Polytope of Type {8,4,6,2}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {8,4,6,2}*768a
if this polytope has a name.
Group : SmallGroup(768,1035859)
Rank : 5
Schlafli Type : {8,4,6,2}
Number of vertices, edges, etc : 8, 16, 12, 6, 2
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 : {4,4,6,2}*384, {8,2,6,2}*384
3-fold quotients : {8,4,2,2}*256a
4-fold quotients : {8,2,3,2}*192, {2,4,6,2}*192a, {4,2,6,2}*192
6-fold quotients : {4,4,2,2}*128, {8,2,2,2}*128
8-fold quotients : {4,2,3,2}*96, {2,2,6,2}*96
12-fold quotients : {2,4,2,2}*64, {4,2,2,2}*64
16-fold quotients : {2,2,3,2}*48
24-fold quotients : {2,2,2,2}*32
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := ( 1, 97)( 2, 98)( 3, 99)( 4,100)( 5,101)( 6,102)( 7,103)( 8,104)
( 9,105)( 10,106)( 11,107)( 12,108)( 13,112)( 14,113)( 15,114)( 16,109)
( 17,110)( 18,111)( 19,118)( 20,119)( 21,120)( 22,115)( 23,116)( 24,117)
( 25,121)( 26,122)( 27,123)( 28,124)( 29,125)( 30,126)( 31,127)( 32,128)
( 33,129)( 34,130)( 35,131)( 36,132)( 37,136)( 38,137)( 39,138)( 40,133)
( 41,134)( 42,135)( 43,142)( 44,143)( 45,144)( 46,139)( 47,140)( 48,141)
( 49,145)( 50,146)( 51,147)( 52,148)( 53,149)( 54,150)( 55,151)( 56,152)
( 57,153)( 58,154)( 59,155)( 60,156)( 61,160)( 62,161)( 63,162)( 64,157)
( 65,158)( 66,159)( 67,166)( 68,167)( 69,168)( 70,163)( 71,164)( 72,165)
( 73,169)( 74,170)( 75,171)( 76,172)( 77,173)( 78,174)( 79,175)( 80,176)
( 81,177)( 82,178)( 83,179)( 84,180)( 85,184)( 86,185)( 87,186)( 88,181)
( 89,182)( 90,183)( 91,190)( 92,191)( 93,192)( 94,187)( 95,188)( 96,189);;
s1 := ( 13, 16)( 14, 17)( 15, 18)( 19, 22)( 20, 23)( 21, 24)( 37, 40)( 38, 41)
( 39, 42)( 43, 46)( 44, 47)( 45, 48)( 49, 55)( 50, 56)( 51, 57)( 52, 58)
( 53, 59)( 54, 60)( 61, 70)( 62, 71)( 63, 72)( 64, 67)( 65, 68)( 66, 69)
( 73, 79)( 74, 80)( 75, 81)( 76, 82)( 77, 83)( 78, 84)( 85, 94)( 86, 95)
( 87, 96)( 88, 91)( 89, 92)( 90, 93)( 97,109)( 98,110)( 99,111)(100,112)
(101,113)(102,114)(103,115)(104,116)(105,117)(106,118)(107,119)(108,120)
(121,133)(122,134)(123,135)(124,136)(125,137)(126,138)(127,139)(128,140)
(129,141)(130,142)(131,143)(132,144)(145,163)(146,164)(147,165)(148,166)
(149,167)(150,168)(151,157)(152,158)(153,159)(154,160)(155,161)(156,162)
(169,187)(170,188)(171,189)(172,190)(173,191)(174,192)(175,181)(176,182)
(177,183)(178,184)(179,185)(180,186);;
s2 := ( 1, 73)( 2, 75)( 3, 74)( 4, 76)( 5, 78)( 6, 77)( 7, 79)( 8, 81)
( 9, 80)( 10, 82)( 11, 84)( 12, 83)( 13, 85)( 14, 87)( 15, 86)( 16, 88)
( 17, 90)( 18, 89)( 19, 91)( 20, 93)( 21, 92)( 22, 94)( 23, 96)( 24, 95)
( 25, 49)( 26, 51)( 27, 50)( 28, 52)( 29, 54)( 30, 53)( 31, 55)( 32, 57)
( 33, 56)( 34, 58)( 35, 60)( 36, 59)( 37, 61)( 38, 63)( 39, 62)( 40, 64)
( 41, 66)( 42, 65)( 43, 67)( 44, 69)( 45, 68)( 46, 70)( 47, 72)( 48, 71)
( 97,169)( 98,171)( 99,170)(100,172)(101,174)(102,173)(103,175)(104,177)
(105,176)(106,178)(107,180)(108,179)(109,181)(110,183)(111,182)(112,184)
(113,186)(114,185)(115,187)(116,189)(117,188)(118,190)(119,192)(120,191)
(121,145)(122,147)(123,146)(124,148)(125,150)(126,149)(127,151)(128,153)
(129,152)(130,154)(131,156)(132,155)(133,157)(134,159)(135,158)(136,160)
(137,162)(138,161)(139,163)(140,165)(141,164)(142,166)(143,168)(144,167);;
s3 := ( 1, 26)( 2, 25)( 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)
( 49, 74)( 50, 73)( 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)
( 97,122)( 98,121)( 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)
(145,170)(146,169)(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);;
s4 := (193,194);;
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*s2*s0*s2,
s0*s3*s0*s3, s1*s3*s1*s3, s0*s4*s0*s4,
s1*s4*s1*s4, s2*s4*s2*s4, s3*s4*s3*s4,
s0*s1*s2*s1*s0*s1*s2*s1, s1*s2*s1*s2*s1*s2*s1*s2,
s1*s2*s3*s2*s1*s2*s3*s2, s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(194)!( 1, 97)( 2, 98)( 3, 99)( 4,100)( 5,101)( 6,102)( 7,103)
( 8,104)( 9,105)( 10,106)( 11,107)( 12,108)( 13,112)( 14,113)( 15,114)
( 16,109)( 17,110)( 18,111)( 19,118)( 20,119)( 21,120)( 22,115)( 23,116)
( 24,117)( 25,121)( 26,122)( 27,123)( 28,124)( 29,125)( 30,126)( 31,127)
( 32,128)( 33,129)( 34,130)( 35,131)( 36,132)( 37,136)( 38,137)( 39,138)
( 40,133)( 41,134)( 42,135)( 43,142)( 44,143)( 45,144)( 46,139)( 47,140)
( 48,141)( 49,145)( 50,146)( 51,147)( 52,148)( 53,149)( 54,150)( 55,151)
( 56,152)( 57,153)( 58,154)( 59,155)( 60,156)( 61,160)( 62,161)( 63,162)
( 64,157)( 65,158)( 66,159)( 67,166)( 68,167)( 69,168)( 70,163)( 71,164)
( 72,165)( 73,169)( 74,170)( 75,171)( 76,172)( 77,173)( 78,174)( 79,175)
( 80,176)( 81,177)( 82,178)( 83,179)( 84,180)( 85,184)( 86,185)( 87,186)
( 88,181)( 89,182)( 90,183)( 91,190)( 92,191)( 93,192)( 94,187)( 95,188)
( 96,189);
s1 := Sym(194)!( 13, 16)( 14, 17)( 15, 18)( 19, 22)( 20, 23)( 21, 24)( 37, 40)
( 38, 41)( 39, 42)( 43, 46)( 44, 47)( 45, 48)( 49, 55)( 50, 56)( 51, 57)
( 52, 58)( 53, 59)( 54, 60)( 61, 70)( 62, 71)( 63, 72)( 64, 67)( 65, 68)
( 66, 69)( 73, 79)( 74, 80)( 75, 81)( 76, 82)( 77, 83)( 78, 84)( 85, 94)
( 86, 95)( 87, 96)( 88, 91)( 89, 92)( 90, 93)( 97,109)( 98,110)( 99,111)
(100,112)(101,113)(102,114)(103,115)(104,116)(105,117)(106,118)(107,119)
(108,120)(121,133)(122,134)(123,135)(124,136)(125,137)(126,138)(127,139)
(128,140)(129,141)(130,142)(131,143)(132,144)(145,163)(146,164)(147,165)
(148,166)(149,167)(150,168)(151,157)(152,158)(153,159)(154,160)(155,161)
(156,162)(169,187)(170,188)(171,189)(172,190)(173,191)(174,192)(175,181)
(176,182)(177,183)(178,184)(179,185)(180,186);
s2 := Sym(194)!( 1, 73)( 2, 75)( 3, 74)( 4, 76)( 5, 78)( 6, 77)( 7, 79)
( 8, 81)( 9, 80)( 10, 82)( 11, 84)( 12, 83)( 13, 85)( 14, 87)( 15, 86)
( 16, 88)( 17, 90)( 18, 89)( 19, 91)( 20, 93)( 21, 92)( 22, 94)( 23, 96)
( 24, 95)( 25, 49)( 26, 51)( 27, 50)( 28, 52)( 29, 54)( 30, 53)( 31, 55)
( 32, 57)( 33, 56)( 34, 58)( 35, 60)( 36, 59)( 37, 61)( 38, 63)( 39, 62)
( 40, 64)( 41, 66)( 42, 65)( 43, 67)( 44, 69)( 45, 68)( 46, 70)( 47, 72)
( 48, 71)( 97,169)( 98,171)( 99,170)(100,172)(101,174)(102,173)(103,175)
(104,177)(105,176)(106,178)(107,180)(108,179)(109,181)(110,183)(111,182)
(112,184)(113,186)(114,185)(115,187)(116,189)(117,188)(118,190)(119,192)
(120,191)(121,145)(122,147)(123,146)(124,148)(125,150)(126,149)(127,151)
(128,153)(129,152)(130,154)(131,156)(132,155)(133,157)(134,159)(135,158)
(136,160)(137,162)(138,161)(139,163)(140,165)(141,164)(142,166)(143,168)
(144,167);
s3 := Sym(194)!( 1, 26)( 2, 25)( 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)( 49, 74)( 50, 73)( 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)( 97,122)( 98,121)( 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)(145,170)(146,169)(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);
s4 := Sym(194)!(193,194);
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*s2*s0*s2, s0*s3*s0*s3,
s1*s3*s1*s3, s0*s4*s0*s4, s1*s4*s1*s4,
s2*s4*s2*s4, s3*s4*s3*s4, s0*s1*s2*s1*s0*s1*s2*s1,
s1*s2*s1*s2*s1*s2*s1*s2, s1*s2*s3*s2*s1*s2*s3*s2,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 >;
to this polytope