Polytope of Type {8,6,18}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {8,6,18}*1728a
Also Known As : {{8,6|2},{6,18|2}}. if this polytope has another name.
Group : SmallGroup(1728,15957)
Rank : 4
Schlafli Type : {8,6,18}
Number of vertices, edges, etc : 8, 24, 54, 18
Order of s0s1s2s3 : 72
Order of s0s1s2s3s2s1 : 2
Special Properties :
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 : {4,6,18}*864a
3-fold quotients : {8,2,18}*576, {8,6,6}*576a
4-fold quotients : {2,6,18}*432a
6-fold quotients : {8,2,9}*288, {4,2,18}*288, {4,6,6}*288a
9-fold quotients : {8,2,6}*192, {8,6,2}*192
12-fold quotients : {4,2,9}*144, {2,2,18}*144, {2,6,6}*144a
18-fold quotients : {8,2,3}*96, {4,2,6}*96, {4,6,2}*96a
24-fold quotients : {2,2,9}*72
27-fold quotients : {8,2,2}*64
36-fold quotients : {4,2,3}*48, {2,2,6}*48, {2,6,2}*48
54-fold quotients : {4,2,2}*32
72-fold quotients : {2,2,3}*24, {2,3,2}*24
108-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := ( 55, 82)( 56, 83)( 57, 84)( 58, 85)( 59, 86)( 60, 87)( 61, 88)( 62, 89)
( 63, 90)( 64, 91)( 65, 92)( 66, 93)( 67, 94)( 68, 95)( 69, 96)( 70, 97)
( 71, 98)( 72, 99)( 73,100)( 74,101)( 75,102)( 76,103)( 77,104)( 78,105)
( 79,106)( 80,107)( 81,108)(109,163)(110,164)(111,165)(112,166)(113,167)
(114,168)(115,169)(116,170)(117,171)(118,172)(119,173)(120,174)(121,175)
(122,176)(123,177)(124,178)(125,179)(126,180)(127,181)(128,182)(129,183)
(130,184)(131,185)(132,186)(133,187)(134,188)(135,189)(136,190)(137,191)
(138,192)(139,193)(140,194)(141,195)(142,196)(143,197)(144,198)(145,199)
(146,200)(147,201)(148,202)(149,203)(150,204)(151,205)(152,206)(153,207)
(154,208)(155,209)(156,210)(157,211)(158,212)(159,213)(160,214)(161,215)
(162,216);;
s1 := ( 1,109)( 2,110)( 3,111)( 4,115)( 5,116)( 6,117)( 7,112)( 8,113)
( 9,114)( 10,118)( 11,119)( 12,120)( 13,124)( 14,125)( 15,126)( 16,121)
( 17,122)( 18,123)( 19,127)( 20,128)( 21,129)( 22,133)( 23,134)( 24,135)
( 25,130)( 26,131)( 27,132)( 28,136)( 29,137)( 30,138)( 31,142)( 32,143)
( 33,144)( 34,139)( 35,140)( 36,141)( 37,145)( 38,146)( 39,147)( 40,151)
( 41,152)( 42,153)( 43,148)( 44,149)( 45,150)( 46,154)( 47,155)( 48,156)
( 49,160)( 50,161)( 51,162)( 52,157)( 53,158)( 54,159)( 55,190)( 56,191)
( 57,192)( 58,196)( 59,197)( 60,198)( 61,193)( 62,194)( 63,195)( 64,199)
( 65,200)( 66,201)( 67,205)( 68,206)( 69,207)( 70,202)( 71,203)( 72,204)
( 73,208)( 74,209)( 75,210)( 76,214)( 77,215)( 78,216)( 79,211)( 80,212)
( 81,213)( 82,163)( 83,164)( 84,165)( 85,169)( 86,170)( 87,171)( 88,166)
( 89,167)( 90,168)( 91,172)( 92,173)( 93,174)( 94,178)( 95,179)( 96,180)
( 97,175)( 98,176)( 99,177)(100,181)(101,182)(102,183)(103,187)(104,188)
(105,189)(106,184)(107,185)(108,186);;
s2 := ( 1, 4)( 2, 6)( 3, 5)( 8, 9)( 10, 24)( 11, 23)( 12, 22)( 13, 21)
( 14, 20)( 15, 19)( 16, 27)( 17, 26)( 18, 25)( 28, 31)( 29, 33)( 30, 32)
( 35, 36)( 37, 51)( 38, 50)( 39, 49)( 40, 48)( 41, 47)( 42, 46)( 43, 54)
( 44, 53)( 45, 52)( 55, 58)( 56, 60)( 57, 59)( 62, 63)( 64, 78)( 65, 77)
( 66, 76)( 67, 75)( 68, 74)( 69, 73)( 70, 81)( 71, 80)( 72, 79)( 82, 85)
( 83, 87)( 84, 86)( 89, 90)( 91,105)( 92,104)( 93,103)( 94,102)( 95,101)
( 96,100)( 97,108)( 98,107)( 99,106)(109,112)(110,114)(111,113)(116,117)
(118,132)(119,131)(120,130)(121,129)(122,128)(123,127)(124,135)(125,134)
(126,133)(136,139)(137,141)(138,140)(143,144)(145,159)(146,158)(147,157)
(148,156)(149,155)(150,154)(151,162)(152,161)(153,160)(163,166)(164,168)
(165,167)(170,171)(172,186)(173,185)(174,184)(175,183)(176,182)(177,181)
(178,189)(179,188)(180,187)(190,193)(191,195)(192,194)(197,198)(199,213)
(200,212)(201,211)(202,210)(203,209)(204,208)(205,216)(206,215)(207,214);;
s3 := ( 1, 10)( 2, 12)( 3, 11)( 4, 13)( 5, 15)( 6, 14)( 7, 16)( 8, 18)
( 9, 17)( 19, 21)( 22, 24)( 25, 27)( 28, 37)( 29, 39)( 30, 38)( 31, 40)
( 32, 42)( 33, 41)( 34, 43)( 35, 45)( 36, 44)( 46, 48)( 49, 51)( 52, 54)
( 55, 64)( 56, 66)( 57, 65)( 58, 67)( 59, 69)( 60, 68)( 61, 70)( 62, 72)
( 63, 71)( 73, 75)( 76, 78)( 79, 81)( 82, 91)( 83, 93)( 84, 92)( 85, 94)
( 86, 96)( 87, 95)( 88, 97)( 89, 99)( 90, 98)(100,102)(103,105)(106,108)
(109,118)(110,120)(111,119)(112,121)(113,123)(114,122)(115,124)(116,126)
(117,125)(127,129)(130,132)(133,135)(136,145)(137,147)(138,146)(139,148)
(140,150)(141,149)(142,151)(143,153)(144,152)(154,156)(157,159)(160,162)
(163,172)(164,174)(165,173)(166,175)(167,177)(168,176)(169,178)(170,180)
(171,179)(181,183)(184,186)(187,189)(190,199)(191,201)(192,200)(193,202)
(194,204)(195,203)(196,205)(197,207)(198,206)(208,210)(211,213)(214,216);;
poly := Group([s0,s1,s2,s3]);;
Finitely Presented Group Representation (GAP) :
F := FreeGroup("s0","s1","s2","s3");;
s0 := F.1;; s1 := F.2;; s2 := F.3;; s3 := F.4;;
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s0*s2*s0*s2,
s0*s3*s0*s3, s1*s3*s1*s3, s0*s1*s2*s1*s0*s1*s2*s1,
s1*s2*s3*s2*s1*s2*s3*s2, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(216)!( 55, 82)( 56, 83)( 57, 84)( 58, 85)( 59, 86)( 60, 87)( 61, 88)
( 62, 89)( 63, 90)( 64, 91)( 65, 92)( 66, 93)( 67, 94)( 68, 95)( 69, 96)
( 70, 97)( 71, 98)( 72, 99)( 73,100)( 74,101)( 75,102)( 76,103)( 77,104)
( 78,105)( 79,106)( 80,107)( 81,108)(109,163)(110,164)(111,165)(112,166)
(113,167)(114,168)(115,169)(116,170)(117,171)(118,172)(119,173)(120,174)
(121,175)(122,176)(123,177)(124,178)(125,179)(126,180)(127,181)(128,182)
(129,183)(130,184)(131,185)(132,186)(133,187)(134,188)(135,189)(136,190)
(137,191)(138,192)(139,193)(140,194)(141,195)(142,196)(143,197)(144,198)
(145,199)(146,200)(147,201)(148,202)(149,203)(150,204)(151,205)(152,206)
(153,207)(154,208)(155,209)(156,210)(157,211)(158,212)(159,213)(160,214)
(161,215)(162,216);
s1 := Sym(216)!( 1,109)( 2,110)( 3,111)( 4,115)( 5,116)( 6,117)( 7,112)
( 8,113)( 9,114)( 10,118)( 11,119)( 12,120)( 13,124)( 14,125)( 15,126)
( 16,121)( 17,122)( 18,123)( 19,127)( 20,128)( 21,129)( 22,133)( 23,134)
( 24,135)( 25,130)( 26,131)( 27,132)( 28,136)( 29,137)( 30,138)( 31,142)
( 32,143)( 33,144)( 34,139)( 35,140)( 36,141)( 37,145)( 38,146)( 39,147)
( 40,151)( 41,152)( 42,153)( 43,148)( 44,149)( 45,150)( 46,154)( 47,155)
( 48,156)( 49,160)( 50,161)( 51,162)( 52,157)( 53,158)( 54,159)( 55,190)
( 56,191)( 57,192)( 58,196)( 59,197)( 60,198)( 61,193)( 62,194)( 63,195)
( 64,199)( 65,200)( 66,201)( 67,205)( 68,206)( 69,207)( 70,202)( 71,203)
( 72,204)( 73,208)( 74,209)( 75,210)( 76,214)( 77,215)( 78,216)( 79,211)
( 80,212)( 81,213)( 82,163)( 83,164)( 84,165)( 85,169)( 86,170)( 87,171)
( 88,166)( 89,167)( 90,168)( 91,172)( 92,173)( 93,174)( 94,178)( 95,179)
( 96,180)( 97,175)( 98,176)( 99,177)(100,181)(101,182)(102,183)(103,187)
(104,188)(105,189)(106,184)(107,185)(108,186);
s2 := Sym(216)!( 1, 4)( 2, 6)( 3, 5)( 8, 9)( 10, 24)( 11, 23)( 12, 22)
( 13, 21)( 14, 20)( 15, 19)( 16, 27)( 17, 26)( 18, 25)( 28, 31)( 29, 33)
( 30, 32)( 35, 36)( 37, 51)( 38, 50)( 39, 49)( 40, 48)( 41, 47)( 42, 46)
( 43, 54)( 44, 53)( 45, 52)( 55, 58)( 56, 60)( 57, 59)( 62, 63)( 64, 78)
( 65, 77)( 66, 76)( 67, 75)( 68, 74)( 69, 73)( 70, 81)( 71, 80)( 72, 79)
( 82, 85)( 83, 87)( 84, 86)( 89, 90)( 91,105)( 92,104)( 93,103)( 94,102)
( 95,101)( 96,100)( 97,108)( 98,107)( 99,106)(109,112)(110,114)(111,113)
(116,117)(118,132)(119,131)(120,130)(121,129)(122,128)(123,127)(124,135)
(125,134)(126,133)(136,139)(137,141)(138,140)(143,144)(145,159)(146,158)
(147,157)(148,156)(149,155)(150,154)(151,162)(152,161)(153,160)(163,166)
(164,168)(165,167)(170,171)(172,186)(173,185)(174,184)(175,183)(176,182)
(177,181)(178,189)(179,188)(180,187)(190,193)(191,195)(192,194)(197,198)
(199,213)(200,212)(201,211)(202,210)(203,209)(204,208)(205,216)(206,215)
(207,214);
s3 := Sym(216)!( 1, 10)( 2, 12)( 3, 11)( 4, 13)( 5, 15)( 6, 14)( 7, 16)
( 8, 18)( 9, 17)( 19, 21)( 22, 24)( 25, 27)( 28, 37)( 29, 39)( 30, 38)
( 31, 40)( 32, 42)( 33, 41)( 34, 43)( 35, 45)( 36, 44)( 46, 48)( 49, 51)
( 52, 54)( 55, 64)( 56, 66)( 57, 65)( 58, 67)( 59, 69)( 60, 68)( 61, 70)
( 62, 72)( 63, 71)( 73, 75)( 76, 78)( 79, 81)( 82, 91)( 83, 93)( 84, 92)
( 85, 94)( 86, 96)( 87, 95)( 88, 97)( 89, 99)( 90, 98)(100,102)(103,105)
(106,108)(109,118)(110,120)(111,119)(112,121)(113,123)(114,122)(115,124)
(116,126)(117,125)(127,129)(130,132)(133,135)(136,145)(137,147)(138,146)
(139,148)(140,150)(141,149)(142,151)(143,153)(144,152)(154,156)(157,159)
(160,162)(163,172)(164,174)(165,173)(166,175)(167,177)(168,176)(169,178)
(170,180)(171,179)(181,183)(184,186)(187,189)(190,199)(191,201)(192,200)
(193,202)(194,204)(195,203)(196,205)(197,207)(198,206)(208,210)(211,213)
(214,216);
poly := sub<Sym(216)|s0,s1,s2,s3>;
Finitely Presented Group Representation (Magma) :
poly<s0,s1,s2,s3> := Group< s0,s1,s2,s3 | s0*s0, s1*s1, s2*s2,
s3*s3, s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3,
s0*s1*s2*s1*s0*s1*s2*s1, s1*s2*s3*s2*s1*s2*s3*s2,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 >;
References : None.
to this polytope