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