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