Polytope of Type {9,4,6}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {9,4,6}*864
if this polytope has a name.
Group : SmallGroup(864,3999)
Rank : 4
Schlafli Type : {9,4,6}
Number of vertices, edges, etc : 18, 36, 24, 6
Order of s0s1s2s3 : 18
Order of s0s1s2s3s2s1 : 2
Special Properties :
Universal
Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{9,4,6,2} of size 1728
Vertex Figure Of :
{2,9,4,6} of size 1728
Quotients (Maximal Quotients in Boldface) :
3-fold quotients : {9,4,2}*288, {3,4,6}*288
4-fold quotients : {9,2,6}*216
6-fold quotients : {9,4,2}*144
8-fold quotients : {9,2,3}*108
9-fold quotients : {3,4,2}*96
12-fold quotients : {9,2,2}*72, {3,2,6}*72
18-fold quotients : {3,4,2}*48
24-fold quotients : {3,2,3}*36
36-fold quotients : {3,2,2}*24
Covers (Minimal Covers in Boldface) :
2-fold covers : {9,4,12}*1728, {9,8,6}*1728, {18,4,6}*1728b
Permutation Representation (GAP) :
s0 := ( 2, 3)( 5, 9)( 6, 11)( 7, 10)( 8, 12)( 14, 15)( 17, 21)( 18, 23)
( 19, 22)( 20, 24)( 26, 27)( 29, 33)( 30, 35)( 31, 34)( 32, 36)( 37, 77)
( 38, 79)( 39, 78)( 40, 80)( 41, 73)( 42, 75)( 43, 74)( 44, 76)( 45, 81)
( 46, 83)( 47, 82)( 48, 84)( 49, 89)( 50, 91)( 51, 90)( 52, 92)( 53, 85)
( 54, 87)( 55, 86)( 56, 88)( 57, 93)( 58, 95)( 59, 94)( 60, 96)( 61,101)
( 62,103)( 63,102)( 64,104)( 65, 97)( 66, 99)( 67, 98)( 68,100)( 69,105)
( 70,107)( 71,106)( 72,108)(110,111)(113,117)(114,119)(115,118)(116,120)
(122,123)(125,129)(126,131)(127,130)(128,132)(134,135)(137,141)(138,143)
(139,142)(140,144)(145,185)(146,187)(147,186)(148,188)(149,181)(150,183)
(151,182)(152,184)(153,189)(154,191)(155,190)(156,192)(157,197)(158,199)
(159,198)(160,200)(161,193)(162,195)(163,194)(164,196)(165,201)(166,203)
(167,202)(168,204)(169,209)(170,211)(171,210)(172,212)(173,205)(174,207)
(175,206)(176,208)(177,213)(178,215)(179,214)(180,216);;
s1 := ( 1, 37)( 2, 38)( 3, 40)( 4, 39)( 5, 45)( 6, 46)( 7, 48)( 8, 47)
( 9, 41)( 10, 42)( 11, 44)( 12, 43)( 13, 49)( 14, 50)( 15, 52)( 16, 51)
( 17, 57)( 18, 58)( 19, 60)( 20, 59)( 21, 53)( 22, 54)( 23, 56)( 24, 55)
( 25, 61)( 26, 62)( 27, 64)( 28, 63)( 29, 69)( 30, 70)( 31, 72)( 32, 71)
( 33, 65)( 34, 66)( 35, 68)( 36, 67)( 73, 77)( 74, 78)( 75, 80)( 76, 79)
( 83, 84)( 85, 89)( 86, 90)( 87, 92)( 88, 91)( 95, 96)( 97,101)( 98,102)
( 99,104)(100,103)(107,108)(109,145)(110,146)(111,148)(112,147)(113,153)
(114,154)(115,156)(116,155)(117,149)(118,150)(119,152)(120,151)(121,157)
(122,158)(123,160)(124,159)(125,165)(126,166)(127,168)(128,167)(129,161)
(130,162)(131,164)(132,163)(133,169)(134,170)(135,172)(136,171)(137,177)
(138,178)(139,180)(140,179)(141,173)(142,174)(143,176)(144,175)(181,185)
(182,186)(183,188)(184,187)(191,192)(193,197)(194,198)(195,200)(196,199)
(203,204)(205,209)(206,210)(207,212)(208,211)(215,216);;
s2 := ( 1, 4)( 2, 3)( 5, 8)( 6, 7)( 9, 12)( 10, 11)( 13, 28)( 14, 27)
( 15, 26)( 16, 25)( 17, 32)( 18, 31)( 19, 30)( 20, 29)( 21, 36)( 22, 35)
( 23, 34)( 24, 33)( 37, 40)( 38, 39)( 41, 44)( 42, 43)( 45, 48)( 46, 47)
( 49, 64)( 50, 63)( 51, 62)( 52, 61)( 53, 68)( 54, 67)( 55, 66)( 56, 65)
( 57, 72)( 58, 71)( 59, 70)( 60, 69)( 73, 76)( 74, 75)( 77, 80)( 78, 79)
( 81, 84)( 82, 83)( 85,100)( 86, 99)( 87, 98)( 88, 97)( 89,104)( 90,103)
( 91,102)( 92,101)( 93,108)( 94,107)( 95,106)( 96,105)(109,112)(110,111)
(113,116)(114,115)(117,120)(118,119)(121,136)(122,135)(123,134)(124,133)
(125,140)(126,139)(127,138)(128,137)(129,144)(130,143)(131,142)(132,141)
(145,148)(146,147)(149,152)(150,151)(153,156)(154,155)(157,172)(158,171)
(159,170)(160,169)(161,176)(162,175)(163,174)(164,173)(165,180)(166,179)
(167,178)(168,177)(181,184)(182,183)(185,188)(186,187)(189,192)(190,191)
(193,208)(194,207)(195,206)(196,205)(197,212)(198,211)(199,210)(200,209)
(201,216)(202,215)(203,214)(204,213);;
s3 := ( 1,121)( 2,122)( 3,123)( 4,124)( 5,125)( 6,126)( 7,127)( 8,128)
( 9,129)( 10,130)( 11,131)( 12,132)( 13,109)( 14,110)( 15,111)( 16,112)
( 17,113)( 18,114)( 19,115)( 20,116)( 21,117)( 22,118)( 23,119)( 24,120)
( 25,133)( 26,134)( 27,135)( 28,136)( 29,137)( 30,138)( 31,139)( 32,140)
( 33,141)( 34,142)( 35,143)( 36,144)( 37,157)( 38,158)( 39,159)( 40,160)
( 41,161)( 42,162)( 43,163)( 44,164)( 45,165)( 46,166)( 47,167)( 48,168)
( 49,145)( 50,146)( 51,147)( 52,148)( 53,149)( 54,150)( 55,151)( 56,152)
( 57,153)( 58,154)( 59,155)( 60,156)( 61,169)( 62,170)( 63,171)( 64,172)
( 65,173)( 66,174)( 67,175)( 68,176)( 69,177)( 70,178)( 71,179)( 72,180)
( 73,193)( 74,194)( 75,195)( 76,196)( 77,197)( 78,198)( 79,199)( 80,200)
( 81,201)( 82,202)( 83,203)( 84,204)( 85,181)( 86,182)( 87,183)( 88,184)
( 89,185)( 90,186)( 91,187)( 92,188)( 93,189)( 94,190)( 95,191)( 96,192)
( 97,205)( 98,206)( 99,207)(100,208)(101,209)(102,210)(103,211)(104,212)
(105,213)(106,214)(107,215)(108,216);;
poly := Group([s0,s1,s2,s3]);;
Finitely Presented Group Representation (GAP) :
F := FreeGroup("s0","s1","s2","s3");;
s0 := F.1;; s1 := F.2;; s2 := F.3;; s3 := F.4;;
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s0*s2*s0*s2,
s0*s3*s0*s3, s1*s3*s1*s3, s1*s2*s1*s2*s1*s2*s1*s2,
s1*s2*s3*s2*s1*s2*s3*s2, s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(216)!( 2, 3)( 5, 9)( 6, 11)( 7, 10)( 8, 12)( 14, 15)( 17, 21)
( 18, 23)( 19, 22)( 20, 24)( 26, 27)( 29, 33)( 30, 35)( 31, 34)( 32, 36)
( 37, 77)( 38, 79)( 39, 78)( 40, 80)( 41, 73)( 42, 75)( 43, 74)( 44, 76)
( 45, 81)( 46, 83)( 47, 82)( 48, 84)( 49, 89)( 50, 91)( 51, 90)( 52, 92)
( 53, 85)( 54, 87)( 55, 86)( 56, 88)( 57, 93)( 58, 95)( 59, 94)( 60, 96)
( 61,101)( 62,103)( 63,102)( 64,104)( 65, 97)( 66, 99)( 67, 98)( 68,100)
( 69,105)( 70,107)( 71,106)( 72,108)(110,111)(113,117)(114,119)(115,118)
(116,120)(122,123)(125,129)(126,131)(127,130)(128,132)(134,135)(137,141)
(138,143)(139,142)(140,144)(145,185)(146,187)(147,186)(148,188)(149,181)
(150,183)(151,182)(152,184)(153,189)(154,191)(155,190)(156,192)(157,197)
(158,199)(159,198)(160,200)(161,193)(162,195)(163,194)(164,196)(165,201)
(166,203)(167,202)(168,204)(169,209)(170,211)(171,210)(172,212)(173,205)
(174,207)(175,206)(176,208)(177,213)(178,215)(179,214)(180,216);
s1 := Sym(216)!( 1, 37)( 2, 38)( 3, 40)( 4, 39)( 5, 45)( 6, 46)( 7, 48)
( 8, 47)( 9, 41)( 10, 42)( 11, 44)( 12, 43)( 13, 49)( 14, 50)( 15, 52)
( 16, 51)( 17, 57)( 18, 58)( 19, 60)( 20, 59)( 21, 53)( 22, 54)( 23, 56)
( 24, 55)( 25, 61)( 26, 62)( 27, 64)( 28, 63)( 29, 69)( 30, 70)( 31, 72)
( 32, 71)( 33, 65)( 34, 66)( 35, 68)( 36, 67)( 73, 77)( 74, 78)( 75, 80)
( 76, 79)( 83, 84)( 85, 89)( 86, 90)( 87, 92)( 88, 91)( 95, 96)( 97,101)
( 98,102)( 99,104)(100,103)(107,108)(109,145)(110,146)(111,148)(112,147)
(113,153)(114,154)(115,156)(116,155)(117,149)(118,150)(119,152)(120,151)
(121,157)(122,158)(123,160)(124,159)(125,165)(126,166)(127,168)(128,167)
(129,161)(130,162)(131,164)(132,163)(133,169)(134,170)(135,172)(136,171)
(137,177)(138,178)(139,180)(140,179)(141,173)(142,174)(143,176)(144,175)
(181,185)(182,186)(183,188)(184,187)(191,192)(193,197)(194,198)(195,200)
(196,199)(203,204)(205,209)(206,210)(207,212)(208,211)(215,216);
s2 := Sym(216)!( 1, 4)( 2, 3)( 5, 8)( 6, 7)( 9, 12)( 10, 11)( 13, 28)
( 14, 27)( 15, 26)( 16, 25)( 17, 32)( 18, 31)( 19, 30)( 20, 29)( 21, 36)
( 22, 35)( 23, 34)( 24, 33)( 37, 40)( 38, 39)( 41, 44)( 42, 43)( 45, 48)
( 46, 47)( 49, 64)( 50, 63)( 51, 62)( 52, 61)( 53, 68)( 54, 67)( 55, 66)
( 56, 65)( 57, 72)( 58, 71)( 59, 70)( 60, 69)( 73, 76)( 74, 75)( 77, 80)
( 78, 79)( 81, 84)( 82, 83)( 85,100)( 86, 99)( 87, 98)( 88, 97)( 89,104)
( 90,103)( 91,102)( 92,101)( 93,108)( 94,107)( 95,106)( 96,105)(109,112)
(110,111)(113,116)(114,115)(117,120)(118,119)(121,136)(122,135)(123,134)
(124,133)(125,140)(126,139)(127,138)(128,137)(129,144)(130,143)(131,142)
(132,141)(145,148)(146,147)(149,152)(150,151)(153,156)(154,155)(157,172)
(158,171)(159,170)(160,169)(161,176)(162,175)(163,174)(164,173)(165,180)
(166,179)(167,178)(168,177)(181,184)(182,183)(185,188)(186,187)(189,192)
(190,191)(193,208)(194,207)(195,206)(196,205)(197,212)(198,211)(199,210)
(200,209)(201,216)(202,215)(203,214)(204,213);
s3 := Sym(216)!( 1,121)( 2,122)( 3,123)( 4,124)( 5,125)( 6,126)( 7,127)
( 8,128)( 9,129)( 10,130)( 11,131)( 12,132)( 13,109)( 14,110)( 15,111)
( 16,112)( 17,113)( 18,114)( 19,115)( 20,116)( 21,117)( 22,118)( 23,119)
( 24,120)( 25,133)( 26,134)( 27,135)( 28,136)( 29,137)( 30,138)( 31,139)
( 32,140)( 33,141)( 34,142)( 35,143)( 36,144)( 37,157)( 38,158)( 39,159)
( 40,160)( 41,161)( 42,162)( 43,163)( 44,164)( 45,165)( 46,166)( 47,167)
( 48,168)( 49,145)( 50,146)( 51,147)( 52,148)( 53,149)( 54,150)( 55,151)
( 56,152)( 57,153)( 58,154)( 59,155)( 60,156)( 61,169)( 62,170)( 63,171)
( 64,172)( 65,173)( 66,174)( 67,175)( 68,176)( 69,177)( 70,178)( 71,179)
( 72,180)( 73,193)( 74,194)( 75,195)( 76,196)( 77,197)( 78,198)( 79,199)
( 80,200)( 81,201)( 82,202)( 83,203)( 84,204)( 85,181)( 86,182)( 87,183)
( 88,184)( 89,185)( 90,186)( 91,187)( 92,188)( 93,189)( 94,190)( 95,191)
( 96,192)( 97,205)( 98,206)( 99,207)(100,208)(101,209)(102,210)(103,211)
(104,212)(105,213)(106,214)(107,215)(108,216);
poly := sub<Sym(216)|s0,s1,s2,s3>;
Finitely Presented Group Representation (Magma) :
poly<s0,s1,s2,s3> := Group< s0,s1,s2,s3 | s0*s0, s1*s1, s2*s2,
s3*s3, s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3,
s1*s2*s1*s2*s1*s2*s1*s2, s1*s2*s3*s2*s1*s2*s3*s2,
s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 >;
References : None.
to this polytope