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