Polytope of Type {2,8,6,3}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,8,6,3}*1728a
if this polytope has a name.
Group : SmallGroup(1728,17452)
Rank : 5
Schlafli Type : {2,8,6,3}
Number of vertices, edges, etc : 2, 8, 72, 27, 9
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 : {2,4,6,3}*864a
3-fold quotients : {2,8,6,3}*576
4-fold quotients : {2,2,6,3}*432
6-fold quotients : {2,4,6,3}*288
9-fold quotients : {2,8,2,3}*192
12-fold quotients : {2,2,6,3}*144
18-fold quotients : {2,4,2,3}*96
36-fold quotients : {2,2,2,3}*48
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := (1,2);;
s1 := ( 3,111)( 4,112)( 5,113)( 6,114)( 7,115)( 8,116)( 9,117)( 10,118)
( 11,119)( 12,120)( 13,121)( 14,122)( 15,123)( 16,124)( 17,125)( 18,126)
( 19,127)( 20,128)( 21,129)( 22,130)( 23,131)( 24,132)( 25,133)( 26,134)
( 27,135)( 28,136)( 29,137)( 30,138)( 31,139)( 32,140)( 33,141)( 34,142)
( 35,143)( 36,144)( 37,145)( 38,146)( 39,147)( 40,148)( 41,149)( 42,150)
( 43,151)( 44,152)( 45,153)( 46,154)( 47,155)( 48,156)( 49,157)( 50,158)
( 51,159)( 52,160)( 53,161)( 54,162)( 55,163)( 56,164)( 57,192)( 58,193)
( 59,194)( 60,195)( 61,196)( 62,197)( 63,198)( 64,199)( 65,200)( 66,201)
( 67,202)( 68,203)( 69,204)( 70,205)( 71,206)( 72,207)( 73,208)( 74,209)
( 75,210)( 76,211)( 77,212)( 78,213)( 79,214)( 80,215)( 81,216)( 82,217)
( 83,218)( 84,165)( 85,166)( 86,167)( 87,168)( 88,169)( 89,170)( 90,171)
( 91,172)( 92,173)( 93,174)( 94,175)( 95,176)( 96,177)( 97,178)( 98,179)
( 99,180)(100,181)(101,182)(102,183)(103,184)(104,185)(105,186)(106,187)
(107,188)(108,189)(109,190)(110,191);;
s2 := ( 4, 5)( 6, 9)( 7, 11)( 8, 10)( 13, 14)( 15, 18)( 16, 20)( 17, 19)
( 22, 23)( 24, 27)( 25, 29)( 26, 28)( 31, 32)( 33, 36)( 34, 38)( 35, 37)
( 40, 41)( 42, 45)( 43, 47)( 44, 46)( 49, 50)( 51, 54)( 52, 56)( 53, 55)
( 57, 84)( 58, 86)( 59, 85)( 60, 90)( 61, 92)( 62, 91)( 63, 87)( 64, 89)
( 65, 88)( 66, 93)( 67, 95)( 68, 94)( 69, 99)( 70,101)( 71,100)( 72, 96)
( 73, 98)( 74, 97)( 75,102)( 76,104)( 77,103)( 78,108)( 79,110)( 80,109)
( 81,105)( 82,107)( 83,106)(111,165)(112,167)(113,166)(114,171)(115,173)
(116,172)(117,168)(118,170)(119,169)(120,174)(121,176)(122,175)(123,180)
(124,182)(125,181)(126,177)(127,179)(128,178)(129,183)(130,185)(131,184)
(132,189)(133,191)(134,190)(135,186)(136,188)(137,187)(138,192)(139,194)
(140,193)(141,198)(142,200)(143,199)(144,195)(145,197)(146,196)(147,201)
(148,203)(149,202)(150,207)(151,209)(152,208)(153,204)(154,206)(155,205)
(156,210)(157,212)(158,211)(159,216)(160,218)(161,217)(162,213)(163,215)
(164,214);;
s3 := ( 3, 6)( 4, 7)( 5, 8)( 12, 24)( 13, 25)( 14, 26)( 15, 21)( 16, 22)
( 17, 23)( 18, 27)( 19, 28)( 20, 29)( 30, 33)( 31, 34)( 32, 35)( 39, 51)
( 40, 52)( 41, 53)( 42, 48)( 43, 49)( 44, 50)( 45, 54)( 46, 55)( 47, 56)
( 57, 60)( 58, 61)( 59, 62)( 66, 78)( 67, 79)( 68, 80)( 69, 75)( 70, 76)
( 71, 77)( 72, 81)( 73, 82)( 74, 83)( 84, 87)( 85, 88)( 86, 89)( 93,105)
( 94,106)( 95,107)( 96,102)( 97,103)( 98,104)( 99,108)(100,109)(101,110)
(111,114)(112,115)(113,116)(120,132)(121,133)(122,134)(123,129)(124,130)
(125,131)(126,135)(127,136)(128,137)(138,141)(139,142)(140,143)(147,159)
(148,160)(149,161)(150,156)(151,157)(152,158)(153,162)(154,163)(155,164)
(165,168)(166,169)(167,170)(174,186)(175,187)(176,188)(177,183)(178,184)
(179,185)(180,189)(181,190)(182,191)(192,195)(193,196)(194,197)(201,213)
(202,214)(203,215)(204,210)(205,211)(206,212)(207,216)(208,217)(209,218);;
s4 := ( 3, 12)( 4, 13)( 5, 14)( 6, 20)( 7, 18)( 8, 19)( 9, 16)( 10, 17)
( 11, 15)( 24, 29)( 25, 27)( 26, 28)( 30, 39)( 31, 40)( 32, 41)( 33, 47)
( 34, 45)( 35, 46)( 36, 43)( 37, 44)( 38, 42)( 51, 56)( 52, 54)( 53, 55)
( 57, 66)( 58, 67)( 59, 68)( 60, 74)( 61, 72)( 62, 73)( 63, 70)( 64, 71)
( 65, 69)( 78, 83)( 79, 81)( 80, 82)( 84, 93)( 85, 94)( 86, 95)( 87,101)
( 88, 99)( 89,100)( 90, 97)( 91, 98)( 92, 96)(105,110)(106,108)(107,109)
(111,120)(112,121)(113,122)(114,128)(115,126)(116,127)(117,124)(118,125)
(119,123)(132,137)(133,135)(134,136)(138,147)(139,148)(140,149)(141,155)
(142,153)(143,154)(144,151)(145,152)(146,150)(159,164)(160,162)(161,163)
(165,174)(166,175)(167,176)(168,182)(169,180)(170,181)(171,178)(172,179)
(173,177)(186,191)(187,189)(188,190)(192,201)(193,202)(194,203)(195,209)
(196,207)(197,208)(198,205)(199,206)(200,204)(213,218)(214,216)(215,217);;
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,
s3*s4*s3*s4*s3*s4, s1*s2*s3*s2*s1*s2*s3*s2,
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,
s2*s4*s3*s2*s4*s3*s2*s4*s3*s2*s4*s3*s2*s4*s3*s2*s4*s3 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(218)!(1,2);
s1 := Sym(218)!( 3,111)( 4,112)( 5,113)( 6,114)( 7,115)( 8,116)( 9,117)
( 10,118)( 11,119)( 12,120)( 13,121)( 14,122)( 15,123)( 16,124)( 17,125)
( 18,126)( 19,127)( 20,128)( 21,129)( 22,130)( 23,131)( 24,132)( 25,133)
( 26,134)( 27,135)( 28,136)( 29,137)( 30,138)( 31,139)( 32,140)( 33,141)
( 34,142)( 35,143)( 36,144)( 37,145)( 38,146)( 39,147)( 40,148)( 41,149)
( 42,150)( 43,151)( 44,152)( 45,153)( 46,154)( 47,155)( 48,156)( 49,157)
( 50,158)( 51,159)( 52,160)( 53,161)( 54,162)( 55,163)( 56,164)( 57,192)
( 58,193)( 59,194)( 60,195)( 61,196)( 62,197)( 63,198)( 64,199)( 65,200)
( 66,201)( 67,202)( 68,203)( 69,204)( 70,205)( 71,206)( 72,207)( 73,208)
( 74,209)( 75,210)( 76,211)( 77,212)( 78,213)( 79,214)( 80,215)( 81,216)
( 82,217)( 83,218)( 84,165)( 85,166)( 86,167)( 87,168)( 88,169)( 89,170)
( 90,171)( 91,172)( 92,173)( 93,174)( 94,175)( 95,176)( 96,177)( 97,178)
( 98,179)( 99,180)(100,181)(101,182)(102,183)(103,184)(104,185)(105,186)
(106,187)(107,188)(108,189)(109,190)(110,191);
s2 := Sym(218)!( 4, 5)( 6, 9)( 7, 11)( 8, 10)( 13, 14)( 15, 18)( 16, 20)
( 17, 19)( 22, 23)( 24, 27)( 25, 29)( 26, 28)( 31, 32)( 33, 36)( 34, 38)
( 35, 37)( 40, 41)( 42, 45)( 43, 47)( 44, 46)( 49, 50)( 51, 54)( 52, 56)
( 53, 55)( 57, 84)( 58, 86)( 59, 85)( 60, 90)( 61, 92)( 62, 91)( 63, 87)
( 64, 89)( 65, 88)( 66, 93)( 67, 95)( 68, 94)( 69, 99)( 70,101)( 71,100)
( 72, 96)( 73, 98)( 74, 97)( 75,102)( 76,104)( 77,103)( 78,108)( 79,110)
( 80,109)( 81,105)( 82,107)( 83,106)(111,165)(112,167)(113,166)(114,171)
(115,173)(116,172)(117,168)(118,170)(119,169)(120,174)(121,176)(122,175)
(123,180)(124,182)(125,181)(126,177)(127,179)(128,178)(129,183)(130,185)
(131,184)(132,189)(133,191)(134,190)(135,186)(136,188)(137,187)(138,192)
(139,194)(140,193)(141,198)(142,200)(143,199)(144,195)(145,197)(146,196)
(147,201)(148,203)(149,202)(150,207)(151,209)(152,208)(153,204)(154,206)
(155,205)(156,210)(157,212)(158,211)(159,216)(160,218)(161,217)(162,213)
(163,215)(164,214);
s3 := Sym(218)!( 3, 6)( 4, 7)( 5, 8)( 12, 24)( 13, 25)( 14, 26)( 15, 21)
( 16, 22)( 17, 23)( 18, 27)( 19, 28)( 20, 29)( 30, 33)( 31, 34)( 32, 35)
( 39, 51)( 40, 52)( 41, 53)( 42, 48)( 43, 49)( 44, 50)( 45, 54)( 46, 55)
( 47, 56)( 57, 60)( 58, 61)( 59, 62)( 66, 78)( 67, 79)( 68, 80)( 69, 75)
( 70, 76)( 71, 77)( 72, 81)( 73, 82)( 74, 83)( 84, 87)( 85, 88)( 86, 89)
( 93,105)( 94,106)( 95,107)( 96,102)( 97,103)( 98,104)( 99,108)(100,109)
(101,110)(111,114)(112,115)(113,116)(120,132)(121,133)(122,134)(123,129)
(124,130)(125,131)(126,135)(127,136)(128,137)(138,141)(139,142)(140,143)
(147,159)(148,160)(149,161)(150,156)(151,157)(152,158)(153,162)(154,163)
(155,164)(165,168)(166,169)(167,170)(174,186)(175,187)(176,188)(177,183)
(178,184)(179,185)(180,189)(181,190)(182,191)(192,195)(193,196)(194,197)
(201,213)(202,214)(203,215)(204,210)(205,211)(206,212)(207,216)(208,217)
(209,218);
s4 := Sym(218)!( 3, 12)( 4, 13)( 5, 14)( 6, 20)( 7, 18)( 8, 19)( 9, 16)
( 10, 17)( 11, 15)( 24, 29)( 25, 27)( 26, 28)( 30, 39)( 31, 40)( 32, 41)
( 33, 47)( 34, 45)( 35, 46)( 36, 43)( 37, 44)( 38, 42)( 51, 56)( 52, 54)
( 53, 55)( 57, 66)( 58, 67)( 59, 68)( 60, 74)( 61, 72)( 62, 73)( 63, 70)
( 64, 71)( 65, 69)( 78, 83)( 79, 81)( 80, 82)( 84, 93)( 85, 94)( 86, 95)
( 87,101)( 88, 99)( 89,100)( 90, 97)( 91, 98)( 92, 96)(105,110)(106,108)
(107,109)(111,120)(112,121)(113,122)(114,128)(115,126)(116,127)(117,124)
(118,125)(119,123)(132,137)(133,135)(134,136)(138,147)(139,148)(140,149)
(141,155)(142,153)(143,154)(144,151)(145,152)(146,150)(159,164)(160,162)
(161,163)(165,174)(166,175)(167,176)(168,182)(169,180)(170,181)(171,178)
(172,179)(173,177)(186,191)(187,189)(188,190)(192,201)(193,202)(194,203)
(195,209)(196,207)(197,208)(198,205)(199,206)(200,204)(213,218)(214,216)
(215,217);
poly := sub<Sym(218)|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, s3*s4*s3*s4*s3*s4,
s1*s2*s3*s2*s1*s2*s3*s2, 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,
s2*s4*s3*s2*s4*s3*s2*s4*s3*s2*s4*s3*s2*s4*s3*s2*s4*s3 >;
to this polytope