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