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}*1728i
if this polytope has a name.
Group : SmallGroup(1728,47410)
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 : {4,6,6,3}*864d, {2,6,6,6}*864c
3-fold quotients : {4,2,6,6}*576b, {4,6,2,6}*576a
4-fold quotients : {2,6,6,3}*432b
6-fold quotients : {4,2,6,3}*288, {4,6,2,3}*288a, {2,2,6,6}*288b, {2,6,2,6}*288
9-fold quotients : {4,2,2,6}*192, {4,6,2,2}*192a
12-fold quotients : {2,2,6,3}*144, {2,3,2,6}*144, {2,6,2,3}*144
18-fold quotients : {4,2,2,3}*96, {2,2,2,6}*96, {2,6,2,2}*96
24-fold quotients : {2,3,2,3}*72
27-fold quotients : {4,2,2,2}*64
36-fold quotients : {2,2,2,3}*48, {2,3,2,2}*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 := ( 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)( 55, 82)( 56, 83)( 57, 84)( 58, 88)( 59, 89)( 60, 90)
( 61, 85)( 62, 86)( 63, 87)( 64, 91)( 65, 92)( 66, 93)( 67, 97)( 68, 98)
( 69, 99)( 70, 94)( 71, 95)( 72, 96)( 73,100)( 74,101)( 75,102)( 76,106)
( 77,107)( 78,108)( 79,103)( 80,104)( 81,105)(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)(163,190)
(164,191)(165,192)(166,196)(167,197)(168,198)(169,193)(170,194)(171,195)
(172,199)(173,200)(174,201)(175,205)(176,206)(177,207)(178,202)(179,203)
(180,204)(181,208)(182,209)(183,210)(184,214)(185,215)(186,216)(187,211)
(188,212)(189,213);;
s2 := ( 1, 4)( 2, 5)( 3, 6)( 10, 22)( 11, 23)( 12, 24)( 13, 19)( 14, 20)
( 15, 21)( 16, 25)( 17, 26)( 18, 27)( 28, 31)( 29, 32)( 30, 33)( 37, 49)
( 38, 50)( 39, 51)( 40, 46)( 41, 47)( 42, 48)( 43, 52)( 44, 53)( 45, 54)
( 55, 58)( 56, 59)( 57, 60)( 64, 76)( 65, 77)( 66, 78)( 67, 73)( 68, 74)
( 69, 75)( 70, 79)( 71, 80)( 72, 81)( 82, 85)( 83, 86)( 84, 87)( 91,103)
( 92,104)( 93,105)( 94,100)( 95,101)( 96,102)( 97,106)( 98,107)( 99,108)
(109,112)(110,113)(111,114)(118,130)(119,131)(120,132)(121,127)(122,128)
(123,129)(124,133)(125,134)(126,135)(136,139)(137,140)(138,141)(145,157)
(146,158)(147,159)(148,154)(149,155)(150,156)(151,160)(152,161)(153,162)
(163,166)(164,167)(165,168)(172,184)(173,185)(174,186)(175,181)(176,182)
(177,183)(178,187)(179,188)(180,189)(190,193)(191,194)(192,195)(199,211)
(200,212)(201,213)(202,208)(203,209)(204,210)(205,214)(206,215)(207,216);;
s3 := ( 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);;
s4 := ( 1,110)( 2,109)( 3,111)( 4,113)( 5,112)( 6,114)( 7,116)( 8,115)
( 9,117)( 10,128)( 11,127)( 12,129)( 13,131)( 14,130)( 15,132)( 16,134)
( 17,133)( 18,135)( 19,119)( 20,118)( 21,120)( 22,122)( 23,121)( 24,123)
( 25,125)( 26,124)( 27,126)( 28,137)( 29,136)( 30,138)( 31,140)( 32,139)
( 33,141)( 34,143)( 35,142)( 36,144)( 37,155)( 38,154)( 39,156)( 40,158)
( 41,157)( 42,159)( 43,161)( 44,160)( 45,162)( 46,146)( 47,145)( 48,147)
( 49,149)( 50,148)( 51,150)( 52,152)( 53,151)( 54,153)( 55,164)( 56,163)
( 57,165)( 58,167)( 59,166)( 60,168)( 61,170)( 62,169)( 63,171)( 64,182)
( 65,181)( 66,183)( 67,185)( 68,184)( 69,186)( 70,188)( 71,187)( 72,189)
( 73,173)( 74,172)( 75,174)( 76,176)( 77,175)( 78,177)( 79,179)( 80,178)
( 81,180)( 82,191)( 83,190)( 84,192)( 85,194)( 86,193)( 87,195)( 88,197)
( 89,196)( 90,198)( 91,209)( 92,208)( 93,210)( 94,212)( 95,211)( 96,213)
( 97,215)( 98,214)( 99,216)(100,200)(101,199)(102,201)(103,203)(104,202)
(105,204)(106,206)(107,205)(108,207);;
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, s1*s2*s3*s2*s1*s2*s3*s2,
s4*s2*s3*s2*s3*s4*s2*s3*s2*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,
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)!( 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)( 55, 82)( 56, 83)( 57, 84)( 58, 88)( 59, 89)
( 60, 90)( 61, 85)( 62, 86)( 63, 87)( 64, 91)( 65, 92)( 66, 93)( 67, 97)
( 68, 98)( 69, 99)( 70, 94)( 71, 95)( 72, 96)( 73,100)( 74,101)( 75,102)
( 76,106)( 77,107)( 78,108)( 79,103)( 80,104)( 81,105)(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)
(163,190)(164,191)(165,192)(166,196)(167,197)(168,198)(169,193)(170,194)
(171,195)(172,199)(173,200)(174,201)(175,205)(176,206)(177,207)(178,202)
(179,203)(180,204)(181,208)(182,209)(183,210)(184,214)(185,215)(186,216)
(187,211)(188,212)(189,213);
s2 := Sym(216)!( 1, 4)( 2, 5)( 3, 6)( 10, 22)( 11, 23)( 12, 24)( 13, 19)
( 14, 20)( 15, 21)( 16, 25)( 17, 26)( 18, 27)( 28, 31)( 29, 32)( 30, 33)
( 37, 49)( 38, 50)( 39, 51)( 40, 46)( 41, 47)( 42, 48)( 43, 52)( 44, 53)
( 45, 54)( 55, 58)( 56, 59)( 57, 60)( 64, 76)( 65, 77)( 66, 78)( 67, 73)
( 68, 74)( 69, 75)( 70, 79)( 71, 80)( 72, 81)( 82, 85)( 83, 86)( 84, 87)
( 91,103)( 92,104)( 93,105)( 94,100)( 95,101)( 96,102)( 97,106)( 98,107)
( 99,108)(109,112)(110,113)(111,114)(118,130)(119,131)(120,132)(121,127)
(122,128)(123,129)(124,133)(125,134)(126,135)(136,139)(137,140)(138,141)
(145,157)(146,158)(147,159)(148,154)(149,155)(150,156)(151,160)(152,161)
(153,162)(163,166)(164,167)(165,168)(172,184)(173,185)(174,186)(175,181)
(176,182)(177,183)(178,187)(179,188)(180,189)(190,193)(191,194)(192,195)
(199,211)(200,212)(201,213)(202,208)(203,209)(204,210)(205,214)(206,215)
(207,216);
s3 := 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);
s4 := Sym(216)!( 1,110)( 2,109)( 3,111)( 4,113)( 5,112)( 6,114)( 7,116)
( 8,115)( 9,117)( 10,128)( 11,127)( 12,129)( 13,131)( 14,130)( 15,132)
( 16,134)( 17,133)( 18,135)( 19,119)( 20,118)( 21,120)( 22,122)( 23,121)
( 24,123)( 25,125)( 26,124)( 27,126)( 28,137)( 29,136)( 30,138)( 31,140)
( 32,139)( 33,141)( 34,143)( 35,142)( 36,144)( 37,155)( 38,154)( 39,156)
( 40,158)( 41,157)( 42,159)( 43,161)( 44,160)( 45,162)( 46,146)( 47,145)
( 48,147)( 49,149)( 50,148)( 51,150)( 52,152)( 53,151)( 54,153)( 55,164)
( 56,163)( 57,165)( 58,167)( 59,166)( 60,168)( 61,170)( 62,169)( 63,171)
( 64,182)( 65,181)( 66,183)( 67,185)( 68,184)( 69,186)( 70,188)( 71,187)
( 72,189)( 73,173)( 74,172)( 75,174)( 76,176)( 77,175)( 78,177)( 79,179)
( 80,178)( 81,180)( 82,191)( 83,190)( 84,192)( 85,194)( 86,193)( 87,195)
( 88,197)( 89,196)( 90,198)( 91,209)( 92,208)( 93,210)( 94,212)( 95,211)
( 96,213)( 97,215)( 98,214)( 99,216)(100,200)(101,199)(102,201)(103,203)
(104,202)(105,204)(106,206)(107,205)(108,207);
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, s1*s2*s3*s2*s1*s2*s3*s2,
s4*s2*s3*s2*s3*s4*s2*s3*s2*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,
s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4 >;
References : None.
to this polytope