Polytope of Type {2,6,9,4}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,6,9,4}*1728
if this polytope has a name.
Group : SmallGroup(1728,46115)
Rank : 5
Schlafli Type : {2,6,9,4}
Number of vertices, edges, etc : 2, 6, 54, 36, 8
Order of s0s1s2s3s4 : 18
Order of s0s1s2s3s4s3s2s1 : 2
Special Properties :
Degenerate
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 : {2,6,9,4}*864
3-fold quotients : {2,2,9,4}*576, {2,6,3,4}*576
4-fold quotients : {2,6,9,2}*432
6-fold quotients : {2,2,9,4}*288, {2,6,3,4}*288
9-fold quotients : {2,2,3,4}*192
12-fold quotients : {2,2,9,2}*144, {2,6,3,2}*144
18-fold quotients : {2,2,3,4}*96
36-fold quotients : {2,2,3,2}*48
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := (1,2);;
s1 := ( 15, 27)( 16, 28)( 17, 29)( 18, 30)( 19, 31)( 20, 32)( 21, 33)( 22, 34)
( 23, 35)( 24, 36)( 25, 37)( 26, 38)( 51, 63)( 52, 64)( 53, 65)( 54, 66)
( 55, 67)( 56, 68)( 57, 69)( 58, 70)( 59, 71)( 60, 72)( 61, 73)( 62, 74)
( 87, 99)( 88,100)( 89,101)( 90,102)( 91,103)( 92,104)( 93,105)( 94,106)
( 95,107)( 96,108)( 97,109)( 98,110)(123,135)(124,136)(125,137)(126,138)
(127,139)(128,140)(129,141)(130,142)(131,143)(132,144)(133,145)(134,146)
(159,171)(160,172)(161,173)(162,174)(163,175)(164,176)(165,177)(166,178)
(167,179)(168,180)(169,181)(170,182)(195,207)(196,208)(197,209)(198,210)
(199,211)(200,212)(201,213)(202,214)(203,215)(204,216)(205,217)(206,218);;
s2 := ( 3, 15)( 4, 17)( 5, 16)( 6, 18)( 7, 23)( 8, 25)( 9, 24)( 10, 26)
( 11, 19)( 12, 21)( 13, 20)( 14, 22)( 28, 29)( 31, 35)( 32, 37)( 33, 36)
( 34, 38)( 39, 91)( 40, 93)( 41, 92)( 42, 94)( 43, 87)( 44, 89)( 45, 88)
( 46, 90)( 47, 95)( 48, 97)( 49, 96)( 50, 98)( 51, 79)( 52, 81)( 53, 80)
( 54, 82)( 55, 75)( 56, 77)( 57, 76)( 58, 78)( 59, 83)( 60, 85)( 61, 84)
( 62, 86)( 63,103)( 64,105)( 65,104)( 66,106)( 67, 99)( 68,101)( 69,100)
( 70,102)( 71,107)( 72,109)( 73,108)( 74,110)(111,123)(112,125)(113,124)
(114,126)(115,131)(116,133)(117,132)(118,134)(119,127)(120,129)(121,128)
(122,130)(136,137)(139,143)(140,145)(141,144)(142,146)(147,199)(148,201)
(149,200)(150,202)(151,195)(152,197)(153,196)(154,198)(155,203)(156,205)
(157,204)(158,206)(159,187)(160,189)(161,188)(162,190)(163,183)(164,185)
(165,184)(166,186)(167,191)(168,193)(169,192)(170,194)(171,211)(172,213)
(173,212)(174,214)(175,207)(176,209)(177,208)(178,210)(179,215)(180,217)
(181,216)(182,218);;
s3 := ( 3, 39)( 4, 40)( 5, 42)( 6, 41)( 7, 47)( 8, 48)( 9, 50)( 10, 49)
( 11, 43)( 12, 44)( 13, 46)( 14, 45)( 15, 63)( 16, 64)( 17, 66)( 18, 65)
( 19, 71)( 20, 72)( 21, 74)( 22, 73)( 23, 67)( 24, 68)( 25, 70)( 26, 69)
( 27, 51)( 28, 52)( 29, 54)( 30, 53)( 31, 59)( 32, 60)( 33, 62)( 34, 61)
( 35, 55)( 36, 56)( 37, 58)( 38, 57)( 75, 79)( 76, 80)( 77, 82)( 78, 81)
( 85, 86)( 87,103)( 88,104)( 89,106)( 90,105)( 91, 99)( 92,100)( 93,102)
( 94,101)( 95,107)( 96,108)( 97,110)( 98,109)(111,147)(112,148)(113,150)
(114,149)(115,155)(116,156)(117,158)(118,157)(119,151)(120,152)(121,154)
(122,153)(123,171)(124,172)(125,174)(126,173)(127,179)(128,180)(129,182)
(130,181)(131,175)(132,176)(133,178)(134,177)(135,159)(136,160)(137,162)
(138,161)(139,167)(140,168)(141,170)(142,169)(143,163)(144,164)(145,166)
(146,165)(183,187)(184,188)(185,190)(186,189)(193,194)(195,211)(196,212)
(197,214)(198,213)(199,207)(200,208)(201,210)(202,209)(203,215)(204,216)
(205,218)(206,217);;
s4 := ( 3,114)( 4,113)( 5,112)( 6,111)( 7,118)( 8,117)( 9,116)( 10,115)
( 11,122)( 12,121)( 13,120)( 14,119)( 15,126)( 16,125)( 17,124)( 18,123)
( 19,130)( 20,129)( 21,128)( 22,127)( 23,134)( 24,133)( 25,132)( 26,131)
( 27,138)( 28,137)( 29,136)( 30,135)( 31,142)( 32,141)( 33,140)( 34,139)
( 35,146)( 36,145)( 37,144)( 38,143)( 39,150)( 40,149)( 41,148)( 42,147)
( 43,154)( 44,153)( 45,152)( 46,151)( 47,158)( 48,157)( 49,156)( 50,155)
( 51,162)( 52,161)( 53,160)( 54,159)( 55,166)( 56,165)( 57,164)( 58,163)
( 59,170)( 60,169)( 61,168)( 62,167)( 63,174)( 64,173)( 65,172)( 66,171)
( 67,178)( 68,177)( 69,176)( 70,175)( 71,182)( 72,181)( 73,180)( 74,179)
( 75,186)( 76,185)( 77,184)( 78,183)( 79,190)( 80,189)( 81,188)( 82,187)
( 83,194)( 84,193)( 85,192)( 86,191)( 87,198)( 88,197)( 89,196)( 90,195)
( 91,202)( 92,201)( 93,200)( 94,199)( 95,206)( 96,205)( 97,204)( 98,203)
( 99,210)(100,209)(101,208)(102,207)(103,214)(104,213)(105,212)(106,211)
(107,218)(108,217)(109,216)(110,215);;
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*s1*s0*s1,
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*s3*s4, s3*s1*s2*s1*s2*s3*s1*s2*s1*s2,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s2*s3*s4*s3*s2*s3*s2*s3*s4*s3*s2*s3,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(218)!(1,2);
s1 := Sym(218)!( 15, 27)( 16, 28)( 17, 29)( 18, 30)( 19, 31)( 20, 32)( 21, 33)
( 22, 34)( 23, 35)( 24, 36)( 25, 37)( 26, 38)( 51, 63)( 52, 64)( 53, 65)
( 54, 66)( 55, 67)( 56, 68)( 57, 69)( 58, 70)( 59, 71)( 60, 72)( 61, 73)
( 62, 74)( 87, 99)( 88,100)( 89,101)( 90,102)( 91,103)( 92,104)( 93,105)
( 94,106)( 95,107)( 96,108)( 97,109)( 98,110)(123,135)(124,136)(125,137)
(126,138)(127,139)(128,140)(129,141)(130,142)(131,143)(132,144)(133,145)
(134,146)(159,171)(160,172)(161,173)(162,174)(163,175)(164,176)(165,177)
(166,178)(167,179)(168,180)(169,181)(170,182)(195,207)(196,208)(197,209)
(198,210)(199,211)(200,212)(201,213)(202,214)(203,215)(204,216)(205,217)
(206,218);
s2 := Sym(218)!( 3, 15)( 4, 17)( 5, 16)( 6, 18)( 7, 23)( 8, 25)( 9, 24)
( 10, 26)( 11, 19)( 12, 21)( 13, 20)( 14, 22)( 28, 29)( 31, 35)( 32, 37)
( 33, 36)( 34, 38)( 39, 91)( 40, 93)( 41, 92)( 42, 94)( 43, 87)( 44, 89)
( 45, 88)( 46, 90)( 47, 95)( 48, 97)( 49, 96)( 50, 98)( 51, 79)( 52, 81)
( 53, 80)( 54, 82)( 55, 75)( 56, 77)( 57, 76)( 58, 78)( 59, 83)( 60, 85)
( 61, 84)( 62, 86)( 63,103)( 64,105)( 65,104)( 66,106)( 67, 99)( 68,101)
( 69,100)( 70,102)( 71,107)( 72,109)( 73,108)( 74,110)(111,123)(112,125)
(113,124)(114,126)(115,131)(116,133)(117,132)(118,134)(119,127)(120,129)
(121,128)(122,130)(136,137)(139,143)(140,145)(141,144)(142,146)(147,199)
(148,201)(149,200)(150,202)(151,195)(152,197)(153,196)(154,198)(155,203)
(156,205)(157,204)(158,206)(159,187)(160,189)(161,188)(162,190)(163,183)
(164,185)(165,184)(166,186)(167,191)(168,193)(169,192)(170,194)(171,211)
(172,213)(173,212)(174,214)(175,207)(176,209)(177,208)(178,210)(179,215)
(180,217)(181,216)(182,218);
s3 := Sym(218)!( 3, 39)( 4, 40)( 5, 42)( 6, 41)( 7, 47)( 8, 48)( 9, 50)
( 10, 49)( 11, 43)( 12, 44)( 13, 46)( 14, 45)( 15, 63)( 16, 64)( 17, 66)
( 18, 65)( 19, 71)( 20, 72)( 21, 74)( 22, 73)( 23, 67)( 24, 68)( 25, 70)
( 26, 69)( 27, 51)( 28, 52)( 29, 54)( 30, 53)( 31, 59)( 32, 60)( 33, 62)
( 34, 61)( 35, 55)( 36, 56)( 37, 58)( 38, 57)( 75, 79)( 76, 80)( 77, 82)
( 78, 81)( 85, 86)( 87,103)( 88,104)( 89,106)( 90,105)( 91, 99)( 92,100)
( 93,102)( 94,101)( 95,107)( 96,108)( 97,110)( 98,109)(111,147)(112,148)
(113,150)(114,149)(115,155)(116,156)(117,158)(118,157)(119,151)(120,152)
(121,154)(122,153)(123,171)(124,172)(125,174)(126,173)(127,179)(128,180)
(129,182)(130,181)(131,175)(132,176)(133,178)(134,177)(135,159)(136,160)
(137,162)(138,161)(139,167)(140,168)(141,170)(142,169)(143,163)(144,164)
(145,166)(146,165)(183,187)(184,188)(185,190)(186,189)(193,194)(195,211)
(196,212)(197,214)(198,213)(199,207)(200,208)(201,210)(202,209)(203,215)
(204,216)(205,218)(206,217);
s4 := Sym(218)!( 3,114)( 4,113)( 5,112)( 6,111)( 7,118)( 8,117)( 9,116)
( 10,115)( 11,122)( 12,121)( 13,120)( 14,119)( 15,126)( 16,125)( 17,124)
( 18,123)( 19,130)( 20,129)( 21,128)( 22,127)( 23,134)( 24,133)( 25,132)
( 26,131)( 27,138)( 28,137)( 29,136)( 30,135)( 31,142)( 32,141)( 33,140)
( 34,139)( 35,146)( 36,145)( 37,144)( 38,143)( 39,150)( 40,149)( 41,148)
( 42,147)( 43,154)( 44,153)( 45,152)( 46,151)( 47,158)( 48,157)( 49,156)
( 50,155)( 51,162)( 52,161)( 53,160)( 54,159)( 55,166)( 56,165)( 57,164)
( 58,163)( 59,170)( 60,169)( 61,168)( 62,167)( 63,174)( 64,173)( 65,172)
( 66,171)( 67,178)( 68,177)( 69,176)( 70,175)( 71,182)( 72,181)( 73,180)
( 74,179)( 75,186)( 76,185)( 77,184)( 78,183)( 79,190)( 80,189)( 81,188)
( 82,187)( 83,194)( 84,193)( 85,192)( 86,191)( 87,198)( 88,197)( 89,196)
( 90,195)( 91,202)( 92,201)( 93,200)( 94,199)( 95,206)( 96,205)( 97,204)
( 98,203)( 99,210)(100,209)(101,208)(102,207)(103,214)(104,213)(105,212)
(106,211)(107,218)(108,217)(109,216)(110,215);
poly := sub<Sym(218)|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*s1*s0*s1, 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*s3*s4,
s3*s1*s2*s1*s2*s3*s1*s2*s1*s2, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s2*s3*s4*s3*s2*s3*s2*s3*s4*s3*s2*s3,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 >;
to this polytope