Polytope of Type {2,8,6,4}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,8,6,4}*768a
if this polytope has a name.
Group : SmallGroup(768,1044756)
Rank : 5
Schlafli Type : {2,8,6,4}
Number of vertices, edges, etc : 2, 8, 24, 12, 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 : {2,4,6,4}*384a, {2,8,6,2}*384
3-fold quotients : {2,8,2,4}*256
4-fold quotients : {2,2,6,4}*192a, {2,4,6,2}*192a
6-fold quotients : {2,4,2,4}*128, {2,8,2,2}*128
8-fold quotients : {2,2,6,2}*96
12-fold quotients : {2,2,2,4}*64, {2,4,2,2}*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 := (1,2);;
s1 := ( 3, 27)( 4, 28)( 5, 29)( 6, 30)( 7, 31)( 8, 32)( 9, 33)( 10, 34)
( 11, 35)( 12, 36)( 13, 37)( 14, 38)( 15, 42)( 16, 43)( 17, 44)( 18, 39)
( 19, 40)( 20, 41)( 21, 48)( 22, 49)( 23, 50)( 24, 45)( 25, 46)( 26, 47)
( 51, 75)( 52, 76)( 53, 77)( 54, 78)( 55, 79)( 56, 80)( 57, 81)( 58, 82)
( 59, 83)( 60, 84)( 61, 85)( 62, 86)( 63, 90)( 64, 91)( 65, 92)( 66, 87)
( 67, 88)( 68, 89)( 69, 96)( 70, 97)( 71, 98)( 72, 93)( 73, 94)( 74, 95)
( 99,123)(100,124)(101,125)(102,126)(103,127)(104,128)(105,129)(106,130)
(107,131)(108,132)(109,133)(110,134)(111,138)(112,139)(113,140)(114,135)
(115,136)(116,137)(117,144)(118,145)(119,146)(120,141)(121,142)(122,143)
(147,171)(148,172)(149,173)(150,174)(151,175)(152,176)(153,177)(154,178)
(155,179)(156,180)(157,181)(158,182)(159,186)(160,187)(161,188)(162,183)
(163,184)(164,185)(165,192)(166,193)(167,194)(168,189)(169,190)(170,191);;
s2 := ( 3, 99)( 4,101)( 5,100)( 6,102)( 7,104)( 8,103)( 9,105)( 10,107)
( 11,106)( 12,108)( 13,110)( 14,109)( 15,114)( 16,116)( 17,115)( 18,111)
( 19,113)( 20,112)( 21,120)( 22,122)( 23,121)( 24,117)( 25,119)( 26,118)
( 27,135)( 28,137)( 29,136)( 30,138)( 31,140)( 32,139)( 33,141)( 34,143)
( 35,142)( 36,144)( 37,146)( 38,145)( 39,123)( 40,125)( 41,124)( 42,126)
( 43,128)( 44,127)( 45,129)( 46,131)( 47,130)( 48,132)( 49,134)( 50,133)
( 51,147)( 52,149)( 53,148)( 54,150)( 55,152)( 56,151)( 57,153)( 58,155)
( 59,154)( 60,156)( 61,158)( 62,157)( 63,162)( 64,164)( 65,163)( 66,159)
( 67,161)( 68,160)( 69,168)( 70,170)( 71,169)( 72,165)( 73,167)( 74,166)
( 75,183)( 76,185)( 77,184)( 78,186)( 79,188)( 80,187)( 81,189)( 82,191)
( 83,190)( 84,192)( 85,194)( 86,193)( 87,171)( 88,173)( 89,172)( 90,174)
( 91,176)( 92,175)( 93,177)( 94,179)( 95,178)( 96,180)( 97,182)( 98,181);;
s3 := ( 3, 40)( 4, 39)( 5, 41)( 6, 43)( 7, 42)( 8, 44)( 9, 46)( 10, 45)
( 11, 47)( 12, 49)( 13, 48)( 14, 50)( 15, 31)( 16, 30)( 17, 32)( 18, 28)
( 19, 27)( 20, 29)( 21, 37)( 22, 36)( 23, 38)( 24, 34)( 25, 33)( 26, 35)
( 51, 94)( 52, 93)( 53, 95)( 54, 97)( 55, 96)( 56, 98)( 57, 88)( 58, 87)
( 59, 89)( 60, 91)( 61, 90)( 62, 92)( 63, 85)( 64, 84)( 65, 86)( 66, 82)
( 67, 81)( 68, 83)( 69, 79)( 70, 78)( 71, 80)( 72, 76)( 73, 75)( 74, 77)
( 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)
(147,178)(148,177)(149,179)(150,181)(151,180)(152,182)(153,172)(154,171)
(155,173)(156,175)(157,174)(158,176)(159,190)(160,189)(161,191)(162,193)
(163,192)(164,194)(165,184)(166,183)(167,185)(168,187)(169,186)(170,188);;
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, s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3,
s1*s2*s1*s2*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, 28)( 5, 29)( 6, 30)( 7, 31)( 8, 32)( 9, 33)
( 10, 34)( 11, 35)( 12, 36)( 13, 37)( 14, 38)( 15, 42)( 16, 43)( 17, 44)
( 18, 39)( 19, 40)( 20, 41)( 21, 48)( 22, 49)( 23, 50)( 24, 45)( 25, 46)
( 26, 47)( 51, 75)( 52, 76)( 53, 77)( 54, 78)( 55, 79)( 56, 80)( 57, 81)
( 58, 82)( 59, 83)( 60, 84)( 61, 85)( 62, 86)( 63, 90)( 64, 91)( 65, 92)
( 66, 87)( 67, 88)( 68, 89)( 69, 96)( 70, 97)( 71, 98)( 72, 93)( 73, 94)
( 74, 95)( 99,123)(100,124)(101,125)(102,126)(103,127)(104,128)(105,129)
(106,130)(107,131)(108,132)(109,133)(110,134)(111,138)(112,139)(113,140)
(114,135)(115,136)(116,137)(117,144)(118,145)(119,146)(120,141)(121,142)
(122,143)(147,171)(148,172)(149,173)(150,174)(151,175)(152,176)(153,177)
(154,178)(155,179)(156,180)(157,181)(158,182)(159,186)(160,187)(161,188)
(162,183)(163,184)(164,185)(165,192)(166,193)(167,194)(168,189)(169,190)
(170,191);
s2 := Sym(194)!( 3, 99)( 4,101)( 5,100)( 6,102)( 7,104)( 8,103)( 9,105)
( 10,107)( 11,106)( 12,108)( 13,110)( 14,109)( 15,114)( 16,116)( 17,115)
( 18,111)( 19,113)( 20,112)( 21,120)( 22,122)( 23,121)( 24,117)( 25,119)
( 26,118)( 27,135)( 28,137)( 29,136)( 30,138)( 31,140)( 32,139)( 33,141)
( 34,143)( 35,142)( 36,144)( 37,146)( 38,145)( 39,123)( 40,125)( 41,124)
( 42,126)( 43,128)( 44,127)( 45,129)( 46,131)( 47,130)( 48,132)( 49,134)
( 50,133)( 51,147)( 52,149)( 53,148)( 54,150)( 55,152)( 56,151)( 57,153)
( 58,155)( 59,154)( 60,156)( 61,158)( 62,157)( 63,162)( 64,164)( 65,163)
( 66,159)( 67,161)( 68,160)( 69,168)( 70,170)( 71,169)( 72,165)( 73,167)
( 74,166)( 75,183)( 76,185)( 77,184)( 78,186)( 79,188)( 80,187)( 81,189)
( 82,191)( 83,190)( 84,192)( 85,194)( 86,193)( 87,171)( 88,173)( 89,172)
( 90,174)( 91,176)( 92,175)( 93,177)( 94,179)( 95,178)( 96,180)( 97,182)
( 98,181);
s3 := Sym(194)!( 3, 40)( 4, 39)( 5, 41)( 6, 43)( 7, 42)( 8, 44)( 9, 46)
( 10, 45)( 11, 47)( 12, 49)( 13, 48)( 14, 50)( 15, 31)( 16, 30)( 17, 32)
( 18, 28)( 19, 27)( 20, 29)( 21, 37)( 22, 36)( 23, 38)( 24, 34)( 25, 33)
( 26, 35)( 51, 94)( 52, 93)( 53, 95)( 54, 97)( 55, 96)( 56, 98)( 57, 88)
( 58, 87)( 59, 89)( 60, 91)( 61, 90)( 62, 92)( 63, 85)( 64, 84)( 65, 86)
( 66, 82)( 67, 81)( 68, 83)( 69, 79)( 70, 78)( 71, 80)( 72, 76)( 73, 75)
( 74, 77)( 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)(147,178)(148,177)(149,179)(150,181)(151,180)(152,182)(153,172)
(154,171)(155,173)(156,175)(157,174)(158,176)(159,190)(160,189)(161,191)
(162,193)(163,192)(164,194)(165,184)(166,183)(167,185)(168,187)(169,186)
(170,188);
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,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 >;
to this polytope