Polytope of Type {6,3,6,8}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,3,6,8}*1728
if this polytope has a name.
Group : SmallGroup(1728,37593)
Rank : 5
Schlafli Type : {6,3,6,8}
Number of vertices, edges, etc : 6, 9, 9, 24, 8
Order of s0s1s2s3s4 : 24
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 : {6,3,6,4}*864
3-fold quotients : {6,3,2,8}*576, {2,3,6,8}*576
4-fold quotients : {6,3,6,2}*432
6-fold quotients : {6,3,2,4}*288, {2,3,6,4}*288
9-fold quotients : {2,3,2,8}*192
12-fold quotients : {2,3,6,2}*144, {6,3,2,2}*144
18-fold quotients : {2,3,2,4}*96
36-fold quotients : {2,3,2,2}*48
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := ( 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)( 58, 61)( 59, 62)( 60, 63)( 67, 70)( 68, 71)( 69, 72)
( 76, 79)( 77, 80)( 78, 81)( 85, 88)( 86, 89)( 87, 90)( 94, 97)( 95, 98)
( 96, 99)(103,106)(104,107)(105,108)(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)(166,169)(167,170)
(168,171)(175,178)(176,179)(177,180)(184,187)(185,188)(186,189)(193,196)
(194,197)(195,198)(202,205)(203,206)(204,207)(211,214)(212,215)(213,216);;
s1 := ( 1, 4)( 2, 6)( 3, 5)( 8, 9)( 10, 22)( 11, 24)( 12, 23)( 13, 19)
( 14, 21)( 15, 20)( 16, 25)( 17, 27)( 18, 26)( 28, 31)( 29, 33)( 30, 32)
( 35, 36)( 37, 49)( 38, 51)( 39, 50)( 40, 46)( 41, 48)( 42, 47)( 43, 52)
( 44, 54)( 45, 53)( 55, 58)( 56, 60)( 57, 59)( 62, 63)( 64, 76)( 65, 78)
( 66, 77)( 67, 73)( 68, 75)( 69, 74)( 70, 79)( 71, 81)( 72, 80)( 82, 85)
( 83, 87)( 84, 86)( 89, 90)( 91,103)( 92,105)( 93,104)( 94,100)( 95,102)
( 96,101)( 97,106)( 98,108)( 99,107)(109,112)(110,114)(111,113)(116,117)
(118,130)(119,132)(120,131)(121,127)(122,129)(123,128)(124,133)(125,135)
(126,134)(136,139)(137,141)(138,140)(143,144)(145,157)(146,159)(147,158)
(148,154)(149,156)(150,155)(151,160)(152,162)(153,161)(163,166)(164,168)
(165,167)(170,171)(172,184)(173,186)(174,185)(175,181)(176,183)(177,182)
(178,187)(179,189)(180,188)(190,193)(191,195)(192,194)(197,198)(199,211)
(200,213)(201,212)(202,208)(203,210)(204,209)(205,214)(206,216)(207,215);;
s2 := ( 1, 11)( 2, 10)( 3, 12)( 4, 17)( 5, 16)( 6, 18)( 7, 14)( 8, 13)
( 9, 15)( 19, 20)( 22, 26)( 23, 25)( 24, 27)( 28, 38)( 29, 37)( 30, 39)
( 31, 44)( 32, 43)( 33, 45)( 34, 41)( 35, 40)( 36, 42)( 46, 47)( 49, 53)
( 50, 52)( 51, 54)( 55, 65)( 56, 64)( 57, 66)( 58, 71)( 59, 70)( 60, 72)
( 61, 68)( 62, 67)( 63, 69)( 73, 74)( 76, 80)( 77, 79)( 78, 81)( 82, 92)
( 83, 91)( 84, 93)( 85, 98)( 86, 97)( 87, 99)( 88, 95)( 89, 94)( 90, 96)
(100,101)(103,107)(104,106)(105,108)(109,119)(110,118)(111,120)(112,125)
(113,124)(114,126)(115,122)(116,121)(117,123)(127,128)(130,134)(131,133)
(132,135)(136,146)(137,145)(138,147)(139,152)(140,151)(141,153)(142,149)
(143,148)(144,150)(154,155)(157,161)(158,160)(159,162)(163,173)(164,172)
(165,174)(166,179)(167,178)(168,180)(169,176)(170,175)(171,177)(181,182)
(184,188)(185,187)(186,189)(190,200)(191,199)(192,201)(193,206)(194,205)
(195,207)(196,203)(197,202)(198,204)(208,209)(211,215)(212,214)(213,216);;
s3 := ( 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)(109,163)(110,164)(111,165)
(112,166)(113,167)(114,168)(115,169)(116,170)(117,171)(118,181)(119,182)
(120,183)(121,184)(122,185)(123,186)(124,187)(125,188)(126,189)(127,172)
(128,173)(129,174)(130,175)(131,176)(132,177)(133,178)(134,179)(135,180)
(136,190)(137,191)(138,192)(139,193)(140,194)(141,195)(142,196)(143,197)
(144,198)(145,208)(146,209)(147,210)(148,211)(149,212)(150,213)(151,214)
(152,215)(153,216)(154,199)(155,200)(156,201)(157,202)(158,203)(159,204)
(160,205)(161,206)(162,207);;
s4 := ( 1,109)( 2,110)( 3,111)( 4,112)( 5,113)( 6,114)( 7,115)( 8,116)
( 9,117)( 10,118)( 11,119)( 12,120)( 13,121)( 14,122)( 15,123)( 16,124)
( 17,125)( 18,126)( 19,127)( 20,128)( 21,129)( 22,130)( 23,131)( 24,132)
( 25,133)( 26,134)( 27,135)( 28,136)( 29,137)( 30,138)( 31,139)( 32,140)
( 33,141)( 34,142)( 35,143)( 36,144)( 37,145)( 38,146)( 39,147)( 40,148)
( 41,149)( 42,150)( 43,151)( 44,152)( 45,153)( 46,154)( 47,155)( 48,156)
( 49,157)( 50,158)( 51,159)( 52,160)( 53,161)( 54,162)( 55,190)( 56,191)
( 57,192)( 58,193)( 59,194)( 60,195)( 61,196)( 62,197)( 63,198)( 64,199)
( 65,200)( 66,201)( 67,202)( 68,203)( 69,204)( 70,205)( 71,206)( 72,207)
( 73,208)( 74,209)( 75,210)( 76,211)( 77,212)( 78,213)( 79,214)( 80,215)
( 81,216)( 82,163)( 83,164)( 84,165)( 85,166)( 86,167)( 87,168)( 88,169)
( 89,170)( 90,171)( 91,172)( 92,173)( 93,174)( 94,175)( 95,176)( 96,177)
( 97,178)( 98,179)( 99,180)(100,181)(101,182)(102,183)(103,184)(104,185)
(105,186)(106,187)(107,188)(108,189);;
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*s4*s3*s2*s3*s4*s3, s2*s0*s1*s0*s1*s2*s0*s1*s0*s1,
s3*s1*s2*s3*s2*s3*s1*s2*s3*s2, s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := 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)( 58, 61)( 59, 62)( 60, 63)( 67, 70)( 68, 71)
( 69, 72)( 76, 79)( 77, 80)( 78, 81)( 85, 88)( 86, 89)( 87, 90)( 94, 97)
( 95, 98)( 96, 99)(103,106)(104,107)(105,108)(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)(166,169)
(167,170)(168,171)(175,178)(176,179)(177,180)(184,187)(185,188)(186,189)
(193,196)(194,197)(195,198)(202,205)(203,206)(204,207)(211,214)(212,215)
(213,216);
s1 := Sym(216)!( 1, 4)( 2, 6)( 3, 5)( 8, 9)( 10, 22)( 11, 24)( 12, 23)
( 13, 19)( 14, 21)( 15, 20)( 16, 25)( 17, 27)( 18, 26)( 28, 31)( 29, 33)
( 30, 32)( 35, 36)( 37, 49)( 38, 51)( 39, 50)( 40, 46)( 41, 48)( 42, 47)
( 43, 52)( 44, 54)( 45, 53)( 55, 58)( 56, 60)( 57, 59)( 62, 63)( 64, 76)
( 65, 78)( 66, 77)( 67, 73)( 68, 75)( 69, 74)( 70, 79)( 71, 81)( 72, 80)
( 82, 85)( 83, 87)( 84, 86)( 89, 90)( 91,103)( 92,105)( 93,104)( 94,100)
( 95,102)( 96,101)( 97,106)( 98,108)( 99,107)(109,112)(110,114)(111,113)
(116,117)(118,130)(119,132)(120,131)(121,127)(122,129)(123,128)(124,133)
(125,135)(126,134)(136,139)(137,141)(138,140)(143,144)(145,157)(146,159)
(147,158)(148,154)(149,156)(150,155)(151,160)(152,162)(153,161)(163,166)
(164,168)(165,167)(170,171)(172,184)(173,186)(174,185)(175,181)(176,183)
(177,182)(178,187)(179,189)(180,188)(190,193)(191,195)(192,194)(197,198)
(199,211)(200,213)(201,212)(202,208)(203,210)(204,209)(205,214)(206,216)
(207,215);
s2 := Sym(216)!( 1, 11)( 2, 10)( 3, 12)( 4, 17)( 5, 16)( 6, 18)( 7, 14)
( 8, 13)( 9, 15)( 19, 20)( 22, 26)( 23, 25)( 24, 27)( 28, 38)( 29, 37)
( 30, 39)( 31, 44)( 32, 43)( 33, 45)( 34, 41)( 35, 40)( 36, 42)( 46, 47)
( 49, 53)( 50, 52)( 51, 54)( 55, 65)( 56, 64)( 57, 66)( 58, 71)( 59, 70)
( 60, 72)( 61, 68)( 62, 67)( 63, 69)( 73, 74)( 76, 80)( 77, 79)( 78, 81)
( 82, 92)( 83, 91)( 84, 93)( 85, 98)( 86, 97)( 87, 99)( 88, 95)( 89, 94)
( 90, 96)(100,101)(103,107)(104,106)(105,108)(109,119)(110,118)(111,120)
(112,125)(113,124)(114,126)(115,122)(116,121)(117,123)(127,128)(130,134)
(131,133)(132,135)(136,146)(137,145)(138,147)(139,152)(140,151)(141,153)
(142,149)(143,148)(144,150)(154,155)(157,161)(158,160)(159,162)(163,173)
(164,172)(165,174)(166,179)(167,178)(168,180)(169,176)(170,175)(171,177)
(181,182)(184,188)(185,187)(186,189)(190,200)(191,199)(192,201)(193,206)
(194,205)(195,207)(196,203)(197,202)(198,204)(208,209)(211,215)(212,214)
(213,216);
s3 := 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)(109,163)(110,164)
(111,165)(112,166)(113,167)(114,168)(115,169)(116,170)(117,171)(118,181)
(119,182)(120,183)(121,184)(122,185)(123,186)(124,187)(125,188)(126,189)
(127,172)(128,173)(129,174)(130,175)(131,176)(132,177)(133,178)(134,179)
(135,180)(136,190)(137,191)(138,192)(139,193)(140,194)(141,195)(142,196)
(143,197)(144,198)(145,208)(146,209)(147,210)(148,211)(149,212)(150,213)
(151,214)(152,215)(153,216)(154,199)(155,200)(156,201)(157,202)(158,203)
(159,204)(160,205)(161,206)(162,207);
s4 := Sym(216)!( 1,109)( 2,110)( 3,111)( 4,112)( 5,113)( 6,114)( 7,115)
( 8,116)( 9,117)( 10,118)( 11,119)( 12,120)( 13,121)( 14,122)( 15,123)
( 16,124)( 17,125)( 18,126)( 19,127)( 20,128)( 21,129)( 22,130)( 23,131)
( 24,132)( 25,133)( 26,134)( 27,135)( 28,136)( 29,137)( 30,138)( 31,139)
( 32,140)( 33,141)( 34,142)( 35,143)( 36,144)( 37,145)( 38,146)( 39,147)
( 40,148)( 41,149)( 42,150)( 43,151)( 44,152)( 45,153)( 46,154)( 47,155)
( 48,156)( 49,157)( 50,158)( 51,159)( 52,160)( 53,161)( 54,162)( 55,190)
( 56,191)( 57,192)( 58,193)( 59,194)( 60,195)( 61,196)( 62,197)( 63,198)
( 64,199)( 65,200)( 66,201)( 67,202)( 68,203)( 69,204)( 70,205)( 71,206)
( 72,207)( 73,208)( 74,209)( 75,210)( 76,211)( 77,212)( 78,213)( 79,214)
( 80,215)( 81,216)( 82,163)( 83,164)( 84,165)( 85,166)( 86,167)( 87,168)
( 88,169)( 89,170)( 90,171)( 91,172)( 92,173)( 93,174)( 94,175)( 95,176)
( 96,177)( 97,178)( 98,179)( 99,180)(100,181)(101,182)(102,183)(103,184)
(104,185)(105,186)(106,187)(107,188)(108,189);
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*s4*s3*s2*s3*s4*s3,
s2*s0*s1*s0*s1*s2*s0*s1*s0*s1, s3*s1*s2*s3*s2*s3*s1*s2*s3*s2,
s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4 >;
References : None.
to this polytope