Polytope of Type {4,12,18}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,12,18}*1728a
Also Known As : {{4,12|2},{12,18|2}}. if this polytope has another name.
Group : SmallGroup(1728,14460)
Rank : 4
Schlafli Type : {4,12,18}
Number of vertices, edges, etc : 4, 24, 108, 18
Order of s0s1s2s3 : 36
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 : {2,12,18}*864a, {4,6,18}*864a
3-fold quotients : {4,4,18}*576, {4,12,6}*576a
4-fold quotients : {2,6,18}*432a
6-fold quotients : {2,4,18}*288a, {4,2,18}*288, {2,12,6}*288a, {4,6,6}*288a
9-fold quotients : {4,12,2}*192a, {4,4,6}*192
12-fold quotients : {4,2,9}*144, {2,2,18}*144, {2,6,6}*144a
18-fold quotients : {2,12,2}*96, {2,4,6}*96a, {4,2,6}*96, {4,6,2}*96a
24-fold quotients : {2,2,9}*72
27-fold quotients : {4,4,2}*64
36-fold quotients : {4,2,3}*48, {2,2,6}*48, {2,6,2}*48
54-fold quotients : {2,4,2}*32, {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 := (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)( 55,163)( 56,164)
( 57,165)( 58,169)( 59,170)( 60,171)( 61,166)( 62,167)( 63,168)( 64,172)
( 65,173)( 66,174)( 67,178)( 68,179)( 69,180)( 70,175)( 71,176)( 72,177)
( 73,181)( 74,182)( 75,183)( 76,187)( 77,188)( 78,189)( 79,184)( 80,185)
( 81,186)( 82,190)( 83,191)( 84,192)( 85,196)( 86,197)( 87,198)( 88,193)
( 89,194)( 90,195)( 91,199)( 92,200)( 93,201)( 94,205)( 95,206)( 96,207)
( 97,202)( 98,203)( 99,204)(100,208)(101,209)(102,210)(103,214)(104,215)
(105,216)(106,211)(107,212)(108,213);;
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)(109,139)(110,141)(111,140)(112,136)
(113,138)(114,137)(115,142)(116,144)(117,143)(118,159)(119,158)(120,157)
(121,156)(122,155)(123,154)(124,162)(125,161)(126,160)(127,150)(128,149)
(129,148)(130,147)(131,146)(132,145)(133,153)(134,152)(135,151)(163,193)
(164,195)(165,194)(166,190)(167,192)(168,191)(169,196)(170,198)(171,197)
(172,213)(173,212)(174,211)(175,210)(176,209)(177,208)(178,216)(179,215)
(180,214)(181,204)(182,203)(183,202)(184,201)(185,200)(186,199)(187,207)
(188,206)(189,205);;
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*s0*s1*s0*s1*s0*s1,
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*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
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)!(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)( 55,163)
( 56,164)( 57,165)( 58,169)( 59,170)( 60,171)( 61,166)( 62,167)( 63,168)
( 64,172)( 65,173)( 66,174)( 67,178)( 68,179)( 69,180)( 70,175)( 71,176)
( 72,177)( 73,181)( 74,182)( 75,183)( 76,187)( 77,188)( 78,189)( 79,184)
( 80,185)( 81,186)( 82,190)( 83,191)( 84,192)( 85,196)( 86,197)( 87,198)
( 88,193)( 89,194)( 90,195)( 91,199)( 92,200)( 93,201)( 94,205)( 95,206)
( 96,207)( 97,202)( 98,203)( 99,204)(100,208)(101,209)(102,210)(103,214)
(104,215)(105,216)(106,211)(107,212)(108,213);
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)(109,139)(110,141)(111,140)
(112,136)(113,138)(114,137)(115,142)(116,144)(117,143)(118,159)(119,158)
(120,157)(121,156)(122,155)(123,154)(124,162)(125,161)(126,160)(127,150)
(128,149)(129,148)(130,147)(131,146)(132,145)(133,153)(134,152)(135,151)
(163,193)(164,195)(165,194)(166,190)(167,192)(168,191)(169,196)(170,198)
(171,197)(172,213)(173,212)(174,211)(175,210)(176,209)(177,208)(178,216)
(179,215)(180,214)(181,204)(182,203)(183,202)(184,201)(185,200)(186,199)
(187,207)(188,206)(189,205);
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*s0*s1*s0*s1*s0*s1, 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*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
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