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