Polytope of Type {6,12,4}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,12,4}*1728q
if this polytope has a name.
Group : SmallGroup(1728,47234)
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 : {6,12,4}*864c, {6,12,2}*864i
   3-fold quotients : {6,4,4}*576
   4-fold quotients : {6,12,2}*432c
   6-fold quotients : {6,4,4}*288, {6,4,2}*288
   9-fold quotients : {2,12,4}*192a
   12-fold quotients : {6,4,2}*144
   18-fold quotients : {2,12,2}*96, {2,6,4}*96a
   27-fold quotients : {2,4,4}*64
   36-fold quotients : {2,6,2}*48
   54-fold quotients : {2,2,4}*32, {2,4,2}*32
   72-fold quotients : {2,3,2}*24
   108-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

s0 := (  1,109)(  2,111)(  3,110)(  4,115)(  5,117)(  6,116)(  7,112)(  8,114)
(  9,113)( 10,118)( 11,120)( 12,119)( 13,124)( 14,126)( 15,125)( 16,121)
( 17,123)( 18,122)( 19,127)( 20,129)( 21,128)( 22,133)( 23,135)( 24,134)
( 25,130)( 26,132)( 27,131)( 28,136)( 29,138)( 30,137)( 31,142)( 32,144)
( 33,143)( 34,139)( 35,141)( 36,140)( 37,145)( 38,147)( 39,146)( 40,151)
( 41,153)( 42,152)( 43,148)( 44,150)( 45,149)( 46,154)( 47,156)( 48,155)
( 49,160)( 50,162)( 51,161)( 52,157)( 53,159)( 54,158)( 55,163)( 56,165)
( 57,164)( 58,169)( 59,171)( 60,170)( 61,166)( 62,168)( 63,167)( 64,172)
( 65,174)( 66,173)( 67,178)( 68,180)( 69,179)( 70,175)( 71,177)( 72,176)
( 73,181)( 74,183)( 75,182)( 76,187)( 77,189)( 78,188)( 79,184)( 80,186)
( 81,185)( 82,190)( 83,192)( 84,191)( 85,196)( 86,198)( 87,197)( 88,193)
( 89,195)( 90,194)( 91,199)( 92,201)( 93,200)( 94,205)( 95,207)( 96,206)
( 97,202)( 98,204)( 99,203)(100,208)(101,210)(102,209)(103,214)(104,216)
(105,215)(106,211)(107,213)(108,212);;
s1 := (  1,  4)(  3,  9)(  5,  8)( 10, 22)( 11, 20)( 12, 27)( 13, 19)( 14, 26)
( 15, 24)( 16, 25)( 17, 23)( 18, 21)( 28, 31)( 30, 36)( 32, 35)( 37, 49)
( 38, 47)( 39, 54)( 40, 46)( 41, 53)( 42, 51)( 43, 52)( 44, 50)( 45, 48)
( 55, 58)( 57, 63)( 59, 62)( 64, 76)( 65, 74)( 66, 81)( 67, 73)( 68, 80)
( 69, 78)( 70, 79)( 71, 77)( 72, 75)( 82, 85)( 84, 90)( 86, 89)( 91,103)
( 92,101)( 93,108)( 94,100)( 95,107)( 96,105)( 97,106)( 98,104)( 99,102)
(109,112)(111,117)(113,116)(118,130)(119,128)(120,135)(121,127)(122,134)
(123,132)(124,133)(125,131)(126,129)(136,139)(138,144)(140,143)(145,157)
(146,155)(147,162)(148,154)(149,161)(150,159)(151,160)(152,158)(153,156)
(163,166)(165,171)(167,170)(172,184)(173,182)(174,189)(175,181)(176,188)
(177,186)(178,187)(179,185)(180,183)(190,193)(192,198)(194,197)(199,211)
(200,209)(201,216)(202,208)(203,215)(204,213)(205,214)(206,212)(207,210);;
s2 := (  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, 91)( 56, 92)( 57, 93)( 58, 99)( 59, 97)( 60, 98)( 61, 95)( 62, 96)
( 63, 94)( 64, 82)( 65, 83)( 66, 84)( 67, 90)( 68, 88)( 69, 89)( 70, 86)
( 71, 87)( 72, 85)( 73,100)( 74,101)( 75,102)( 76,108)( 77,106)( 78,107)
( 79,104)( 80,105)( 81,103)(109,118)(110,119)(111,120)(112,126)(113,124)
(114,125)(115,122)(116,123)(117,121)(130,135)(131,133)(132,134)(136,145)
(137,146)(138,147)(139,153)(140,151)(141,152)(142,149)(143,150)(144,148)
(157,162)(158,160)(159,161)(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);;
s3 := (  1, 55)(  2, 56)(  3, 57)(  4, 58)(  5, 59)(  6, 60)(  7, 61)(  8, 62)
(  9, 63)( 10, 64)( 11, 65)( 12, 66)( 13, 67)( 14, 68)( 15, 69)( 16, 70)
( 17, 71)( 18, 72)( 19, 73)( 20, 74)( 21, 75)( 22, 76)( 23, 77)( 24, 78)
( 25, 79)( 26, 80)( 27, 81)( 28, 82)( 29, 83)( 30, 84)( 31, 85)( 32, 86)
( 33, 87)( 34, 88)( 35, 89)( 36, 90)( 37, 91)( 38, 92)( 39, 93)( 40, 94)
( 41, 95)( 42, 96)( 43, 97)( 44, 98)( 45, 99)( 46,100)( 47,101)( 48,102)
( 49,103)( 50,104)( 51,105)( 52,106)( 53,107)( 54,108)(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);;
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, 
s2*s0*s1*s2*s0*s1*s0*s1*s2*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)!(  1,109)(  2,111)(  3,110)(  4,115)(  5,117)(  6,116)(  7,112)
(  8,114)(  9,113)( 10,118)( 11,120)( 12,119)( 13,124)( 14,126)( 15,125)
( 16,121)( 17,123)( 18,122)( 19,127)( 20,129)( 21,128)( 22,133)( 23,135)
( 24,134)( 25,130)( 26,132)( 27,131)( 28,136)( 29,138)( 30,137)( 31,142)
( 32,144)( 33,143)( 34,139)( 35,141)( 36,140)( 37,145)( 38,147)( 39,146)
( 40,151)( 41,153)( 42,152)( 43,148)( 44,150)( 45,149)( 46,154)( 47,156)
( 48,155)( 49,160)( 50,162)( 51,161)( 52,157)( 53,159)( 54,158)( 55,163)
( 56,165)( 57,164)( 58,169)( 59,171)( 60,170)( 61,166)( 62,168)( 63,167)
( 64,172)( 65,174)( 66,173)( 67,178)( 68,180)( 69,179)( 70,175)( 71,177)
( 72,176)( 73,181)( 74,183)( 75,182)( 76,187)( 77,189)( 78,188)( 79,184)
( 80,186)( 81,185)( 82,190)( 83,192)( 84,191)( 85,196)( 86,198)( 87,197)
( 88,193)( 89,195)( 90,194)( 91,199)( 92,201)( 93,200)( 94,205)( 95,207)
( 96,206)( 97,202)( 98,204)( 99,203)(100,208)(101,210)(102,209)(103,214)
(104,216)(105,215)(106,211)(107,213)(108,212);
s1 := Sym(216)!(  1,  4)(  3,  9)(  5,  8)( 10, 22)( 11, 20)( 12, 27)( 13, 19)
( 14, 26)( 15, 24)( 16, 25)( 17, 23)( 18, 21)( 28, 31)( 30, 36)( 32, 35)
( 37, 49)( 38, 47)( 39, 54)( 40, 46)( 41, 53)( 42, 51)( 43, 52)( 44, 50)
( 45, 48)( 55, 58)( 57, 63)( 59, 62)( 64, 76)( 65, 74)( 66, 81)( 67, 73)
( 68, 80)( 69, 78)( 70, 79)( 71, 77)( 72, 75)( 82, 85)( 84, 90)( 86, 89)
( 91,103)( 92,101)( 93,108)( 94,100)( 95,107)( 96,105)( 97,106)( 98,104)
( 99,102)(109,112)(111,117)(113,116)(118,130)(119,128)(120,135)(121,127)
(122,134)(123,132)(124,133)(125,131)(126,129)(136,139)(138,144)(140,143)
(145,157)(146,155)(147,162)(148,154)(149,161)(150,159)(151,160)(152,158)
(153,156)(163,166)(165,171)(167,170)(172,184)(173,182)(174,189)(175,181)
(176,188)(177,186)(178,187)(179,185)(180,183)(190,193)(192,198)(194,197)
(199,211)(200,209)(201,216)(202,208)(203,215)(204,213)(205,214)(206,212)
(207,210);
s2 := 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, 91)( 56, 92)( 57, 93)( 58, 99)( 59, 97)( 60, 98)( 61, 95)
( 62, 96)( 63, 94)( 64, 82)( 65, 83)( 66, 84)( 67, 90)( 68, 88)( 69, 89)
( 70, 86)( 71, 87)( 72, 85)( 73,100)( 74,101)( 75,102)( 76,108)( 77,106)
( 78,107)( 79,104)( 80,105)( 81,103)(109,118)(110,119)(111,120)(112,126)
(113,124)(114,125)(115,122)(116,123)(117,121)(130,135)(131,133)(132,134)
(136,145)(137,146)(138,147)(139,153)(140,151)(141,152)(142,149)(143,150)
(144,148)(157,162)(158,160)(159,161)(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);
s3 := Sym(216)!(  1, 55)(  2, 56)(  3, 57)(  4, 58)(  5, 59)(  6, 60)(  7, 61)
(  8, 62)(  9, 63)( 10, 64)( 11, 65)( 12, 66)( 13, 67)( 14, 68)( 15, 69)
( 16, 70)( 17, 71)( 18, 72)( 19, 73)( 20, 74)( 21, 75)( 22, 76)( 23, 77)
( 24, 78)( 25, 79)( 26, 80)( 27, 81)( 28, 82)( 29, 83)( 30, 84)( 31, 85)
( 32, 86)( 33, 87)( 34, 88)( 35, 89)( 36, 90)( 37, 91)( 38, 92)( 39, 93)
( 40, 94)( 41, 95)( 42, 96)( 43, 97)( 44, 98)( 45, 99)( 46,100)( 47,101)
( 48,102)( 49,103)( 50,104)( 51,105)( 52,106)( 53,107)( 54,108)(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);
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, 
s2*s0*s1*s2*s0*s1*s0*s1*s2*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