Polytope of Type {18,12,4}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {18,12,4}*1728a
Also Known As : {{18,12|2},{12,4|2}}. if this polytope has another name.
Group : SmallGroup(1728,14460)
Rank : 4
Schlafli Type : {18,12,4}
Number of vertices, edges, etc : 18, 108, 24, 4
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 : {18,12,2}*864a, {18,6,4}*864a
3-fold quotients : {18,4,4}*576, {6,12,4}*576a
4-fold quotients : {18,6,2}*432a
6-fold quotients : {18,2,4}*288, {18,4,2}*288a, {6,12,2}*288a, {6,6,4}*288a
9-fold quotients : {2,12,4}*192a, {6,4,4}*192
12-fold quotients : {9,2,4}*144, {18,2,2}*144, {6,6,2}*144a
18-fold quotients : {2,12,2}*96, {2,6,4}*96a, {6,2,4}*96, {6,4,2}*96a
24-fold quotients : {9,2,2}*72
27-fold quotients : {2,4,4}*64
36-fold quotients : {3,2,4}*48, {2,6,2}*48, {6,2,2}*48
54-fold quotients : {2,2,4}*32, {2,4,2}*32
72-fold quotients : {2,3,2}*24, {3,2,2}*24
108-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := ( 2, 3)( 5, 6)( 8, 9)( 10, 21)( 11, 20)( 12, 19)( 13, 24)( 14, 23)
( 15, 22)( 16, 27)( 17, 26)( 18, 25)( 29, 30)( 32, 33)( 35, 36)( 37, 48)
( 38, 47)( 39, 46)( 40, 51)( 41, 50)( 42, 49)( 43, 54)( 44, 53)( 45, 52)
( 56, 57)( 59, 60)( 62, 63)( 64, 75)( 65, 74)( 66, 73)( 67, 78)( 68, 77)
( 69, 76)( 70, 81)( 71, 80)( 72, 79)( 83, 84)( 86, 87)( 89, 90)( 91,102)
( 92,101)( 93,100)( 94,105)( 95,104)( 96,103)( 97,108)( 98,107)( 99,106)
(110,111)(113,114)(116,117)(118,129)(119,128)(120,127)(121,132)(122,131)
(123,130)(124,135)(125,134)(126,133)(137,138)(140,141)(143,144)(145,156)
(146,155)(147,154)(148,159)(149,158)(150,157)(151,162)(152,161)(153,160)
(164,165)(167,168)(170,171)(172,183)(173,182)(174,181)(175,186)(176,185)
(177,184)(178,189)(179,188)(180,187)(191,192)(194,195)(197,198)(199,210)
(200,209)(201,208)(202,213)(203,212)(204,211)(205,216)(206,215)(207,214);;
s1 := ( 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)(109,145)(110,147)(111,146)(112,151)
(113,153)(114,152)(115,148)(116,150)(117,149)(118,136)(119,138)(120,137)
(121,142)(122,144)(123,143)(124,139)(125,141)(126,140)(127,156)(128,155)
(129,154)(130,162)(131,161)(132,160)(133,159)(134,158)(135,157)(163,199)
(164,201)(165,200)(166,205)(167,207)(168,206)(169,202)(170,204)(171,203)
(172,190)(173,192)(174,191)(175,196)(176,198)(177,197)(178,193)(179,195)
(180,194)(181,210)(182,209)(183,208)(184,216)(185,215)(186,214)(187,213)
(188,212)(189,211);;
s2 := ( 1,112)( 2,113)( 3,114)( 4,109)( 5,110)( 6,111)( 7,115)( 8,116)
( 9,117)( 10,121)( 11,122)( 12,123)( 13,118)( 14,119)( 15,120)( 16,124)
( 17,125)( 18,126)( 19,130)( 20,131)( 21,132)( 22,127)( 23,128)( 24,129)
( 25,133)( 26,134)( 27,135)( 28,139)( 29,140)( 30,141)( 31,136)( 32,137)
( 33,138)( 34,142)( 35,143)( 36,144)( 37,148)( 38,149)( 39,150)( 40,145)
( 41,146)( 42,147)( 43,151)( 44,152)( 45,153)( 46,157)( 47,158)( 48,159)
( 49,154)( 50,155)( 51,156)( 52,160)( 53,161)( 54,162)( 55,166)( 56,167)
( 57,168)( 58,163)( 59,164)( 60,165)( 61,169)( 62,170)( 63,171)( 64,175)
( 65,176)( 66,177)( 67,172)( 68,173)( 69,174)( 70,178)( 71,179)( 72,180)
( 73,184)( 74,185)( 75,186)( 76,181)( 77,182)( 78,183)( 79,187)( 80,188)
( 81,189)( 82,193)( 83,194)( 84,195)( 85,190)( 86,191)( 87,192)( 88,196)
( 89,197)( 90,198)( 91,202)( 92,203)( 93,204)( 94,199)( 95,200)( 96,201)
( 97,205)( 98,206)( 99,207)(100,211)(101,212)(102,213)(103,208)(104,209)
(105,210)(106,214)(107,215)(108,216);;
s3 := (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);;
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,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*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*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(216)!( 2, 3)( 5, 6)( 8, 9)( 10, 21)( 11, 20)( 12, 19)( 13, 24)
( 14, 23)( 15, 22)( 16, 27)( 17, 26)( 18, 25)( 29, 30)( 32, 33)( 35, 36)
( 37, 48)( 38, 47)( 39, 46)( 40, 51)( 41, 50)( 42, 49)( 43, 54)( 44, 53)
( 45, 52)( 56, 57)( 59, 60)( 62, 63)( 64, 75)( 65, 74)( 66, 73)( 67, 78)
( 68, 77)( 69, 76)( 70, 81)( 71, 80)( 72, 79)( 83, 84)( 86, 87)( 89, 90)
( 91,102)( 92,101)( 93,100)( 94,105)( 95,104)( 96,103)( 97,108)( 98,107)
( 99,106)(110,111)(113,114)(116,117)(118,129)(119,128)(120,127)(121,132)
(122,131)(123,130)(124,135)(125,134)(126,133)(137,138)(140,141)(143,144)
(145,156)(146,155)(147,154)(148,159)(149,158)(150,157)(151,162)(152,161)
(153,160)(164,165)(167,168)(170,171)(172,183)(173,182)(174,181)(175,186)
(176,185)(177,184)(178,189)(179,188)(180,187)(191,192)(194,195)(197,198)
(199,210)(200,209)(201,208)(202,213)(203,212)(204,211)(205,216)(206,215)
(207,214);
s1 := 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)(109,145)(110,147)(111,146)
(112,151)(113,153)(114,152)(115,148)(116,150)(117,149)(118,136)(119,138)
(120,137)(121,142)(122,144)(123,143)(124,139)(125,141)(126,140)(127,156)
(128,155)(129,154)(130,162)(131,161)(132,160)(133,159)(134,158)(135,157)
(163,199)(164,201)(165,200)(166,205)(167,207)(168,206)(169,202)(170,204)
(171,203)(172,190)(173,192)(174,191)(175,196)(176,198)(177,197)(178,193)
(179,195)(180,194)(181,210)(182,209)(183,208)(184,216)(185,215)(186,214)
(187,213)(188,212)(189,211);
s2 := Sym(216)!( 1,112)( 2,113)( 3,114)( 4,109)( 5,110)( 6,111)( 7,115)
( 8,116)( 9,117)( 10,121)( 11,122)( 12,123)( 13,118)( 14,119)( 15,120)
( 16,124)( 17,125)( 18,126)( 19,130)( 20,131)( 21,132)( 22,127)( 23,128)
( 24,129)( 25,133)( 26,134)( 27,135)( 28,139)( 29,140)( 30,141)( 31,136)
( 32,137)( 33,138)( 34,142)( 35,143)( 36,144)( 37,148)( 38,149)( 39,150)
( 40,145)( 41,146)( 42,147)( 43,151)( 44,152)( 45,153)( 46,157)( 47,158)
( 48,159)( 49,154)( 50,155)( 51,156)( 52,160)( 53,161)( 54,162)( 55,166)
( 56,167)( 57,168)( 58,163)( 59,164)( 60,165)( 61,169)( 62,170)( 63,171)
( 64,175)( 65,176)( 66,177)( 67,172)( 68,173)( 69,174)( 70,178)( 71,179)
( 72,180)( 73,184)( 74,185)( 75,186)( 76,181)( 77,182)( 78,183)( 79,187)
( 80,188)( 81,189)( 82,193)( 83,194)( 84,195)( 85,190)( 86,191)( 87,192)
( 88,196)( 89,197)( 90,198)( 91,202)( 92,203)( 93,204)( 94,199)( 95,200)
( 96,201)( 97,205)( 98,206)( 99,207)(100,211)(101,212)(102,213)(103,208)
(104,209)(105,210)(106,214)(107,215)(108,216);
s3 := 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);
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, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*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*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 >;
References : None.
to this polytope