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