Polytope of Type {6,12,4}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,12,4}*1728b
if this polytope has a name.
Group : SmallGroup(1728,14771)
Rank : 4
Schlafli Type : {6,12,4}
Number of vertices, edges, etc : 18, 108, 72, 4
Order of s0s1s2s3 : 12
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 : {6,12,2}*864b, {6,6,4}*864b
3-fold quotients : {6,12,4}*576a
4-fold quotients : {6,6,2}*432b
6-fold quotients : {6,12,2}*288a, {6,6,4}*288a
8-fold quotients : {6,6,2}*216
9-fold quotients : {2,12,4}*192a, {6,4,4}*192
12-fold quotients : {6,6,2}*144a
18-fold quotients : {2,12,2}*96, {2,6,4}*96a, {6,2,4}*96, {6,4,2}*96a
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.
Irregular Quotients (of which this is a minimal cover):
P/N, where N=<s0*s1*s2*s1*s0*s2*s1*s2*s1*s2> of order 3.
4 facets:
4 of 3-fold non-regular quotient of {6,12}*432b
10 vertex figures:
4 of {12,4}*96a
6 of {4,4}*32
P/N, where N=<s0*s1*s0*s1> of order 3.
4 facets:
4 of 3-fold non-regular quotient of {6,12}*432b
6 vertex figures:
6 of {12,4}*96a
Permutation Representation (GAP) :
s0 := ( 2, 3)( 5, 6)( 8, 9)( 10, 19)( 11, 21)( 12, 20)( 13, 22)( 14, 24)( 15, 23)( 16, 25)( 17, 27)( 18, 26)( 29, 30)( 32, 33)( 35, 36)( 37, 46)( 38, 48)( 39, 47)( 40, 49)( 41, 51)( 42, 50)( 43, 52)( 44, 54)( 45, 53)( 56, 57)( 59, 60)( 62, 63)( 64, 73)( 65, 75)( 66, 74)( 67, 76)( 68, 78)( 69, 77)( 70, 79)( 71, 81)( 72, 80)( 83, 84)( 86, 87)( 89, 90)( 91,100)( 92,102)( 93,101)( 94,103)( 95,105)( 96,104)( 97,106)( 98,108)( 99,107)(110,111)(113,114)(116,117)(118,127)(119,129)(120,128)(121,130)(122,132)(123,131)(124,133)(125,135)(126,134)(137,138)(140,141)(143,144)(145,154)(146,156)(147,155)(148,157)(149,159)(150,158)(151,160)(152,162)(153,161)(164,165)(167,168)(170,171)(172,181)(173,183)(174,182)(175,184)(176,186)(177,185)(178,187)(179,189)(180,188)(191,192)(194,195)(197,198)(199,208)(200,210)(201,209)(202,211)(203,213)(204,212)(205,214)(206,216)(207,215);;
s1 := ( 1, 10)( 2, 11)( 3, 12)( 4, 18)( 5, 16)( 6, 17)( 7, 14)( 8, 15)( 9, 13)( 22, 27)( 23, 25)( 24, 26)( 28, 37)( 29, 38)( 30, 39)( 31, 45)( 32, 43)( 33, 44)( 34, 41)( 35, 42)( 36, 40)( 49, 54)( 50, 52)( 51, 53)( 55, 64)( 56, 65)( 57, 66)( 58, 72)( 59, 70)( 60, 71)( 61, 68)( 62, 69)( 63, 67)( 76, 81)( 77, 79)( 78, 80)( 82, 91)( 83, 92)( 84, 93)( 85, 99)( 86, 97)( 87, 98)( 88, 95)( 89, 96)( 90, 94)(103,108)(104,106)(105,107)(109,145)(110,146)(111,147)(112,153)(113,151)(114,152)(115,149)(116,150)(117,148)(118,136)(119,137)(120,138)(121,144)(122,142)(123,143)(124,140)(125,141)(126,139)(127,154)(128,155)(129,156)(130,162)(131,160)(132,161)(133,158)(134,159)(135,157)(163,199)(164,200)(165,201)(166,207)(167,205)(168,206)(169,203)(170,204)(171,202)(172,190)(173,191)(174,192)(175,198)(176,196)(177,197)(178,194)(179,195)(180,193)(181,208)(182,209)(183,210)(184,216)(185,214)(186,215)(187,212)(188,213)(189,211);;
s2 := ( 1,112)( 2,114)( 3,113)( 4,109)( 5,111)( 6,110)( 7,115)( 8,117)( 9,116)( 10,121)( 11,123)( 12,122)( 13,118)( 14,120)( 15,119)( 16,124)( 17,126)( 18,125)( 19,130)( 20,132)( 21,131)( 22,127)( 23,129)( 24,128)( 25,133)( 26,135)( 27,134)( 28,139)( 29,141)( 30,140)( 31,136)( 32,138)( 33,137)( 34,142)( 35,144)( 36,143)( 37,148)( 38,150)( 39,149)( 40,145)( 41,147)( 42,146)( 43,151)( 44,153)( 45,152)( 46,157)( 47,159)( 48,158)( 49,154)( 50,156)( 51,155)( 52,160)( 53,162)( 54,161)( 55,166)( 56,168)( 57,167)( 58,163)( 59,165)( 60,164)( 61,169)( 62,171)( 63,170)( 64,175)( 65,177)( 66,176)( 67,172)( 68,174)( 69,173)( 70,178)( 71,180)( 72,179)( 73,184)( 74,186)( 75,185)( 76,181)( 77,183)( 78,182)( 79,187)( 80,189)( 81,188)( 82,193)( 83,195)( 84,194)( 85,190)( 86,192)( 87,191)( 88,196)( 89,198)( 90,197)( 91,202)( 92,204)( 93,203)( 94,199)( 95,201)( 96,200)( 97,205)( 98,207)( 99,206)(100,211)(101,213)(102,212)(103,208)(104,210)(105,209)(106,214)(107,216)(108,215);;
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, s1*s2*s3*s2*s1*s2*s3*s2,
s2*s3*s2*s3*s2*s3*s2*s3, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s0*s1*s2*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s2*s0*s1,
s0*s1*s2*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2*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 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(216)!( 2, 3)( 5, 6)( 8, 9)( 10, 19)( 11, 21)( 12, 20)( 13, 22)( 14, 24)( 15, 23)( 16, 25)( 17, 27)( 18, 26)( 29, 30)( 32, 33)( 35, 36)( 37, 46)( 38, 48)( 39, 47)( 40, 49)( 41, 51)( 42, 50)( 43, 52)( 44, 54)( 45, 53)( 56, 57)( 59, 60)( 62, 63)( 64, 73)( 65, 75)( 66, 74)( 67, 76)( 68, 78)( 69, 77)( 70, 79)( 71, 81)( 72, 80)( 83, 84)( 86, 87)( 89, 90)( 91,100)( 92,102)( 93,101)( 94,103)( 95,105)( 96,104)( 97,106)( 98,108)( 99,107)(110,111)(113,114)(116,117)(118,127)(119,129)(120,128)(121,130)(122,132)(123,131)(124,133)(125,135)(126,134)(137,138)(140,141)(143,144)(145,154)(146,156)(147,155)(148,157)(149,159)(150,158)(151,160)(152,162)(153,161)(164,165)(167,168)(170,171)(172,181)(173,183)(174,182)(175,184)(176,186)(177,185)(178,187)(179,189)(180,188)(191,192)(194,195)(197,198)(199,208)(200,210)(201,209)(202,211)(203,213)(204,212)(205,214)(206,216)(207,215);
s1 := Sym(216)!( 1, 10)( 2, 11)( 3, 12)( 4, 18)( 5, 16)( 6, 17)( 7, 14)( 8, 15)( 9, 13)( 22, 27)( 23, 25)( 24, 26)( 28, 37)( 29, 38)( 30, 39)( 31, 45)( 32, 43)( 33, 44)( 34, 41)( 35, 42)( 36, 40)( 49, 54)( 50, 52)( 51, 53)( 55, 64)( 56, 65)( 57, 66)( 58, 72)( 59, 70)( 60, 71)( 61, 68)( 62, 69)( 63, 67)( 76, 81)( 77, 79)( 78, 80)( 82, 91)( 83, 92)( 84, 93)( 85, 99)( 86, 97)( 87, 98)( 88, 95)( 89, 96)( 90, 94)(103,108)(104,106)(105,107)(109,145)(110,146)(111,147)(112,153)(113,151)(114,152)(115,149)(116,150)(117,148)(118,136)(119,137)(120,138)(121,144)(122,142)(123,143)(124,140)(125,141)(126,139)(127,154)(128,155)(129,156)(130,162)(131,160)(132,161)(133,158)(134,159)(135,157)(163,199)(164,200)(165,201)(166,207)(167,205)(168,206)(169,203)(170,204)(171,202)(172,190)(173,191)(174,192)(175,198)(176,196)(177,197)(178,194)(179,195)(180,193)(181,208)(182,209)(183,210)(184,216)(185,214)(186,215)(187,212)(188,213)(189,211);
s2 := Sym(216)!( 1,112)( 2,114)( 3,113)( 4,109)( 5,111)( 6,110)( 7,115)( 8,117)( 9,116)( 10,121)( 11,123)( 12,122)( 13,118)( 14,120)( 15,119)( 16,124)( 17,126)( 18,125)( 19,130)( 20,132)( 21,131)( 22,127)( 23,129)( 24,128)( 25,133)( 26,135)( 27,134)( 28,139)( 29,141)( 30,140)( 31,136)( 32,138)( 33,137)( 34,142)( 35,144)( 36,143)( 37,148)( 38,150)( 39,149)( 40,145)( 41,147)( 42,146)( 43,151)( 44,153)( 45,152)( 46,157)( 47,159)( 48,158)( 49,154)( 50,156)( 51,155)( 52,160)( 53,162)( 54,161)( 55,166)( 56,168)( 57,167)( 58,163)( 59,165)( 60,164)( 61,169)( 62,171)( 63,170)( 64,175)( 65,177)( 66,176)( 67,172)( 68,174)( 69,173)( 70,178)( 71,180)( 72,179)( 73,184)( 74,186)( 75,185)( 76,181)( 77,183)( 78,182)( 79,187)( 80,189)( 81,188)( 82,193)( 83,195)( 84,194)( 85,190)( 86,192)( 87,191)( 88,196)( 89,198)( 90,197)( 91,202)( 92,204)( 93,203)( 94,199)( 95,201)( 96,200)( 97,205)( 98,207)( 99,206)(100,211)(101,213)(102,212)(103,208)(104,210)(105,209)(106,214)(107,216)(108,215);
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,
s1*s2*s3*s2*s1*s2*s3*s2, s2*s3*s2*s3*s2*s3*s2*s3,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s0*s1*s2*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s2*s0*s1,
s0*s1*s2*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2*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 >;
References : None.
to this polytope