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Polytope of Type {4,6,6,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,6,6,6}*1728h
if this polytope has a name.
Group : SmallGroup(1728,47409)
Rank : 5
Schlafli Type : {4,6,6,6}
Number of vertices, edges, etc : 4, 12, 18, 18, 6
Order of s0s1s2s3s4 : 12
Order of s0s1s2s3s4s3s2s1 : 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 : {4,6,3,6}*864, {2,6,6,6}*864f
   3-fold quotients : {4,2,6,6}*576c, {4,6,6,2}*576c
   4-fold quotients : {2,6,3,6}*432
   6-fold quotients : {4,2,3,6}*288, {4,6,3,2}*288, {2,2,6,6}*288c, {2,6,6,2}*288b
   9-fold quotients : {4,2,6,2}*192
   12-fold quotients : {2,2,3,6}*144, {2,6,3,2}*144
   18-fold quotients : {4,2,3,2}*96, {2,2,6,2}*96
   27-fold quotients : {4,2,2,2}*64
   36-fold quotients : {2,2,3,2}*48
   54-fold quotients : {2,2,2,2}*32
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :
s0 := (  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);;
s1 := ( 10, 19)( 11, 20)( 12, 21)( 13, 22)( 14, 23)( 15, 24)( 16, 25)( 17, 26)
( 18, 27)( 37, 46)( 38, 47)( 39, 48)( 40, 49)( 41, 50)( 42, 51)( 43, 52)
( 44, 53)( 45, 54)( 55, 82)( 56, 83)( 57, 84)( 58, 85)( 59, 86)( 60, 87)
( 61, 88)( 62, 89)( 63, 90)( 64,100)( 65,101)( 66,102)( 67,103)( 68,104)
( 69,105)( 70,106)( 71,107)( 72,108)( 73, 91)( 74, 92)( 75, 93)( 76, 94)
( 77, 95)( 78, 96)( 79, 97)( 80, 98)( 81, 99)(118,127)(119,128)(120,129)
(121,130)(122,131)(123,132)(124,133)(125,134)(126,135)(145,154)(146,155)
(147,156)(148,157)(149,158)(150,159)(151,160)(152,161)(153,162)(163,190)
(164,191)(165,192)(166,193)(167,194)(168,195)(169,196)(170,197)(171,198)
(172,208)(173,209)(174,210)(175,211)(176,212)(177,213)(178,214)(179,215)
(180,216)(181,199)(182,200)(183,201)(184,202)(185,203)(186,204)(187,205)
(188,206)(189,207);;
s2 := (  1, 10)(  2, 12)(  3, 11)(  4, 16)(  5, 18)(  6, 17)(  7, 13)(  8, 15)
(  9, 14)( 20, 21)( 22, 25)( 23, 27)( 24, 26)( 28, 37)( 29, 39)( 30, 38)
( 31, 43)( 32, 45)( 33, 44)( 34, 40)( 35, 42)( 36, 41)( 47, 48)( 49, 52)
( 50, 54)( 51, 53)( 55, 64)( 56, 66)( 57, 65)( 58, 70)( 59, 72)( 60, 71)
( 61, 67)( 62, 69)( 63, 68)( 74, 75)( 76, 79)( 77, 81)( 78, 80)( 82, 91)
( 83, 93)( 84, 92)( 85, 97)( 86, 99)( 87, 98)( 88, 94)( 89, 96)( 90, 95)
(101,102)(103,106)(104,108)(105,107)(109,118)(110,120)(111,119)(112,124)
(113,126)(114,125)(115,121)(116,123)(117,122)(128,129)(130,133)(131,135)
(132,134)(136,145)(137,147)(138,146)(139,151)(140,153)(141,152)(142,148)
(143,150)(144,149)(155,156)(157,160)(158,162)(159,161)(163,172)(164,174)
(165,173)(166,178)(167,180)(168,179)(169,175)(170,177)(171,176)(182,183)
(184,187)(185,189)(186,188)(190,199)(191,201)(192,200)(193,205)(194,207)
(195,206)(196,202)(197,204)(198,203)(209,210)(211,214)(212,216)(213,215);;
s3 := (  1,113)(  2,112)(  3,114)(  4,110)(  5,109)(  6,111)(  7,116)(  8,115)
(  9,117)( 10,131)( 11,130)( 12,132)( 13,128)( 14,127)( 15,129)( 16,134)
( 17,133)( 18,135)( 19,122)( 20,121)( 21,123)( 22,119)( 23,118)( 24,120)
( 25,125)( 26,124)( 27,126)( 28,140)( 29,139)( 30,141)( 31,137)( 32,136)
( 33,138)( 34,143)( 35,142)( 36,144)( 37,158)( 38,157)( 39,159)( 40,155)
( 41,154)( 42,156)( 43,161)( 44,160)( 45,162)( 46,149)( 47,148)( 48,150)
( 49,146)( 50,145)( 51,147)( 52,152)( 53,151)( 54,153)( 55,167)( 56,166)
( 57,168)( 58,164)( 59,163)( 60,165)( 61,170)( 62,169)( 63,171)( 64,185)
( 65,184)( 66,186)( 67,182)( 68,181)( 69,183)( 70,188)( 71,187)( 72,189)
( 73,176)( 74,175)( 75,177)( 76,173)( 77,172)( 78,174)( 79,179)( 80,178)
( 81,180)( 82,194)( 83,193)( 84,195)( 85,191)( 86,190)( 87,192)( 88,197)
( 89,196)( 90,198)( 91,212)( 92,211)( 93,213)( 94,209)( 95,208)( 96,210)
( 97,215)( 98,214)( 99,216)(100,203)(101,202)(102,204)(103,200)(104,199)
(105,201)(106,206)(107,205)(108,207);;
s4 := (  4,  7)(  5,  8)(  6,  9)( 13, 16)( 14, 17)( 15, 18)( 22, 25)( 23, 26)
( 24, 27)( 31, 34)( 32, 35)( 33, 36)( 40, 43)( 41, 44)( 42, 45)( 49, 52)
( 50, 53)( 51, 54)( 58, 61)( 59, 62)( 60, 63)( 67, 70)( 68, 71)( 69, 72)
( 76, 79)( 77, 80)( 78, 81)( 85, 88)( 86, 89)( 87, 90)( 94, 97)( 95, 98)
( 96, 99)(103,106)(104,107)(105,108)(112,115)(113,116)(114,117)(121,124)
(122,125)(123,126)(130,133)(131,134)(132,135)(139,142)(140,143)(141,144)
(148,151)(149,152)(150,153)(157,160)(158,161)(159,162)(166,169)(167,170)
(168,171)(175,178)(176,179)(177,180)(184,187)(185,188)(186,189)(193,196)
(194,197)(195,198)(202,205)(203,206)(204,207)(211,214)(212,215)(213,216);;
poly := Group([s0,s1,s2,s3,s4]);;
 
Finitely Presented Group Representation (GAP) :
F := FreeGroup("s0","s1","s2","s3","s4");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;  s4 := F.5;;  
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s4*s4, s0*s2*s0*s2, 
s0*s3*s0*s3, s1*s3*s1*s3, s0*s4*s0*s4, 
s1*s4*s1*s4, s2*s4*s2*s4, s0*s1*s0*s1*s0*s1*s0*s1, 
s0*s1*s2*s1*s0*s1*s2*s1, s3*s1*s2*s1*s2*s3*s1*s2*s1*s2, 
s4*s2*s3*s4*s3*s4*s2*s3*s4*s3, 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*s4*s3*s2*s3*s2*s3*s4*s3*s2*s3 ];;
poly := F / rels;;
 
Permutation Representation (Magma) :
s0 := 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);
s1 := Sym(216)!( 10, 19)( 11, 20)( 12, 21)( 13, 22)( 14, 23)( 15, 24)( 16, 25)
( 17, 26)( 18, 27)( 37, 46)( 38, 47)( 39, 48)( 40, 49)( 41, 50)( 42, 51)
( 43, 52)( 44, 53)( 45, 54)( 55, 82)( 56, 83)( 57, 84)( 58, 85)( 59, 86)
( 60, 87)( 61, 88)( 62, 89)( 63, 90)( 64,100)( 65,101)( 66,102)( 67,103)
( 68,104)( 69,105)( 70,106)( 71,107)( 72,108)( 73, 91)( 74, 92)( 75, 93)
( 76, 94)( 77, 95)( 78, 96)( 79, 97)( 80, 98)( 81, 99)(118,127)(119,128)
(120,129)(121,130)(122,131)(123,132)(124,133)(125,134)(126,135)(145,154)
(146,155)(147,156)(148,157)(149,158)(150,159)(151,160)(152,161)(153,162)
(163,190)(164,191)(165,192)(166,193)(167,194)(168,195)(169,196)(170,197)
(171,198)(172,208)(173,209)(174,210)(175,211)(176,212)(177,213)(178,214)
(179,215)(180,216)(181,199)(182,200)(183,201)(184,202)(185,203)(186,204)
(187,205)(188,206)(189,207);
s2 := Sym(216)!(  1, 10)(  2, 12)(  3, 11)(  4, 16)(  5, 18)(  6, 17)(  7, 13)
(  8, 15)(  9, 14)( 20, 21)( 22, 25)( 23, 27)( 24, 26)( 28, 37)( 29, 39)
( 30, 38)( 31, 43)( 32, 45)( 33, 44)( 34, 40)( 35, 42)( 36, 41)( 47, 48)
( 49, 52)( 50, 54)( 51, 53)( 55, 64)( 56, 66)( 57, 65)( 58, 70)( 59, 72)
( 60, 71)( 61, 67)( 62, 69)( 63, 68)( 74, 75)( 76, 79)( 77, 81)( 78, 80)
( 82, 91)( 83, 93)( 84, 92)( 85, 97)( 86, 99)( 87, 98)( 88, 94)( 89, 96)
( 90, 95)(101,102)(103,106)(104,108)(105,107)(109,118)(110,120)(111,119)
(112,124)(113,126)(114,125)(115,121)(116,123)(117,122)(128,129)(130,133)
(131,135)(132,134)(136,145)(137,147)(138,146)(139,151)(140,153)(141,152)
(142,148)(143,150)(144,149)(155,156)(157,160)(158,162)(159,161)(163,172)
(164,174)(165,173)(166,178)(167,180)(168,179)(169,175)(170,177)(171,176)
(182,183)(184,187)(185,189)(186,188)(190,199)(191,201)(192,200)(193,205)
(194,207)(195,206)(196,202)(197,204)(198,203)(209,210)(211,214)(212,216)
(213,215);
s3 := Sym(216)!(  1,113)(  2,112)(  3,114)(  4,110)(  5,109)(  6,111)(  7,116)
(  8,115)(  9,117)( 10,131)( 11,130)( 12,132)( 13,128)( 14,127)( 15,129)
( 16,134)( 17,133)( 18,135)( 19,122)( 20,121)( 21,123)( 22,119)( 23,118)
( 24,120)( 25,125)( 26,124)( 27,126)( 28,140)( 29,139)( 30,141)( 31,137)
( 32,136)( 33,138)( 34,143)( 35,142)( 36,144)( 37,158)( 38,157)( 39,159)
( 40,155)( 41,154)( 42,156)( 43,161)( 44,160)( 45,162)( 46,149)( 47,148)
( 48,150)( 49,146)( 50,145)( 51,147)( 52,152)( 53,151)( 54,153)( 55,167)
( 56,166)( 57,168)( 58,164)( 59,163)( 60,165)( 61,170)( 62,169)( 63,171)
( 64,185)( 65,184)( 66,186)( 67,182)( 68,181)( 69,183)( 70,188)( 71,187)
( 72,189)( 73,176)( 74,175)( 75,177)( 76,173)( 77,172)( 78,174)( 79,179)
( 80,178)( 81,180)( 82,194)( 83,193)( 84,195)( 85,191)( 86,190)( 87,192)
( 88,197)( 89,196)( 90,198)( 91,212)( 92,211)( 93,213)( 94,209)( 95,208)
( 96,210)( 97,215)( 98,214)( 99,216)(100,203)(101,202)(102,204)(103,200)
(104,199)(105,201)(106,206)(107,205)(108,207);
s4 := Sym(216)!(  4,  7)(  5,  8)(  6,  9)( 13, 16)( 14, 17)( 15, 18)( 22, 25)
( 23, 26)( 24, 27)( 31, 34)( 32, 35)( 33, 36)( 40, 43)( 41, 44)( 42, 45)
( 49, 52)( 50, 53)( 51, 54)( 58, 61)( 59, 62)( 60, 63)( 67, 70)( 68, 71)
( 69, 72)( 76, 79)( 77, 80)( 78, 81)( 85, 88)( 86, 89)( 87, 90)( 94, 97)
( 95, 98)( 96, 99)(103,106)(104,107)(105,108)(112,115)(113,116)(114,117)
(121,124)(122,125)(123,126)(130,133)(131,134)(132,135)(139,142)(140,143)
(141,144)(148,151)(149,152)(150,153)(157,160)(158,161)(159,162)(166,169)
(167,170)(168,171)(175,178)(176,179)(177,180)(184,187)(185,188)(186,189)
(193,196)(194,197)(195,198)(202,205)(203,206)(204,207)(211,214)(212,215)
(213,216);
poly := sub<Sym(216)|s0,s1,s2,s3,s4>;
 
Finitely Presented Group Representation (Magma) :
poly<s0,s1,s2,s3,s4> := Group< s0,s1,s2,s3,s4 | s0*s0, s1*s1, s2*s2, 
s3*s3, s4*s4, s0*s2*s0*s2, s0*s3*s0*s3, 
s1*s3*s1*s3, s0*s4*s0*s4, s1*s4*s1*s4, 
s2*s4*s2*s4, s0*s1*s0*s1*s0*s1*s0*s1, 
s0*s1*s2*s1*s0*s1*s2*s1, s3*s1*s2*s1*s2*s3*s1*s2*s1*s2, 
s4*s2*s3*s4*s3*s4*s2*s3*s4*s3, 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*s4*s3*s2*s3*s2*s3*s4*s3*s2*s3 >; 
 
References : None.
to this polytope