Polytope of Type {2,4,8,6}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,4,8,6}*768a
if this polytope has a name.
Group : SmallGroup(768,1035859)
Rank : 5
Schlafli Type : {2,4,8,6}
Number of vertices, edges, etc : 2, 4, 16, 24, 6
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 : {2,4,4,6}*384, {2,2,8,6}*384
3-fold quotients : {2,4,8,2}*256a
4-fold quotients : {2,2,4,6}*192a, {2,4,2,6}*192
6-fold quotients : {2,4,4,2}*128, {2,2,8,2}*128
8-fold quotients : {2,4,2,3}*96, {2,2,2,6}*96
12-fold quotients : {2,2,4,2}*64, {2,4,2,2}*64
16-fold quotients : {2,2,2,3}*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, 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);;
s2 := ( 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);;
s3 := ( 3,123)( 4,125)( 5,124)( 6,126)( 7,128)( 8,127)( 9,129)( 10,131)
( 11,130)( 12,132)( 13,134)( 14,133)( 15,138)( 16,140)( 17,139)( 18,135)
( 19,137)( 20,136)( 21,144)( 22,146)( 23,145)( 24,141)( 25,143)( 26,142)
( 27, 99)( 28,101)( 29,100)( 30,102)( 31,104)( 32,103)( 33,105)( 34,107)
( 35,106)( 36,108)( 37,110)( 38,109)( 39,114)( 40,116)( 41,115)( 42,111)
( 43,113)( 44,112)( 45,120)( 46,122)( 47,121)( 48,117)( 49,119)( 50,118)
( 51,171)( 52,173)( 53,172)( 54,174)( 55,176)( 56,175)( 57,177)( 58,179)
( 59,178)( 60,180)( 61,182)( 62,181)( 63,186)( 64,188)( 65,187)( 66,183)
( 67,185)( 68,184)( 69,192)( 70,194)( 71,193)( 72,189)( 73,191)( 74,190)
( 75,147)( 76,149)( 77,148)( 78,150)( 79,152)( 80,151)( 81,153)( 82,155)
( 83,154)( 84,156)( 85,158)( 86,157)( 87,162)( 88,164)( 89,163)( 90,159)
( 91,161)( 92,160)( 93,168)( 94,170)( 95,169)( 96,165)( 97,167)( 98,166);;
s4 := ( 3, 28)( 4, 27)( 5, 29)( 6, 31)( 7, 30)( 8, 32)( 9, 34)( 10, 33)
( 11, 35)( 12, 37)( 13, 36)( 14, 38)( 15, 40)( 16, 39)( 17, 41)( 18, 43)
( 19, 42)( 20, 44)( 21, 46)( 22, 45)( 23, 47)( 24, 49)( 25, 48)( 26, 50)
( 51, 76)( 52, 75)( 53, 77)( 54, 79)( 55, 78)( 56, 80)( 57, 82)( 58, 81)
( 59, 83)( 60, 85)( 61, 84)( 62, 86)( 63, 88)( 64, 87)( 65, 89)( 66, 91)
( 67, 90)( 68, 92)( 69, 94)( 70, 93)( 71, 95)( 72, 97)( 73, 96)( 74, 98)
( 99,124)(100,123)(101,125)(102,127)(103,126)(104,128)(105,130)(106,129)
(107,131)(108,133)(109,132)(110,134)(111,136)(112,135)(113,137)(114,139)
(115,138)(116,140)(117,142)(118,141)(119,143)(120,145)(121,144)(122,146)
(147,172)(148,171)(149,173)(150,175)(151,174)(152,176)(153,178)(154,177)
(155,179)(156,181)(157,180)(158,182)(159,184)(160,183)(161,185)(162,187)
(163,186)(164,188)(165,190)(166,189)(167,191)(168,193)(169,192)(170,194);;
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*s1*s2*s1*s2*s1*s2, s1*s2*s3*s2*s1*s2*s3*s2,
s2*s3*s4*s3*s2*s3*s4*s3, s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4,
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, 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);
s2 := 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);
s3 := Sym(194)!( 3,123)( 4,125)( 5,124)( 6,126)( 7,128)( 8,127)( 9,129)
( 10,131)( 11,130)( 12,132)( 13,134)( 14,133)( 15,138)( 16,140)( 17,139)
( 18,135)( 19,137)( 20,136)( 21,144)( 22,146)( 23,145)( 24,141)( 25,143)
( 26,142)( 27, 99)( 28,101)( 29,100)( 30,102)( 31,104)( 32,103)( 33,105)
( 34,107)( 35,106)( 36,108)( 37,110)( 38,109)( 39,114)( 40,116)( 41,115)
( 42,111)( 43,113)( 44,112)( 45,120)( 46,122)( 47,121)( 48,117)( 49,119)
( 50,118)( 51,171)( 52,173)( 53,172)( 54,174)( 55,176)( 56,175)( 57,177)
( 58,179)( 59,178)( 60,180)( 61,182)( 62,181)( 63,186)( 64,188)( 65,187)
( 66,183)( 67,185)( 68,184)( 69,192)( 70,194)( 71,193)( 72,189)( 73,191)
( 74,190)( 75,147)( 76,149)( 77,148)( 78,150)( 79,152)( 80,151)( 81,153)
( 82,155)( 83,154)( 84,156)( 85,158)( 86,157)( 87,162)( 88,164)( 89,163)
( 90,159)( 91,161)( 92,160)( 93,168)( 94,170)( 95,169)( 96,165)( 97,167)
( 98,166);
s4 := Sym(194)!( 3, 28)( 4, 27)( 5, 29)( 6, 31)( 7, 30)( 8, 32)( 9, 34)
( 10, 33)( 11, 35)( 12, 37)( 13, 36)( 14, 38)( 15, 40)( 16, 39)( 17, 41)
( 18, 43)( 19, 42)( 20, 44)( 21, 46)( 22, 45)( 23, 47)( 24, 49)( 25, 48)
( 26, 50)( 51, 76)( 52, 75)( 53, 77)( 54, 79)( 55, 78)( 56, 80)( 57, 82)
( 58, 81)( 59, 83)( 60, 85)( 61, 84)( 62, 86)( 63, 88)( 64, 87)( 65, 89)
( 66, 91)( 67, 90)( 68, 92)( 69, 94)( 70, 93)( 71, 95)( 72, 97)( 73, 96)
( 74, 98)( 99,124)(100,123)(101,125)(102,127)(103,126)(104,128)(105,130)
(106,129)(107,131)(108,133)(109,132)(110,134)(111,136)(112,135)(113,137)
(114,139)(115,138)(116,140)(117,142)(118,141)(119,143)(120,145)(121,144)
(122,146)(147,172)(148,171)(149,173)(150,175)(151,174)(152,176)(153,178)
(154,177)(155,179)(156,181)(157,180)(158,182)(159,184)(160,183)(161,185)
(162,187)(163,186)(164,188)(165,190)(166,189)(167,191)(168,193)(169,192)
(170,194);
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*s1*s2*s1*s2*s1*s2,
s1*s2*s3*s2*s1*s2*s3*s2, s2*s3*s4*s3*s2*s3*s4*s3,
s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 >;
to this polytope