Polytope of Type {3,6,18}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {3,6,18}*864
Also Known As : {{3,6}4,{6,18|2}}. if this polytope has another name.
Group : SmallGroup(864,3998)
Rank : 4
Schlafli Type : {3,6,18}
Number of vertices, edges, etc : 4, 12, 72, 18
Order of s0s1s2s3 : 36
Order of s0s1s2s3s2s1 : 2
Special Properties :
Universal
Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{3,6,18,2} of size 1728
Vertex Figure Of :
{2,3,6,18} of size 1728
Quotients (Maximal Quotients in Boldface) :
3-fold quotients : {3,6,6}*288
9-fold quotients : {3,6,2}*96
18-fold quotients : {3,3,2}*48
Covers (Minimal Covers in Boldface) :
2-fold covers : {3,6,36}*1728, {3,12,18}*1728, {6,6,18}*1728
Permutation Representation (GAP) :
s0 := ( 2, 3)( 6, 7)( 10, 11)( 14, 15)( 18, 19)( 22, 23)( 26, 27)( 30, 31)
( 34, 35)( 37, 73)( 38, 75)( 39, 74)( 40, 76)( 41, 77)( 42, 79)( 43, 78)
( 44, 80)( 45, 81)( 46, 83)( 47, 82)( 48, 84)( 49, 85)( 50, 87)( 51, 86)
( 52, 88)( 53, 89)( 54, 91)( 55, 90)( 56, 92)( 57, 93)( 58, 95)( 59, 94)
( 60, 96)( 61, 97)( 62, 99)( 63, 98)( 64,100)( 65,101)( 66,103)( 67,102)
( 68,104)( 69,105)( 70,107)( 71,106)( 72,108)(110,111)(114,115)(118,119)
(122,123)(126,127)(130,131)(134,135)(138,139)(142,143)(145,181)(146,183)
(147,182)(148,184)(149,185)(150,187)(151,186)(152,188)(153,189)(154,191)
(155,190)(156,192)(157,193)(158,195)(159,194)(160,196)(161,197)(162,199)
(163,198)(164,200)(165,201)(166,203)(167,202)(168,204)(169,205)(170,207)
(171,206)(172,208)(173,209)(174,211)(175,210)(176,212)(177,213)(178,215)
(179,214)(180,216);;
s1 := ( 1, 37)( 2, 38)( 3, 40)( 4, 39)( 5, 41)( 6, 42)( 7, 44)( 8, 43)
( 9, 45)( 10, 46)( 11, 48)( 12, 47)( 13, 49)( 14, 50)( 15, 52)( 16, 51)
( 17, 53)( 18, 54)( 19, 56)( 20, 55)( 21, 57)( 22, 58)( 23, 60)( 24, 59)
( 25, 61)( 26, 62)( 27, 64)( 28, 63)( 29, 65)( 30, 66)( 31, 68)( 32, 67)
( 33, 69)( 34, 70)( 35, 72)( 36, 71)( 75, 76)( 79, 80)( 83, 84)( 87, 88)
( 91, 92)( 95, 96)( 99,100)(103,104)(107,108)(109,145)(110,146)(111,148)
(112,147)(113,149)(114,150)(115,152)(116,151)(117,153)(118,154)(119,156)
(120,155)(121,157)(122,158)(123,160)(124,159)(125,161)(126,162)(127,164)
(128,163)(129,165)(130,166)(131,168)(132,167)(133,169)(134,170)(135,172)
(136,171)(137,173)(138,174)(139,176)(140,175)(141,177)(142,178)(143,180)
(144,179)(183,184)(187,188)(191,192)(195,196)(199,200)(203,204)(207,208)
(211,212)(215,216);;
s2 := ( 1, 4)( 5, 12)( 6, 10)( 7, 11)( 8, 9)( 13, 32)( 14, 30)( 15, 31)
( 16, 29)( 17, 28)( 18, 26)( 19, 27)( 20, 25)( 21, 36)( 22, 34)( 23, 35)
( 24, 33)( 37, 76)( 38, 74)( 39, 75)( 40, 73)( 41, 84)( 42, 82)( 43, 83)
( 44, 81)( 45, 80)( 46, 78)( 47, 79)( 48, 77)( 49,104)( 50,102)( 51,103)
( 52,101)( 53,100)( 54, 98)( 55, 99)( 56, 97)( 57,108)( 58,106)( 59,107)
( 60,105)( 61, 92)( 62, 90)( 63, 91)( 64, 89)( 65, 88)( 66, 86)( 67, 87)
( 68, 85)( 69, 96)( 70, 94)( 71, 95)( 72, 93)(109,112)(113,120)(114,118)
(115,119)(116,117)(121,140)(122,138)(123,139)(124,137)(125,136)(126,134)
(127,135)(128,133)(129,144)(130,142)(131,143)(132,141)(145,184)(146,182)
(147,183)(148,181)(149,192)(150,190)(151,191)(152,189)(153,188)(154,186)
(155,187)(156,185)(157,212)(158,210)(159,211)(160,209)(161,208)(162,206)
(163,207)(164,205)(165,216)(166,214)(167,215)(168,213)(169,200)(170,198)
(171,199)(172,197)(173,196)(174,194)(175,195)(176,193)(177,204)(178,202)
(179,203)(180,201);;
s3 := ( 1,121)( 2,122)( 3,123)( 4,124)( 5,129)( 6,130)( 7,131)( 8,132)
( 9,125)( 10,126)( 11,127)( 12,128)( 13,109)( 14,110)( 15,111)( 16,112)
( 17,117)( 18,118)( 19,119)( 20,120)( 21,113)( 22,114)( 23,115)( 24,116)
( 25,137)( 26,138)( 27,139)( 28,140)( 29,133)( 30,134)( 31,135)( 32,136)
( 33,141)( 34,142)( 35,143)( 36,144)( 37,157)( 38,158)( 39,159)( 40,160)
( 41,165)( 42,166)( 43,167)( 44,168)( 45,161)( 46,162)( 47,163)( 48,164)
( 49,145)( 50,146)( 51,147)( 52,148)( 53,153)( 54,154)( 55,155)( 56,156)
( 57,149)( 58,150)( 59,151)( 60,152)( 61,173)( 62,174)( 63,175)( 64,176)
( 65,169)( 66,170)( 67,171)( 68,172)( 69,177)( 70,178)( 71,179)( 72,180)
( 73,193)( 74,194)( 75,195)( 76,196)( 77,201)( 78,202)( 79,203)( 80,204)
( 81,197)( 82,198)( 83,199)( 84,200)( 85,181)( 86,182)( 87,183)( 88,184)
( 89,189)( 90,190)( 91,191)( 92,192)( 93,185)( 94,186)( 95,187)( 96,188)
( 97,209)( 98,210)( 99,211)(100,212)(101,205)(102,206)(103,207)(104,208)
(105,213)(106,214)(107,215)(108,216);;
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*s0*s1*s0*s1,
s1*s2*s3*s2*s1*s2*s3*s2, s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*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)!( 2, 3)( 6, 7)( 10, 11)( 14, 15)( 18, 19)( 22, 23)( 26, 27)
( 30, 31)( 34, 35)( 37, 73)( 38, 75)( 39, 74)( 40, 76)( 41, 77)( 42, 79)
( 43, 78)( 44, 80)( 45, 81)( 46, 83)( 47, 82)( 48, 84)( 49, 85)( 50, 87)
( 51, 86)( 52, 88)( 53, 89)( 54, 91)( 55, 90)( 56, 92)( 57, 93)( 58, 95)
( 59, 94)( 60, 96)( 61, 97)( 62, 99)( 63, 98)( 64,100)( 65,101)( 66,103)
( 67,102)( 68,104)( 69,105)( 70,107)( 71,106)( 72,108)(110,111)(114,115)
(118,119)(122,123)(126,127)(130,131)(134,135)(138,139)(142,143)(145,181)
(146,183)(147,182)(148,184)(149,185)(150,187)(151,186)(152,188)(153,189)
(154,191)(155,190)(156,192)(157,193)(158,195)(159,194)(160,196)(161,197)
(162,199)(163,198)(164,200)(165,201)(166,203)(167,202)(168,204)(169,205)
(170,207)(171,206)(172,208)(173,209)(174,211)(175,210)(176,212)(177,213)
(178,215)(179,214)(180,216);
s1 := Sym(216)!( 1, 37)( 2, 38)( 3, 40)( 4, 39)( 5, 41)( 6, 42)( 7, 44)
( 8, 43)( 9, 45)( 10, 46)( 11, 48)( 12, 47)( 13, 49)( 14, 50)( 15, 52)
( 16, 51)( 17, 53)( 18, 54)( 19, 56)( 20, 55)( 21, 57)( 22, 58)( 23, 60)
( 24, 59)( 25, 61)( 26, 62)( 27, 64)( 28, 63)( 29, 65)( 30, 66)( 31, 68)
( 32, 67)( 33, 69)( 34, 70)( 35, 72)( 36, 71)( 75, 76)( 79, 80)( 83, 84)
( 87, 88)( 91, 92)( 95, 96)( 99,100)(103,104)(107,108)(109,145)(110,146)
(111,148)(112,147)(113,149)(114,150)(115,152)(116,151)(117,153)(118,154)
(119,156)(120,155)(121,157)(122,158)(123,160)(124,159)(125,161)(126,162)
(127,164)(128,163)(129,165)(130,166)(131,168)(132,167)(133,169)(134,170)
(135,172)(136,171)(137,173)(138,174)(139,176)(140,175)(141,177)(142,178)
(143,180)(144,179)(183,184)(187,188)(191,192)(195,196)(199,200)(203,204)
(207,208)(211,212)(215,216);
s2 := Sym(216)!( 1, 4)( 5, 12)( 6, 10)( 7, 11)( 8, 9)( 13, 32)( 14, 30)
( 15, 31)( 16, 29)( 17, 28)( 18, 26)( 19, 27)( 20, 25)( 21, 36)( 22, 34)
( 23, 35)( 24, 33)( 37, 76)( 38, 74)( 39, 75)( 40, 73)( 41, 84)( 42, 82)
( 43, 83)( 44, 81)( 45, 80)( 46, 78)( 47, 79)( 48, 77)( 49,104)( 50,102)
( 51,103)( 52,101)( 53,100)( 54, 98)( 55, 99)( 56, 97)( 57,108)( 58,106)
( 59,107)( 60,105)( 61, 92)( 62, 90)( 63, 91)( 64, 89)( 65, 88)( 66, 86)
( 67, 87)( 68, 85)( 69, 96)( 70, 94)( 71, 95)( 72, 93)(109,112)(113,120)
(114,118)(115,119)(116,117)(121,140)(122,138)(123,139)(124,137)(125,136)
(126,134)(127,135)(128,133)(129,144)(130,142)(131,143)(132,141)(145,184)
(146,182)(147,183)(148,181)(149,192)(150,190)(151,191)(152,189)(153,188)
(154,186)(155,187)(156,185)(157,212)(158,210)(159,211)(160,209)(161,208)
(162,206)(163,207)(164,205)(165,216)(166,214)(167,215)(168,213)(169,200)
(170,198)(171,199)(172,197)(173,196)(174,194)(175,195)(176,193)(177,204)
(178,202)(179,203)(180,201);
s3 := Sym(216)!( 1,121)( 2,122)( 3,123)( 4,124)( 5,129)( 6,130)( 7,131)
( 8,132)( 9,125)( 10,126)( 11,127)( 12,128)( 13,109)( 14,110)( 15,111)
( 16,112)( 17,117)( 18,118)( 19,119)( 20,120)( 21,113)( 22,114)( 23,115)
( 24,116)( 25,137)( 26,138)( 27,139)( 28,140)( 29,133)( 30,134)( 31,135)
( 32,136)( 33,141)( 34,142)( 35,143)( 36,144)( 37,157)( 38,158)( 39,159)
( 40,160)( 41,165)( 42,166)( 43,167)( 44,168)( 45,161)( 46,162)( 47,163)
( 48,164)( 49,145)( 50,146)( 51,147)( 52,148)( 53,153)( 54,154)( 55,155)
( 56,156)( 57,149)( 58,150)( 59,151)( 60,152)( 61,173)( 62,174)( 63,175)
( 64,176)( 65,169)( 66,170)( 67,171)( 68,172)( 69,177)( 70,178)( 71,179)
( 72,180)( 73,193)( 74,194)( 75,195)( 76,196)( 77,201)( 78,202)( 79,203)
( 80,204)( 81,197)( 82,198)( 83,199)( 84,200)( 85,181)( 86,182)( 87,183)
( 88,184)( 89,189)( 90,190)( 91,191)( 92,192)( 93,185)( 94,186)( 95,187)
( 96,188)( 97,209)( 98,210)( 99,211)(100,212)(101,205)(102,206)(103,207)
(104,208)(105,213)(106,214)(107,215)(108,216);
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*s0*s1*s0*s1, s1*s2*s3*s2*s1*s2*s3*s2,
s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*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