Polytope of Type {3,6,6,8}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {3,6,6,8}*1728b
if this polytope has a name.
Group : SmallGroup(1728,37597)
Rank : 5
Schlafli Type : {3,6,6,8}
Number of vertices, edges, etc : 3, 9, 18, 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 : {3,6,6,4}*864d
3-fold quotients : {3,2,6,8}*576, {3,6,2,8}*576
4-fold quotients : {3,6,6,2}*432b
6-fold quotients : {3,2,6,4}*288a, {3,6,2,4}*288
9-fold quotients : {3,2,2,8}*192
12-fold quotients : {3,2,6,2}*144, {3,6,2,2}*144
18-fold quotients : {3,2,2,4}*96
24-fold quotients : {3,2,3,2}*72
36-fold quotients : {3,2,2,2}*48
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := ( 4, 7)( 5, 8)( 6, 9)( 10, 19)( 11, 20)( 12, 21)( 13, 25)( 14, 26)
( 15, 27)( 16, 22)( 17, 23)( 18, 24)( 31, 34)( 32, 35)( 33, 36)( 37, 46)
( 38, 47)( 39, 48)( 40, 52)( 41, 53)( 42, 54)( 43, 49)( 44, 50)( 45, 51)
( 58, 61)( 59, 62)( 60, 63)( 64, 73)( 65, 74)( 66, 75)( 67, 79)( 68, 80)
( 69, 81)( 70, 76)( 71, 77)( 72, 78)( 85, 88)( 86, 89)( 87, 90)( 91,100)
( 92,101)( 93,102)( 94,106)( 95,107)( 96,108)( 97,103)( 98,104)( 99,105)
(112,115)(113,116)(114,117)(118,127)(119,128)(120,129)(121,133)(122,134)
(123,135)(124,130)(125,131)(126,132)(139,142)(140,143)(141,144)(145,154)
(146,155)(147,156)(148,160)(149,161)(150,162)(151,157)(152,158)(153,159)
(166,169)(167,170)(168,171)(172,181)(173,182)(174,183)(175,187)(176,188)
(177,189)(178,184)(179,185)(180,186)(193,196)(194,197)(195,198)(199,208)
(200,209)(201,210)(202,214)(203,215)(204,216)(205,211)(206,212)(207,213);;
s1 := ( 1, 13)( 2, 14)( 3, 15)( 4, 10)( 5, 11)( 6, 12)( 7, 16)( 8, 17)
( 9, 18)( 19, 22)( 20, 23)( 21, 24)( 28, 40)( 29, 41)( 30, 42)( 31, 37)
( 32, 38)( 33, 39)( 34, 43)( 35, 44)( 36, 45)( 46, 49)( 47, 50)( 48, 51)
( 55, 67)( 56, 68)( 57, 69)( 58, 64)( 59, 65)( 60, 66)( 61, 70)( 62, 71)
( 63, 72)( 73, 76)( 74, 77)( 75, 78)( 82, 94)( 83, 95)( 84, 96)( 85, 91)
( 86, 92)( 87, 93)( 88, 97)( 89, 98)( 90, 99)(100,103)(101,104)(102,105)
(109,121)(110,122)(111,123)(112,118)(113,119)(114,120)(115,124)(116,125)
(117,126)(127,130)(128,131)(129,132)(136,148)(137,149)(138,150)(139,145)
(140,146)(141,147)(142,151)(143,152)(144,153)(154,157)(155,158)(156,159)
(163,175)(164,176)(165,177)(166,172)(167,173)(168,174)(169,178)(170,179)
(171,180)(181,184)(182,185)(183,186)(190,202)(191,203)(192,204)(193,199)
(194,200)(195,201)(196,205)(197,206)(198,207)(208,211)(209,212)(210,213);;
s2 := ( 2, 3)( 4, 7)( 5, 9)( 6, 8)( 11, 12)( 13, 16)( 14, 18)( 15, 17)
( 20, 21)( 22, 25)( 23, 27)( 24, 26)( 29, 30)( 31, 34)( 32, 36)( 33, 35)
( 38, 39)( 40, 43)( 41, 45)( 42, 44)( 47, 48)( 49, 52)( 50, 54)( 51, 53)
( 56, 57)( 58, 61)( 59, 63)( 60, 62)( 65, 66)( 67, 70)( 68, 72)( 69, 71)
( 74, 75)( 76, 79)( 77, 81)( 78, 80)( 83, 84)( 85, 88)( 86, 90)( 87, 89)
( 92, 93)( 94, 97)( 95, 99)( 96, 98)(101,102)(103,106)(104,108)(105,107)
(110,111)(112,115)(113,117)(114,116)(119,120)(121,124)(122,126)(123,125)
(128,129)(130,133)(131,135)(132,134)(137,138)(139,142)(140,144)(141,143)
(146,147)(148,151)(149,153)(150,152)(155,156)(157,160)(158,162)(159,161)
(164,165)(166,169)(167,171)(168,170)(173,174)(175,178)(176,180)(177,179)
(182,183)(184,187)(185,189)(186,188)(191,192)(193,196)(194,198)(195,197)
(200,201)(202,205)(203,207)(204,206)(209,210)(211,214)(212,216)(213,215);;
s3 := ( 1, 2)( 4, 5)( 7, 8)( 10, 11)( 13, 14)( 16, 17)( 19, 20)( 22, 23)
( 25, 26)( 28, 29)( 31, 32)( 34, 35)( 37, 38)( 40, 41)( 43, 44)( 46, 47)
( 49, 50)( 52, 53)( 55, 83)( 56, 82)( 57, 84)( 58, 86)( 59, 85)( 60, 87)
( 61, 89)( 62, 88)( 63, 90)( 64, 92)( 65, 91)( 66, 93)( 67, 95)( 68, 94)
( 69, 96)( 70, 98)( 71, 97)( 72, 99)( 73,101)( 74,100)( 75,102)( 76,104)
( 77,103)( 78,105)( 79,107)( 80,106)( 81,108)(109,164)(110,163)(111,165)
(112,167)(113,166)(114,168)(115,170)(116,169)(117,171)(118,173)(119,172)
(120,174)(121,176)(122,175)(123,177)(124,179)(125,178)(126,180)(127,182)
(128,181)(129,183)(130,185)(131,184)(132,186)(133,188)(134,187)(135,189)
(136,191)(137,190)(138,192)(139,194)(140,193)(141,195)(142,197)(143,196)
(144,198)(145,200)(146,199)(147,201)(148,203)(149,202)(150,204)(151,206)
(152,205)(153,207)(154,209)(155,208)(156,210)(157,212)(158,211)(159,213)
(160,215)(161,214)(162,216);;
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, 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, s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3,
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)( 10, 19)( 11, 20)( 12, 21)( 13, 25)
( 14, 26)( 15, 27)( 16, 22)( 17, 23)( 18, 24)( 31, 34)( 32, 35)( 33, 36)
( 37, 46)( 38, 47)( 39, 48)( 40, 52)( 41, 53)( 42, 54)( 43, 49)( 44, 50)
( 45, 51)( 58, 61)( 59, 62)( 60, 63)( 64, 73)( 65, 74)( 66, 75)( 67, 79)
( 68, 80)( 69, 81)( 70, 76)( 71, 77)( 72, 78)( 85, 88)( 86, 89)( 87, 90)
( 91,100)( 92,101)( 93,102)( 94,106)( 95,107)( 96,108)( 97,103)( 98,104)
( 99,105)(112,115)(113,116)(114,117)(118,127)(119,128)(120,129)(121,133)
(122,134)(123,135)(124,130)(125,131)(126,132)(139,142)(140,143)(141,144)
(145,154)(146,155)(147,156)(148,160)(149,161)(150,162)(151,157)(152,158)
(153,159)(166,169)(167,170)(168,171)(172,181)(173,182)(174,183)(175,187)
(176,188)(177,189)(178,184)(179,185)(180,186)(193,196)(194,197)(195,198)
(199,208)(200,209)(201,210)(202,214)(203,215)(204,216)(205,211)(206,212)
(207,213);
s1 := Sym(216)!( 1, 13)( 2, 14)( 3, 15)( 4, 10)( 5, 11)( 6, 12)( 7, 16)
( 8, 17)( 9, 18)( 19, 22)( 20, 23)( 21, 24)( 28, 40)( 29, 41)( 30, 42)
( 31, 37)( 32, 38)( 33, 39)( 34, 43)( 35, 44)( 36, 45)( 46, 49)( 47, 50)
( 48, 51)( 55, 67)( 56, 68)( 57, 69)( 58, 64)( 59, 65)( 60, 66)( 61, 70)
( 62, 71)( 63, 72)( 73, 76)( 74, 77)( 75, 78)( 82, 94)( 83, 95)( 84, 96)
( 85, 91)( 86, 92)( 87, 93)( 88, 97)( 89, 98)( 90, 99)(100,103)(101,104)
(102,105)(109,121)(110,122)(111,123)(112,118)(113,119)(114,120)(115,124)
(116,125)(117,126)(127,130)(128,131)(129,132)(136,148)(137,149)(138,150)
(139,145)(140,146)(141,147)(142,151)(143,152)(144,153)(154,157)(155,158)
(156,159)(163,175)(164,176)(165,177)(166,172)(167,173)(168,174)(169,178)
(170,179)(171,180)(181,184)(182,185)(183,186)(190,202)(191,203)(192,204)
(193,199)(194,200)(195,201)(196,205)(197,206)(198,207)(208,211)(209,212)
(210,213);
s2 := Sym(216)!( 2, 3)( 4, 7)( 5, 9)( 6, 8)( 11, 12)( 13, 16)( 14, 18)
( 15, 17)( 20, 21)( 22, 25)( 23, 27)( 24, 26)( 29, 30)( 31, 34)( 32, 36)
( 33, 35)( 38, 39)( 40, 43)( 41, 45)( 42, 44)( 47, 48)( 49, 52)( 50, 54)
( 51, 53)( 56, 57)( 58, 61)( 59, 63)( 60, 62)( 65, 66)( 67, 70)( 68, 72)
( 69, 71)( 74, 75)( 76, 79)( 77, 81)( 78, 80)( 83, 84)( 85, 88)( 86, 90)
( 87, 89)( 92, 93)( 94, 97)( 95, 99)( 96, 98)(101,102)(103,106)(104,108)
(105,107)(110,111)(112,115)(113,117)(114,116)(119,120)(121,124)(122,126)
(123,125)(128,129)(130,133)(131,135)(132,134)(137,138)(139,142)(140,144)
(141,143)(146,147)(148,151)(149,153)(150,152)(155,156)(157,160)(158,162)
(159,161)(164,165)(166,169)(167,171)(168,170)(173,174)(175,178)(176,180)
(177,179)(182,183)(184,187)(185,189)(186,188)(191,192)(193,196)(194,198)
(195,197)(200,201)(202,205)(203,207)(204,206)(209,210)(211,214)(212,216)
(213,215);
s3 := Sym(216)!( 1, 2)( 4, 5)( 7, 8)( 10, 11)( 13, 14)( 16, 17)( 19, 20)
( 22, 23)( 25, 26)( 28, 29)( 31, 32)( 34, 35)( 37, 38)( 40, 41)( 43, 44)
( 46, 47)( 49, 50)( 52, 53)( 55, 83)( 56, 82)( 57, 84)( 58, 86)( 59, 85)
( 60, 87)( 61, 89)( 62, 88)( 63, 90)( 64, 92)( 65, 91)( 66, 93)( 67, 95)
( 68, 94)( 69, 96)( 70, 98)( 71, 97)( 72, 99)( 73,101)( 74,100)( 75,102)
( 76,104)( 77,103)( 78,105)( 79,107)( 80,106)( 81,108)(109,164)(110,163)
(111,165)(112,167)(113,166)(114,168)(115,170)(116,169)(117,171)(118,173)
(119,172)(120,174)(121,176)(122,175)(123,177)(124,179)(125,178)(126,180)
(127,182)(128,181)(129,183)(130,185)(131,184)(132,186)(133,188)(134,187)
(135,189)(136,191)(137,190)(138,192)(139,194)(140,193)(141,195)(142,197)
(143,196)(144,198)(145,200)(146,199)(147,201)(148,203)(149,202)(150,204)
(151,206)(152,205)(153,207)(154,209)(155,208)(156,210)(157,212)(158,211)
(159,213)(160,215)(161,214)(162,216);
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, 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,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3,
s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4 >;
References : None.
to this polytope