Polytope of Type {3,6,6,6}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {3,6,6,6}*1728a
if this polytope has a name.
Group : SmallGroup(1728,47874)
Rank : 5
Schlafli Type : {3,6,6,6}
Number of vertices, edges, etc : 3, 9, 24, 24, 8
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 : {3,6,6,3}*864a
3-fold quotients : {3,2,6,6}*576
6-fold quotients : {3,2,3,6}*288, {3,2,6,3}*288
12-fold quotients : {3,2,3,3}*144, {3,6,2,2}*144
36-fold quotients : {3,2,2,2}*48
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := ( 5, 9)( 6, 10)( 7, 11)( 8, 12)( 13, 25)( 14, 26)( 15, 27)( 16, 28)
( 17, 33)( 18, 34)( 19, 35)( 20, 36)( 21, 29)( 22, 30)( 23, 31)( 24, 32)
( 41, 45)( 42, 46)( 43, 47)( 44, 48)( 49, 61)( 50, 62)( 51, 63)( 52, 64)
( 53, 69)( 54, 70)( 55, 71)( 56, 72)( 57, 65)( 58, 66)( 59, 67)( 60, 68)
( 77, 81)( 78, 82)( 79, 83)( 80, 84)( 85, 97)( 86, 98)( 87, 99)( 88,100)
( 89,105)( 90,106)( 91,107)( 92,108)( 93,101)( 94,102)( 95,103)( 96,104)
(113,117)(114,118)(115,119)(116,120)(121,133)(122,134)(123,135)(124,136)
(125,141)(126,142)(127,143)(128,144)(129,137)(130,138)(131,139)(132,140)
(149,153)(150,154)(151,155)(152,156)(157,169)(158,170)(159,171)(160,172)
(161,177)(162,178)(163,179)(164,180)(165,173)(166,174)(167,175)(168,176)
(185,189)(186,190)(187,191)(188,192)(193,205)(194,206)(195,207)(196,208)
(197,213)(198,214)(199,215)(200,216)(201,209)(202,210)(203,211)(204,212);;
s1 := ( 1, 17)( 2, 18)( 3, 19)( 4, 20)( 5, 13)( 6, 14)( 7, 15)( 8, 16)
( 9, 21)( 10, 22)( 11, 23)( 12, 24)( 25, 29)( 26, 30)( 27, 31)( 28, 32)
( 37, 53)( 38, 54)( 39, 55)( 40, 56)( 41, 49)( 42, 50)( 43, 51)( 44, 52)
( 45, 57)( 46, 58)( 47, 59)( 48, 60)( 61, 65)( 62, 66)( 63, 67)( 64, 68)
( 73, 89)( 74, 90)( 75, 91)( 76, 92)( 77, 85)( 78, 86)( 79, 87)( 80, 88)
( 81, 93)( 82, 94)( 83, 95)( 84, 96)( 97,101)( 98,102)( 99,103)(100,104)
(109,125)(110,126)(111,127)(112,128)(113,121)(114,122)(115,123)(116,124)
(117,129)(118,130)(119,131)(120,132)(133,137)(134,138)(135,139)(136,140)
(145,161)(146,162)(147,163)(148,164)(149,157)(150,158)(151,159)(152,160)
(153,165)(154,166)(155,167)(156,168)(169,173)(170,174)(171,175)(172,176)
(181,197)(182,198)(183,199)(184,200)(185,193)(186,194)(187,195)(188,196)
(189,201)(190,202)(191,203)(192,204)(205,209)(206,210)(207,211)(208,212);;
s2 := ( 3, 4)( 7, 8)( 11, 12)( 13, 25)( 14, 26)( 15, 28)( 16, 27)( 17, 29)
( 18, 30)( 19, 32)( 20, 31)( 21, 33)( 22, 34)( 23, 36)( 24, 35)( 37, 73)
( 38, 74)( 39, 76)( 40, 75)( 41, 77)( 42, 78)( 43, 80)( 44, 79)( 45, 81)
( 46, 82)( 47, 84)( 48, 83)( 49, 97)( 50, 98)( 51,100)( 52, 99)( 53,101)
( 54,102)( 55,104)( 56,103)( 57,105)( 58,106)( 59,108)( 60,107)( 61, 85)
( 62, 86)( 63, 88)( 64, 87)( 65, 89)( 66, 90)( 67, 92)( 68, 91)( 69, 93)
( 70, 94)( 71, 96)( 72, 95)(111,112)(115,116)(119,120)(121,133)(122,134)
(123,136)(124,135)(125,137)(126,138)(127,140)(128,139)(129,141)(130,142)
(131,144)(132,143)(145,181)(146,182)(147,184)(148,183)(149,185)(150,186)
(151,188)(152,187)(153,189)(154,190)(155,192)(156,191)(157,205)(158,206)
(159,208)(160,207)(161,209)(162,210)(163,212)(164,211)(165,213)(166,214)
(167,216)(168,215)(169,193)(170,194)(171,196)(172,195)(173,197)(174,198)
(175,200)(176,199)(177,201)(178,202)(179,204)(180,203);;
s3 := ( 1, 37)( 2, 40)( 3, 39)( 4, 38)( 5, 41)( 6, 44)( 7, 43)( 8, 42)
( 9, 45)( 10, 48)( 11, 47)( 12, 46)( 13, 49)( 14, 52)( 15, 51)( 16, 50)
( 17, 53)( 18, 56)( 19, 55)( 20, 54)( 21, 57)( 22, 60)( 23, 59)( 24, 58)
( 25, 61)( 26, 64)( 27, 63)( 28, 62)( 29, 65)( 30, 68)( 31, 67)( 32, 66)
( 33, 69)( 34, 72)( 35, 71)( 36, 70)( 74, 76)( 78, 80)( 82, 84)( 86, 88)
( 90, 92)( 94, 96)( 98,100)(102,104)(106,108)(109,145)(110,148)(111,147)
(112,146)(113,149)(114,152)(115,151)(116,150)(117,153)(118,156)(119,155)
(120,154)(121,157)(122,160)(123,159)(124,158)(125,161)(126,164)(127,163)
(128,162)(129,165)(130,168)(131,167)(132,166)(133,169)(134,172)(135,171)
(136,170)(137,173)(138,176)(139,175)(140,174)(141,177)(142,180)(143,179)
(144,178)(182,184)(186,188)(190,192)(194,196)(198,200)(202,204)(206,208)
(210,212)(214,216);;
s4 := ( 1,110)( 2,109)( 3,111)( 4,112)( 5,114)( 6,113)( 7,115)( 8,116)
( 9,118)( 10,117)( 11,119)( 12,120)( 13,122)( 14,121)( 15,123)( 16,124)
( 17,126)( 18,125)( 19,127)( 20,128)( 21,130)( 22,129)( 23,131)( 24,132)
( 25,134)( 26,133)( 27,135)( 28,136)( 29,138)( 30,137)( 31,139)( 32,140)
( 33,142)( 34,141)( 35,143)( 36,144)( 37,182)( 38,181)( 39,183)( 40,184)
( 41,186)( 42,185)( 43,187)( 44,188)( 45,190)( 46,189)( 47,191)( 48,192)
( 49,194)( 50,193)( 51,195)( 52,196)( 53,198)( 54,197)( 55,199)( 56,200)
( 57,202)( 58,201)( 59,203)( 60,204)( 61,206)( 62,205)( 63,207)( 64,208)
( 65,210)( 66,209)( 67,211)( 68,212)( 69,214)( 70,213)( 71,215)( 72,216)
( 73,146)( 74,145)( 75,147)( 76,148)( 77,150)( 78,149)( 79,151)( 80,152)
( 81,154)( 82,153)( 83,155)( 84,156)( 85,158)( 86,157)( 87,159)( 88,160)
( 89,162)( 90,161)( 91,163)( 92,164)( 93,166)( 94,165)( 95,167)( 96,168)
( 97,170)( 98,169)( 99,171)(100,172)(101,174)(102,173)(103,175)(104,176)
(105,178)(106,177)(107,179)(108,180);;
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,
s1*s2*s3*s2*s1*s2*s3*s2, s2*s0*s1*s2*s1*s2*s0*s1*s2*s1,
s4*s2*s3*s4*s2*s3*s4*s2*s3*s4*s2*s3,
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)!( 5, 9)( 6, 10)( 7, 11)( 8, 12)( 13, 25)( 14, 26)( 15, 27)
( 16, 28)( 17, 33)( 18, 34)( 19, 35)( 20, 36)( 21, 29)( 22, 30)( 23, 31)
( 24, 32)( 41, 45)( 42, 46)( 43, 47)( 44, 48)( 49, 61)( 50, 62)( 51, 63)
( 52, 64)( 53, 69)( 54, 70)( 55, 71)( 56, 72)( 57, 65)( 58, 66)( 59, 67)
( 60, 68)( 77, 81)( 78, 82)( 79, 83)( 80, 84)( 85, 97)( 86, 98)( 87, 99)
( 88,100)( 89,105)( 90,106)( 91,107)( 92,108)( 93,101)( 94,102)( 95,103)
( 96,104)(113,117)(114,118)(115,119)(116,120)(121,133)(122,134)(123,135)
(124,136)(125,141)(126,142)(127,143)(128,144)(129,137)(130,138)(131,139)
(132,140)(149,153)(150,154)(151,155)(152,156)(157,169)(158,170)(159,171)
(160,172)(161,177)(162,178)(163,179)(164,180)(165,173)(166,174)(167,175)
(168,176)(185,189)(186,190)(187,191)(188,192)(193,205)(194,206)(195,207)
(196,208)(197,213)(198,214)(199,215)(200,216)(201,209)(202,210)(203,211)
(204,212);
s1 := Sym(216)!( 1, 17)( 2, 18)( 3, 19)( 4, 20)( 5, 13)( 6, 14)( 7, 15)
( 8, 16)( 9, 21)( 10, 22)( 11, 23)( 12, 24)( 25, 29)( 26, 30)( 27, 31)
( 28, 32)( 37, 53)( 38, 54)( 39, 55)( 40, 56)( 41, 49)( 42, 50)( 43, 51)
( 44, 52)( 45, 57)( 46, 58)( 47, 59)( 48, 60)( 61, 65)( 62, 66)( 63, 67)
( 64, 68)( 73, 89)( 74, 90)( 75, 91)( 76, 92)( 77, 85)( 78, 86)( 79, 87)
( 80, 88)( 81, 93)( 82, 94)( 83, 95)( 84, 96)( 97,101)( 98,102)( 99,103)
(100,104)(109,125)(110,126)(111,127)(112,128)(113,121)(114,122)(115,123)
(116,124)(117,129)(118,130)(119,131)(120,132)(133,137)(134,138)(135,139)
(136,140)(145,161)(146,162)(147,163)(148,164)(149,157)(150,158)(151,159)
(152,160)(153,165)(154,166)(155,167)(156,168)(169,173)(170,174)(171,175)
(172,176)(181,197)(182,198)(183,199)(184,200)(185,193)(186,194)(187,195)
(188,196)(189,201)(190,202)(191,203)(192,204)(205,209)(206,210)(207,211)
(208,212);
s2 := Sym(216)!( 3, 4)( 7, 8)( 11, 12)( 13, 25)( 14, 26)( 15, 28)( 16, 27)
( 17, 29)( 18, 30)( 19, 32)( 20, 31)( 21, 33)( 22, 34)( 23, 36)( 24, 35)
( 37, 73)( 38, 74)( 39, 76)( 40, 75)( 41, 77)( 42, 78)( 43, 80)( 44, 79)
( 45, 81)( 46, 82)( 47, 84)( 48, 83)( 49, 97)( 50, 98)( 51,100)( 52, 99)
( 53,101)( 54,102)( 55,104)( 56,103)( 57,105)( 58,106)( 59,108)( 60,107)
( 61, 85)( 62, 86)( 63, 88)( 64, 87)( 65, 89)( 66, 90)( 67, 92)( 68, 91)
( 69, 93)( 70, 94)( 71, 96)( 72, 95)(111,112)(115,116)(119,120)(121,133)
(122,134)(123,136)(124,135)(125,137)(126,138)(127,140)(128,139)(129,141)
(130,142)(131,144)(132,143)(145,181)(146,182)(147,184)(148,183)(149,185)
(150,186)(151,188)(152,187)(153,189)(154,190)(155,192)(156,191)(157,205)
(158,206)(159,208)(160,207)(161,209)(162,210)(163,212)(164,211)(165,213)
(166,214)(167,216)(168,215)(169,193)(170,194)(171,196)(172,195)(173,197)
(174,198)(175,200)(176,199)(177,201)(178,202)(179,204)(180,203);
s3 := Sym(216)!( 1, 37)( 2, 40)( 3, 39)( 4, 38)( 5, 41)( 6, 44)( 7, 43)
( 8, 42)( 9, 45)( 10, 48)( 11, 47)( 12, 46)( 13, 49)( 14, 52)( 15, 51)
( 16, 50)( 17, 53)( 18, 56)( 19, 55)( 20, 54)( 21, 57)( 22, 60)( 23, 59)
( 24, 58)( 25, 61)( 26, 64)( 27, 63)( 28, 62)( 29, 65)( 30, 68)( 31, 67)
( 32, 66)( 33, 69)( 34, 72)( 35, 71)( 36, 70)( 74, 76)( 78, 80)( 82, 84)
( 86, 88)( 90, 92)( 94, 96)( 98,100)(102,104)(106,108)(109,145)(110,148)
(111,147)(112,146)(113,149)(114,152)(115,151)(116,150)(117,153)(118,156)
(119,155)(120,154)(121,157)(122,160)(123,159)(124,158)(125,161)(126,164)
(127,163)(128,162)(129,165)(130,168)(131,167)(132,166)(133,169)(134,172)
(135,171)(136,170)(137,173)(138,176)(139,175)(140,174)(141,177)(142,180)
(143,179)(144,178)(182,184)(186,188)(190,192)(194,196)(198,200)(202,204)
(206,208)(210,212)(214,216);
s4 := Sym(216)!( 1,110)( 2,109)( 3,111)( 4,112)( 5,114)( 6,113)( 7,115)
( 8,116)( 9,118)( 10,117)( 11,119)( 12,120)( 13,122)( 14,121)( 15,123)
( 16,124)( 17,126)( 18,125)( 19,127)( 20,128)( 21,130)( 22,129)( 23,131)
( 24,132)( 25,134)( 26,133)( 27,135)( 28,136)( 29,138)( 30,137)( 31,139)
( 32,140)( 33,142)( 34,141)( 35,143)( 36,144)( 37,182)( 38,181)( 39,183)
( 40,184)( 41,186)( 42,185)( 43,187)( 44,188)( 45,190)( 46,189)( 47,191)
( 48,192)( 49,194)( 50,193)( 51,195)( 52,196)( 53,198)( 54,197)( 55,199)
( 56,200)( 57,202)( 58,201)( 59,203)( 60,204)( 61,206)( 62,205)( 63,207)
( 64,208)( 65,210)( 66,209)( 67,211)( 68,212)( 69,214)( 70,213)( 71,215)
( 72,216)( 73,146)( 74,145)( 75,147)( 76,148)( 77,150)( 78,149)( 79,151)
( 80,152)( 81,154)( 82,153)( 83,155)( 84,156)( 85,158)( 86,157)( 87,159)
( 88,160)( 89,162)( 90,161)( 91,163)( 92,164)( 93,166)( 94,165)( 95,167)
( 96,168)( 97,170)( 98,169)( 99,171)(100,172)(101,174)(102,173)(103,175)
(104,176)(105,178)(106,177)(107,179)(108,180);
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, s1*s2*s3*s2*s1*s2*s3*s2,
s2*s0*s1*s2*s1*s2*s0*s1*s2*s1, s4*s2*s3*s4*s2*s3*s4*s2*s3*s4*s2*s3,
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