Polytope of Type {2,3,6,9}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,3,6,9}*1944a
if this polytope has a name.
Group : SmallGroup(1944,949)
Rank : 5
Schlafli Type : {2,3,6,9}
Number of vertices, edges, etc : 2, 3, 27, 81, 27
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) :
3-fold quotients : {2,3,6,9}*648, {2,3,6,3}*648a
9-fold quotients : {2,3,2,9}*216, {2,3,6,3}*216
27-fold quotients : {2,3,2,3}*72
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := (1,2);;
s1 := ( 4, 5)( 6, 9)( 7, 11)( 8, 10)( 13, 14)( 15, 18)( 16, 20)( 17, 19)
( 22, 23)( 24, 27)( 25, 29)( 26, 28)( 30, 57)( 31, 59)( 32, 58)( 33, 63)
( 34, 65)( 35, 64)( 36, 60)( 37, 62)( 38, 61)( 39, 66)( 40, 68)( 41, 67)
( 42, 72)( 43, 74)( 44, 73)( 45, 69)( 46, 71)( 47, 70)( 48, 75)( 49, 77)
( 50, 76)( 51, 81)( 52, 83)( 53, 82)( 54, 78)( 55, 80)( 56, 79)( 85, 86)
( 87, 90)( 88, 92)( 89, 91)( 94, 95)( 96, 99)( 97,101)( 98,100)(103,104)
(105,108)(106,110)(107,109)(111,138)(112,140)(113,139)(114,144)(115,146)
(116,145)(117,141)(118,143)(119,142)(120,147)(121,149)(122,148)(123,153)
(124,155)(125,154)(126,150)(127,152)(128,151)(129,156)(130,158)(131,157)
(132,162)(133,164)(134,163)(135,159)(136,161)(137,160)(166,167)(168,171)
(169,173)(170,172)(175,176)(177,180)(178,182)(179,181)(184,185)(186,189)
(187,191)(188,190)(192,219)(193,221)(194,220)(195,225)(196,227)(197,226)
(198,222)(199,224)(200,223)(201,228)(202,230)(203,229)(204,234)(205,236)
(206,235)(207,231)(208,233)(209,232)(210,237)(211,239)(212,238)(213,243)
(214,245)(215,244)(216,240)(217,242)(218,241);;
s2 := ( 3, 30)( 4, 32)( 5, 31)( 6, 36)( 7, 38)( 8, 37)( 9, 33)( 10, 35)
( 11, 34)( 12, 39)( 13, 41)( 14, 40)( 15, 45)( 16, 47)( 17, 46)( 18, 42)
( 19, 44)( 20, 43)( 21, 48)( 22, 50)( 23, 49)( 24, 54)( 25, 56)( 26, 55)
( 27, 51)( 28, 53)( 29, 52)( 58, 59)( 60, 63)( 61, 65)( 62, 64)( 67, 68)
( 69, 72)( 70, 74)( 71, 73)( 76, 77)( 78, 81)( 79, 83)( 80, 82)( 84,111)
( 85,113)( 86,112)( 87,117)( 88,119)( 89,118)( 90,114)( 91,116)( 92,115)
( 93,120)( 94,122)( 95,121)( 96,126)( 97,128)( 98,127)( 99,123)(100,125)
(101,124)(102,129)(103,131)(104,130)(105,135)(106,137)(107,136)(108,132)
(109,134)(110,133)(139,140)(141,144)(142,146)(143,145)(148,149)(150,153)
(151,155)(152,154)(157,158)(159,162)(160,164)(161,163)(165,192)(166,194)
(167,193)(168,198)(169,200)(170,199)(171,195)(172,197)(173,196)(174,201)
(175,203)(176,202)(177,207)(178,209)(179,208)(180,204)(181,206)(182,205)
(183,210)(184,212)(185,211)(186,216)(187,218)(188,217)(189,213)(190,215)
(191,214)(220,221)(222,225)(223,227)(224,226)(229,230)(231,234)(232,236)
(233,235)(238,239)(240,243)(241,245)(242,244);;
s3 := ( 3, 84)( 4, 85)( 5, 86)( 6, 90)( 7, 91)( 8, 92)( 9, 87)( 10, 88)
( 11, 89)( 12,102)( 13,103)( 14,104)( 15,108)( 16,109)( 17,110)( 18,105)
( 19,106)( 20,107)( 21, 93)( 22, 94)( 23, 95)( 24, 99)( 25,100)( 26,101)
( 27, 96)( 28, 97)( 29, 98)( 30,114)( 31,115)( 32,116)( 33,111)( 34,112)
( 35,113)( 36,117)( 37,118)( 38,119)( 39,132)( 40,133)( 41,134)( 42,129)
( 43,130)( 44,131)( 45,135)( 46,136)( 47,137)( 48,123)( 49,124)( 50,125)
( 51,120)( 52,121)( 53,122)( 54,126)( 55,127)( 56,128)( 57,144)( 58,145)
( 59,146)( 60,141)( 61,142)( 62,143)( 63,138)( 64,139)( 65,140)( 66,162)
( 67,163)( 68,164)( 69,159)( 70,160)( 71,161)( 72,156)( 73,157)( 74,158)
( 75,153)( 76,154)( 77,155)( 78,150)( 79,151)( 80,152)( 81,147)( 82,148)
( 83,149)(165,183)(166,184)(167,185)(168,189)(169,190)(170,191)(171,186)
(172,187)(173,188)(177,180)(178,181)(179,182)(192,213)(193,214)(194,215)
(195,210)(196,211)(197,212)(198,216)(199,217)(200,218)(201,204)(202,205)
(203,206)(219,243)(220,244)(221,245)(222,240)(223,241)(224,242)(225,237)
(226,238)(227,239)(228,234)(229,235)(230,236);;
s4 := ( 6, 10)( 7, 11)( 8, 9)( 12, 21)( 13, 22)( 14, 23)( 15, 28)( 16, 29)
( 17, 27)( 18, 26)( 19, 24)( 20, 25)( 33, 37)( 34, 38)( 35, 36)( 39, 48)
( 40, 49)( 41, 50)( 42, 55)( 43, 56)( 44, 54)( 45, 53)( 46, 51)( 47, 52)
( 60, 64)( 61, 65)( 62, 63)( 66, 75)( 67, 76)( 68, 77)( 69, 82)( 70, 83)
( 71, 81)( 72, 80)( 73, 78)( 74, 79)( 84,183)( 85,184)( 86,185)( 87,190)
( 88,191)( 89,189)( 90,188)( 91,186)( 92,187)( 93,174)( 94,175)( 95,176)
( 96,181)( 97,182)( 98,180)( 99,179)(100,177)(101,178)(102,165)(103,166)
(104,167)(105,172)(106,173)(107,171)(108,170)(109,168)(110,169)(111,210)
(112,211)(113,212)(114,217)(115,218)(116,216)(117,215)(118,213)(119,214)
(120,201)(121,202)(122,203)(123,208)(124,209)(125,207)(126,206)(127,204)
(128,205)(129,192)(130,193)(131,194)(132,199)(133,200)(134,198)(135,197)
(136,195)(137,196)(138,237)(139,238)(140,239)(141,244)(142,245)(143,243)
(144,242)(145,240)(146,241)(147,228)(148,229)(149,230)(150,235)(151,236)
(152,234)(153,233)(154,231)(155,232)(156,219)(157,220)(158,221)(159,226)
(160,227)(161,225)(162,224)(163,222)(164,223);;
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,
s1*s2*s1*s2*s1*s2, s3*s1*s2*s3*s2*s3*s1*s2*s3*s2,
s2*s3*s4*s3*s4*s3*s2*s3*s4*s3*s4*s3,
s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(245)!(1,2);
s1 := Sym(245)!( 4, 5)( 6, 9)( 7, 11)( 8, 10)( 13, 14)( 15, 18)( 16, 20)
( 17, 19)( 22, 23)( 24, 27)( 25, 29)( 26, 28)( 30, 57)( 31, 59)( 32, 58)
( 33, 63)( 34, 65)( 35, 64)( 36, 60)( 37, 62)( 38, 61)( 39, 66)( 40, 68)
( 41, 67)( 42, 72)( 43, 74)( 44, 73)( 45, 69)( 46, 71)( 47, 70)( 48, 75)
( 49, 77)( 50, 76)( 51, 81)( 52, 83)( 53, 82)( 54, 78)( 55, 80)( 56, 79)
( 85, 86)( 87, 90)( 88, 92)( 89, 91)( 94, 95)( 96, 99)( 97,101)( 98,100)
(103,104)(105,108)(106,110)(107,109)(111,138)(112,140)(113,139)(114,144)
(115,146)(116,145)(117,141)(118,143)(119,142)(120,147)(121,149)(122,148)
(123,153)(124,155)(125,154)(126,150)(127,152)(128,151)(129,156)(130,158)
(131,157)(132,162)(133,164)(134,163)(135,159)(136,161)(137,160)(166,167)
(168,171)(169,173)(170,172)(175,176)(177,180)(178,182)(179,181)(184,185)
(186,189)(187,191)(188,190)(192,219)(193,221)(194,220)(195,225)(196,227)
(197,226)(198,222)(199,224)(200,223)(201,228)(202,230)(203,229)(204,234)
(205,236)(206,235)(207,231)(208,233)(209,232)(210,237)(211,239)(212,238)
(213,243)(214,245)(215,244)(216,240)(217,242)(218,241);
s2 := Sym(245)!( 3, 30)( 4, 32)( 5, 31)( 6, 36)( 7, 38)( 8, 37)( 9, 33)
( 10, 35)( 11, 34)( 12, 39)( 13, 41)( 14, 40)( 15, 45)( 16, 47)( 17, 46)
( 18, 42)( 19, 44)( 20, 43)( 21, 48)( 22, 50)( 23, 49)( 24, 54)( 25, 56)
( 26, 55)( 27, 51)( 28, 53)( 29, 52)( 58, 59)( 60, 63)( 61, 65)( 62, 64)
( 67, 68)( 69, 72)( 70, 74)( 71, 73)( 76, 77)( 78, 81)( 79, 83)( 80, 82)
( 84,111)( 85,113)( 86,112)( 87,117)( 88,119)( 89,118)( 90,114)( 91,116)
( 92,115)( 93,120)( 94,122)( 95,121)( 96,126)( 97,128)( 98,127)( 99,123)
(100,125)(101,124)(102,129)(103,131)(104,130)(105,135)(106,137)(107,136)
(108,132)(109,134)(110,133)(139,140)(141,144)(142,146)(143,145)(148,149)
(150,153)(151,155)(152,154)(157,158)(159,162)(160,164)(161,163)(165,192)
(166,194)(167,193)(168,198)(169,200)(170,199)(171,195)(172,197)(173,196)
(174,201)(175,203)(176,202)(177,207)(178,209)(179,208)(180,204)(181,206)
(182,205)(183,210)(184,212)(185,211)(186,216)(187,218)(188,217)(189,213)
(190,215)(191,214)(220,221)(222,225)(223,227)(224,226)(229,230)(231,234)
(232,236)(233,235)(238,239)(240,243)(241,245)(242,244);
s3 := Sym(245)!( 3, 84)( 4, 85)( 5, 86)( 6, 90)( 7, 91)( 8, 92)( 9, 87)
( 10, 88)( 11, 89)( 12,102)( 13,103)( 14,104)( 15,108)( 16,109)( 17,110)
( 18,105)( 19,106)( 20,107)( 21, 93)( 22, 94)( 23, 95)( 24, 99)( 25,100)
( 26,101)( 27, 96)( 28, 97)( 29, 98)( 30,114)( 31,115)( 32,116)( 33,111)
( 34,112)( 35,113)( 36,117)( 37,118)( 38,119)( 39,132)( 40,133)( 41,134)
( 42,129)( 43,130)( 44,131)( 45,135)( 46,136)( 47,137)( 48,123)( 49,124)
( 50,125)( 51,120)( 52,121)( 53,122)( 54,126)( 55,127)( 56,128)( 57,144)
( 58,145)( 59,146)( 60,141)( 61,142)( 62,143)( 63,138)( 64,139)( 65,140)
( 66,162)( 67,163)( 68,164)( 69,159)( 70,160)( 71,161)( 72,156)( 73,157)
( 74,158)( 75,153)( 76,154)( 77,155)( 78,150)( 79,151)( 80,152)( 81,147)
( 82,148)( 83,149)(165,183)(166,184)(167,185)(168,189)(169,190)(170,191)
(171,186)(172,187)(173,188)(177,180)(178,181)(179,182)(192,213)(193,214)
(194,215)(195,210)(196,211)(197,212)(198,216)(199,217)(200,218)(201,204)
(202,205)(203,206)(219,243)(220,244)(221,245)(222,240)(223,241)(224,242)
(225,237)(226,238)(227,239)(228,234)(229,235)(230,236);
s4 := Sym(245)!( 6, 10)( 7, 11)( 8, 9)( 12, 21)( 13, 22)( 14, 23)( 15, 28)
( 16, 29)( 17, 27)( 18, 26)( 19, 24)( 20, 25)( 33, 37)( 34, 38)( 35, 36)
( 39, 48)( 40, 49)( 41, 50)( 42, 55)( 43, 56)( 44, 54)( 45, 53)( 46, 51)
( 47, 52)( 60, 64)( 61, 65)( 62, 63)( 66, 75)( 67, 76)( 68, 77)( 69, 82)
( 70, 83)( 71, 81)( 72, 80)( 73, 78)( 74, 79)( 84,183)( 85,184)( 86,185)
( 87,190)( 88,191)( 89,189)( 90,188)( 91,186)( 92,187)( 93,174)( 94,175)
( 95,176)( 96,181)( 97,182)( 98,180)( 99,179)(100,177)(101,178)(102,165)
(103,166)(104,167)(105,172)(106,173)(107,171)(108,170)(109,168)(110,169)
(111,210)(112,211)(113,212)(114,217)(115,218)(116,216)(117,215)(118,213)
(119,214)(120,201)(121,202)(122,203)(123,208)(124,209)(125,207)(126,206)
(127,204)(128,205)(129,192)(130,193)(131,194)(132,199)(133,200)(134,198)
(135,197)(136,195)(137,196)(138,237)(139,238)(140,239)(141,244)(142,245)
(143,243)(144,242)(145,240)(146,241)(147,228)(148,229)(149,230)(150,235)
(151,236)(152,234)(153,233)(154,231)(155,232)(156,219)(157,220)(158,221)
(159,226)(160,227)(161,225)(162,224)(163,222)(164,223);
poly := sub<Sym(245)|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, s1*s2*s1*s2*s1*s2,
s3*s1*s2*s3*s2*s3*s1*s2*s3*s2, s2*s3*s4*s3*s4*s3*s2*s3*s4*s3*s4*s3,
s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4 >;
to this polytope