( 88,162)( 89,164)( 90,160)
( 91,159)( 92,161)( 93,169)( 94,168)( 95,170)( 96,166)( 97,165)( 98,167);;
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)(147,165)(148,166)(149,167)(150,168)
(151,169)(152,170)(153,159)(154,160)(155,161)(156,162)(157,163)(158,164)
(171,189)(172,190)(173,191)(174,192)(175,193)(176,194)(177,183)(178,184)
(179,185)(180,186)(181,187)(182,188);;
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, 63)( 16, 64)( 17, 65)( 18, 66)
( 19, 67)( 20, 68)( 21, 69)( 22, 70)( 23, 71)( 24, 72)( 25, 73)( 26, 74)
( 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, 87)( 40, 88)( 41, 89)( 42, 90)
( 43, 91)( 44, 92)( 45, 93)( 46, 94)( 47, 95)( 48, 96)( 49, 97)( 50, 98)
( 99,147)(100,148)(101,149)(102,150)(103,151)(104,152)(105,153)(106,154)
(107,155)(108,156)(109,157)(110,158)(111,159)(112,160)(113,161)(114,162)
(115,163)(116,164)(117,165)(118,166)(119,167)(120,168)(121,169)(122,170)
(123,171)(124,172)(125,173)(126,174)(127,175)(128,176)(129,177)(130,178)
(131,179)(132,180)(133,181)(134,182)(135,183)(136,184)(137,185)(138,186)
(139,187)(140,188)(141,189)(142,190)(143,191)(144,192)(145,193)(146,194);;
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, s2*s3*s4*s3*s2*s3*s4*s3,
s3*s4*s3*s4*s3*s4*s3*s4, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s2*s3*s2*s3*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, 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)( 9,130)
( 10,129)( 11,131)( 12,133)( 13,132)( 14,134)( 15,139)( 16,138)( 17,140)
( 18,136)( 19,135)( 20,137)( 21,145)( 22,144)( 23,146)( 24,142)( 25,141)
( 26,143)( 27,100)( 28, 99)( 29,101)( 30,103)( 31,102)( 32,104)( 33,106)
( 34,105)( 35,107)( 36,109)( 37,108)( 38,110)( 39,115)( 40,114)( 41,116)
( 42,112)( 43,111)( 44,113)( 45,121)( 46,120)( 47,122)( 48,118)( 49,117)
( 50,119)( 51,172)( 52,171)( 53,173)( 54,175)( 55,174)( 56,176)( 57,178)
( 58,177)( 59,179)( 60,181)( 61,180)( 62,182)( 63,187)( 64,186)( 65,188)
( 66,184)( 67,183)( 68,185)( 69,193)( 70,192)( 71,194)( 72,190)( 73,189)
( 74,191)( 75,148)( 76,147)( 77,149)( 78,151)( 79,150)( 80,152)( 81,154)
( 82,153)( 83,155)( 84,157)( 85,156)( 86,158)( 87,163)( 88,162)( 89,164)
( 90,160)( 91,159)( 92,161)( 93,169)( 94,168)( 95,170)( 96,166)( 97,165)
( 98,167);
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)(147,165)(148,166)(149,167)
(150,168)(151,169)(152,170)(153,159)(154,160)(155,161)(156,162)(157,163)
(158,164)(171,189)(172,190)(173,191)(174,192)(175,193)(176,194)(177,183)
(178,184)(179,185)(180,186)(181,187)(182,188);
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, 63)( 16, 64)( 17, 65)
( 18, 66)( 19, 67)( 20, 68)( 21, 69)( 22, 70)( 23, 71)( 24, 72)( 25, 73)
( 26, 74)( 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, 87)( 40, 88)( 41, 89)
( 42, 90)( 43, 91)( 44, 92)( 45, 93)( 46, 94)( 47, 95)( 48, 96)( 49, 97)
( 50, 98)( 99,147)(100,148)(101,149)(102,150)(103,151)(104,152)(105,153)
(106,154)(107,155)(108,156)(109,157)(110,158)(111,159)(112,160)(113,161)
(114,162)(115,163)(116,164)(117,165)(118,166)(119,167)(120,168)(121,169)
(122,170)(123,171)(124,172)(125,173)(126,174)(127,175)(128,176)(129,177)
(130,178)(131,179)(132,180)(133,181)(134,182)(135,183)(136,184)(137,185)
(138,186)(139,187)(140,188)(141,189)(142,190)(143,191)(144,192)(145,193)
(146,194);
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,
s2*s3*s4*s3*s2*s3*s4*s3, s3*s4*s3*s4*s3*s4*s3*s4,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 >;
to this polytope