Polytope of Type {6,4,3,6}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,4,3,6}*1728
if this polytope has a name.
Group : SmallGroup(1728,47874)
Rank : 5
Schlafli Type : {6,4,3,6}
Number of vertices, edges, etc : 6, 24, 12, 18, 6
Order of s0s1s2s3s4 : 6
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) :
3-fold quotients : {2,4,3,6}*576, {6,4,3,2}*576
4-fold quotients : {6,2,3,6}*432
6-fold quotients : {2,4,3,6}*288
8-fold quotients : {3,2,3,6}*216
9-fold quotients : {2,4,3,2}*192
12-fold quotients : {2,2,3,6}*144, {6,2,3,2}*144
18-fold quotients : {2,4,3,2}*96
24-fold quotients : {3,2,3,2}*72
36-fold quotients : {2,2,3,2}*48
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := ( 13, 25)( 14, 26)( 15, 27)( 16, 28)( 17, 29)( 18, 30)( 19, 31)( 20, 32)
( 21, 33)( 22, 34)( 23, 35)( 24, 36)( 49, 61)( 50, 62)( 51, 63)( 52, 64)
( 53, 65)( 54, 66)( 55, 67)( 56, 68)( 57, 69)( 58, 70)( 59, 71)( 60, 72)
( 85, 97)( 86, 98)( 87, 99)( 88,100)( 89,101)( 90,102)( 91,103)( 92,104)
( 93,105)( 94,106)( 95,107)( 96,108)(121,133)(122,134)(123,135)(124,136)
(125,137)(126,138)(127,139)(128,140)(129,141)(130,142)(131,143)(132,144)
(157,169)(158,170)(159,171)(160,172)(161,173)(162,174)(163,175)(164,176)
(165,177)(166,178)(167,179)(168,180)(193,205)(194,206)(195,207)(196,208)
(197,209)(198,210)(199,211)(200,212)(201,213)(202,214)(203,215)(204,216);;
s1 := ( 1,123)( 2,124)( 3,121)( 4,122)( 5,127)( 6,128)( 7,125)( 8,126)
( 9,131)( 10,132)( 11,129)( 12,130)( 13,111)( 14,112)( 15,109)( 16,110)
( 17,115)( 18,116)( 19,113)( 20,114)( 21,119)( 22,120)( 23,117)( 24,118)
( 25,135)( 26,136)( 27,133)( 28,134)( 29,139)( 30,140)( 31,137)( 32,138)
( 33,143)( 34,144)( 35,141)( 36,142)( 37,159)( 38,160)( 39,157)( 40,158)
( 41,163)( 42,164)( 43,161)( 44,162)( 45,167)( 46,168)( 47,165)( 48,166)
( 49,147)( 50,148)( 51,145)( 52,146)( 53,151)( 54,152)( 55,149)( 56,150)
( 57,155)( 58,156)( 59,153)( 60,154)( 61,171)( 62,172)( 63,169)( 64,170)
( 65,175)( 66,176)( 67,173)( 68,174)( 69,179)( 70,180)( 71,177)( 72,178)
( 73,195)( 74,196)( 75,193)( 76,194)( 77,199)( 78,200)( 79,197)( 80,198)
( 81,203)( 82,204)( 83,201)( 84,202)( 85,183)( 86,184)( 87,181)( 88,182)
( 89,187)( 90,188)( 91,185)( 92,186)( 93,191)( 94,192)( 95,189)( 96,190)
( 97,207)( 98,208)( 99,205)(100,206)(101,211)(102,212)(103,209)(104,210)
(105,215)(106,216)(107,213)(108,214);;
s2 := ( 3, 4)( 5, 9)( 6, 10)( 7, 12)( 8, 11)( 15, 16)( 17, 21)( 18, 22)
( 19, 24)( 20, 23)( 27, 28)( 29, 33)( 30, 34)( 31, 36)( 32, 35)( 37, 73)
( 38, 74)( 39, 76)( 40, 75)( 41, 81)( 42, 82)( 43, 84)( 44, 83)( 45, 77)
( 46, 78)( 47, 80)( 48, 79)( 49, 85)( 50, 86)( 51, 88)( 52, 87)( 53, 93)
( 54, 94)( 55, 96)( 56, 95)( 57, 89)( 58, 90)( 59, 92)( 60, 91)( 61, 97)
( 62, 98)( 63,100)( 64, 99)( 65,105)( 66,106)( 67,108)( 68,107)( 69,101)
( 70,102)( 71,104)( 72,103)(111,112)(113,117)(114,118)(115,120)(116,119)
(123,124)(125,129)(126,130)(127,132)(128,131)(135,136)(137,141)(138,142)
(139,144)(140,143)(145,181)(146,182)(147,184)(148,183)(149,189)(150,190)
(151,192)(152,191)(153,185)(154,186)(155,188)(156,187)(157,193)(158,194)
(159,196)(160,195)(161,201)(162,202)(163,204)(164,203)(165,197)(166,198)
(167,200)(168,199)(169,205)(170,206)(171,208)(172,207)(173,213)(174,214)
(175,216)(176,215)(177,209)(178,210)(179,212)(180,211);;
s3 := ( 1, 41)( 2, 44)( 3, 43)( 4, 42)( 5, 37)( 6, 40)( 7, 39)( 8, 38)
( 9, 45)( 10, 48)( 11, 47)( 12, 46)( 13, 53)( 14, 56)( 15, 55)( 16, 54)
( 17, 49)( 18, 52)( 19, 51)( 20, 50)( 21, 57)( 22, 60)( 23, 59)( 24, 58)
( 25, 65)( 26, 68)( 27, 67)( 28, 66)( 29, 61)( 30, 64)( 31, 63)( 32, 62)
( 33, 69)( 34, 72)( 35, 71)( 36, 70)( 73, 77)( 74, 80)( 75, 79)( 76, 78)
( 82, 84)( 85, 89)( 86, 92)( 87, 91)( 88, 90)( 94, 96)( 97,101)( 98,104)
( 99,103)(100,102)(106,108)(109,149)(110,152)(111,151)(112,150)(113,145)
(114,148)(115,147)(116,146)(117,153)(118,156)(119,155)(120,154)(121,161)
(122,164)(123,163)(124,162)(125,157)(126,160)(127,159)(128,158)(129,165)
(130,168)(131,167)(132,166)(133,173)(134,176)(135,175)(136,174)(137,169)
(138,172)(139,171)(140,170)(141,177)(142,180)(143,179)(144,178)(181,185)
(182,188)(183,187)(184,186)(190,192)(193,197)(194,200)(195,199)(196,198)
(202,204)(205,209)(206,212)(207,211)(208,210)(214,216);;
s4 := ( 5, 9)( 6, 10)( 7, 11)( 8, 12)( 17, 21)( 18, 22)( 19, 23)( 20, 24)
( 29, 33)( 30, 34)( 31, 35)( 32, 36)( 41, 45)( 42, 46)( 43, 47)( 44, 48)
( 53, 57)( 54, 58)( 55, 59)( 56, 60)( 65, 69)( 66, 70)( 67, 71)( 68, 72)
( 77, 81)( 78, 82)( 79, 83)( 80, 84)( 89, 93)( 90, 94)( 91, 95)( 92, 96)
(101,105)(102,106)(103,107)(104,108)(113,117)(114,118)(115,119)(116,120)
(125,129)(126,130)(127,131)(128,132)(137,141)(138,142)(139,143)(140,144)
(149,153)(150,154)(151,155)(152,156)(161,165)(162,166)(163,167)(164,168)
(173,177)(174,178)(175,179)(176,180)(185,189)(186,190)(187,191)(188,192)
(197,201)(198,202)(199,203)(200,204)(209,213)(210,214)(211,215)(212,216);;
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, s2*s3*s2*s3*s2*s3,
s0*s1*s2*s1*s0*s1*s2*s1, s1*s2*s1*s2*s1*s2*s1*s2,
s4*s2*s3*s4*s3*s4*s2*s3*s4*s3, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(216)!( 13, 25)( 14, 26)( 15, 27)( 16, 28)( 17, 29)( 18, 30)( 19, 31)
( 20, 32)( 21, 33)( 22, 34)( 23, 35)( 24, 36)( 49, 61)( 50, 62)( 51, 63)
( 52, 64)( 53, 65)( 54, 66)( 55, 67)( 56, 68)( 57, 69)( 58, 70)( 59, 71)
( 60, 72)( 85, 97)( 86, 98)( 87, 99)( 88,100)( 89,101)( 90,102)( 91,103)
( 92,104)( 93,105)( 94,106)( 95,107)( 96,108)(121,133)(122,134)(123,135)
(124,136)(125,137)(126,138)(127,139)(128,140)(129,141)(130,142)(131,143)
(132,144)(157,169)(158,170)(159,171)(160,172)(161,173)(162,174)(163,175)
(164,176)(165,177)(166,178)(167,179)(168,180)(193,205)(194,206)(195,207)
(196,208)(197,209)(198,210)(199,211)(200,212)(201,213)(202,214)(203,215)
(204,216);
s1 := Sym(216)!( 1,123)( 2,124)( 3,121)( 4,122)( 5,127)( 6,128)( 7,125)
( 8,126)( 9,131)( 10,132)( 11,129)( 12,130)( 13,111)( 14,112)( 15,109)
( 16,110)( 17,115)( 18,116)( 19,113)( 20,114)( 21,119)( 22,120)( 23,117)
( 24,118)( 25,135)( 26,136)( 27,133)( 28,134)( 29,139)( 30,140)( 31,137)
( 32,138)( 33,143)( 34,144)( 35,141)( 36,142)( 37,159)( 38,160)( 39,157)
( 40,158)( 41,163)( 42,164)( 43,161)( 44,162)( 45,167)( 46,168)( 47,165)
( 48,166)( 49,147)( 50,148)( 51,145)( 52,146)( 53,151)( 54,152)( 55,149)
( 56,150)( 57,155)( 58,156)( 59,153)( 60,154)( 61,171)( 62,172)( 63,169)
( 64,170)( 65,175)( 66,176)( 67,173)( 68,174)( 69,179)( 70,180)( 71,177)
( 72,178)( 73,195)( 74,196)( 75,193)( 76,194)( 77,199)( 78,200)( 79,197)
( 80,198)( 81,203)( 82,204)( 83,201)( 84,202)( 85,183)( 86,184)( 87,181)
( 88,182)( 89,187)( 90,188)( 91,185)( 92,186)( 93,191)( 94,192)( 95,189)
( 96,190)( 97,207)( 98,208)( 99,205)(100,206)(101,211)(102,212)(103,209)
(104,210)(105,215)(106,216)(107,213)(108,214);
s2 := Sym(216)!( 3, 4)( 5, 9)( 6, 10)( 7, 12)( 8, 11)( 15, 16)( 17, 21)
( 18, 22)( 19, 24)( 20, 23)( 27, 28)( 29, 33)( 30, 34)( 31, 36)( 32, 35)
( 37, 73)( 38, 74)( 39, 76)( 40, 75)( 41, 81)( 42, 82)( 43, 84)( 44, 83)
( 45, 77)( 46, 78)( 47, 80)( 48, 79)( 49, 85)( 50, 86)( 51, 88)( 52, 87)
( 53, 93)( 54, 94)( 55, 96)( 56, 95)( 57, 89)( 58, 90)( 59, 92)( 60, 91)
( 61, 97)( 62, 98)( 63,100)( 64, 99)( 65,105)( 66,106)( 67,108)( 68,107)
( 69,101)( 70,102)( 71,104)( 72,103)(111,112)(113,117)(114,118)(115,120)
(116,119)(123,124)(125,129)(126,130)(127,132)(128,131)(135,136)(137,141)
(138,142)(139,144)(140,143)(145,181)(146,182)(147,184)(148,183)(149,189)
(150,190)(151,192)(152,191)(153,185)(154,186)(155,188)(156,187)(157,193)
(158,194)(159,196)(160,195)(161,201)(162,202)(163,204)(164,203)(165,197)
(166,198)(167,200)(168,199)(169,205)(170,206)(171,208)(172,207)(173,213)
(174,214)(175,216)(176,215)(177,209)(178,210)(179,212)(180,211);
s3 := Sym(216)!( 1, 41)( 2, 44)( 3, 43)( 4, 42)( 5, 37)( 6, 40)( 7, 39)
( 8, 38)( 9, 45)( 10, 48)( 11, 47)( 12, 46)( 13, 53)( 14, 56)( 15, 55)
( 16, 54)( 17, 49)( 18, 52)( 19, 51)( 20, 50)( 21, 57)( 22, 60)( 23, 59)
( 24, 58)( 25, 65)( 26, 68)( 27, 67)( 28, 66)( 29, 61)( 30, 64)( 31, 63)
( 32, 62)( 33, 69)( 34, 72)( 35, 71)( 36, 70)( 73, 77)( 74, 80)( 75, 79)
( 76, 78)( 82, 84)( 85, 89)( 86, 92)( 87, 91)( 88, 90)( 94, 96)( 97,101)
( 98,104)( 99,103)(100,102)(106,108)(109,149)(110,152)(111,151)(112,150)
(113,145)(114,148)(115,147)(116,146)(117,153)(118,156)(119,155)(120,154)
(121,161)(122,164)(123,163)(124,162)(125,157)(126,160)(127,159)(128,158)
(129,165)(130,168)(131,167)(132,166)(133,173)(134,176)(135,175)(136,174)
(137,169)(138,172)(139,171)(140,170)(141,177)(142,180)(143,179)(144,178)
(181,185)(182,188)(183,187)(184,186)(190,192)(193,197)(194,200)(195,199)
(196,198)(202,204)(205,209)(206,212)(207,211)(208,210)(214,216);
s4 := Sym(216)!( 5, 9)( 6, 10)( 7, 11)( 8, 12)( 17, 21)( 18, 22)( 19, 23)
( 20, 24)( 29, 33)( 30, 34)( 31, 35)( 32, 36)( 41, 45)( 42, 46)( 43, 47)
( 44, 48)( 53, 57)( 54, 58)( 55, 59)( 56, 60)( 65, 69)( 66, 70)( 67, 71)
( 68, 72)( 77, 81)( 78, 82)( 79, 83)( 80, 84)( 89, 93)( 90, 94)( 91, 95)
( 92, 96)(101,105)(102,106)(103,107)(104,108)(113,117)(114,118)(115,119)
(116,120)(125,129)(126,130)(127,131)(128,132)(137,141)(138,142)(139,143)
(140,144)(149,153)(150,154)(151,155)(152,156)(161,165)(162,166)(163,167)
(164,168)(173,177)(174,178)(175,179)(176,180)(185,189)(186,190)(187,191)
(188,192)(197,201)(198,202)(199,203)(200,204)(209,213)(210,214)(211,215)
(212,216);
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, s2*s3*s2*s3*s2*s3, s0*s1*s2*s1*s0*s1*s2*s1,
s1*s2*s1*s2*s1*s2*s1*s2, s4*s2*s3*s4*s3*s4*s2*s3*s4*s3,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 >;
References : None.
to this polytope