Polytope of Type {2,9,6,9}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,9,6,9}*1944
if this polytope has a name.
Group : SmallGroup(1944,940)
Rank : 5
Schlafli Type : {2,9,6,9}
Number of vertices, edges, etc : 2, 9, 27, 27, 9
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 : {2,9,2,9}*648, {2,3,6,9}*648, {2,9,6,3}*648
9-fold quotients : {2,3,2,9}*216, {2,9,2,3}*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, 63)( 31, 65)( 32, 64)( 33, 60)
( 34, 62)( 35, 61)( 36, 57)( 37, 59)( 38, 58)( 39, 72)( 40, 74)( 41, 73)
( 42, 69)( 43, 71)( 44, 70)( 45, 66)( 46, 68)( 47, 67)( 48, 81)( 49, 83)
( 50, 82)( 51, 78)( 52, 80)( 53, 79)( 54, 75)( 55, 77)( 56, 76)( 85, 86)
( 87, 90)( 88, 92)( 89, 91)( 94, 95)( 96, 99)( 97,101)( 98,100)(103,104)
(105,108)(106,110)(107,109)(111,144)(112,146)(113,145)(114,141)(115,143)
(116,142)(117,138)(118,140)(119,139)(120,153)(121,155)(122,154)(123,150)
(124,152)(125,151)(126,147)(127,149)(128,148)(129,162)(130,164)(131,163)
(132,159)(133,161)(134,160)(135,156)(136,158)(137,157)(166,167)(168,171)
(169,173)(170,172)(175,176)(177,180)(178,182)(179,181)(184,185)(186,189)
(187,191)(188,190)(192,225)(193,227)(194,226)(195,222)(196,224)(197,223)
(198,219)(199,221)(200,220)(201,234)(202,236)(203,235)(204,231)(205,233)
(206,232)(207,228)(208,230)(209,229)(210,243)(211,245)(212,244)(213,240)
(214,242)(215,241)(216,237)(217,239)(218,238);;
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)( 57, 63)( 58, 65)( 59, 64)( 61, 62)( 66, 72)
( 67, 74)( 68, 73)( 70, 71)( 75, 81)( 76, 83)( 77, 82)( 79, 80)( 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)(138,144)(139,146)(140,145)(142,143)(147,153)(148,155)
(149,154)(151,152)(156,162)(157,164)(158,163)(160,161)(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)(219,225)(220,227)(221,226)(223,224)(228,234)(229,236)(230,235)
(232,233)(237,243)(238,245)(239,244)(241,242);;
s3 := ( 3, 84)( 4, 86)( 5, 85)( 6, 87)( 7, 89)( 8, 88)( 9, 90)( 10, 92)
( 11, 91)( 12,102)( 13,104)( 14,103)( 15,105)( 16,107)( 17,106)( 18,108)
( 19,110)( 20,109)( 21, 93)( 22, 95)( 23, 94)( 24, 96)( 25, 98)( 26, 97)
( 27, 99)( 28,101)( 29,100)( 30,112)( 31,111)( 32,113)( 33,115)( 34,114)
( 35,116)( 36,118)( 37,117)( 38,119)( 39,130)( 40,129)( 41,131)( 42,133)
( 43,132)( 44,134)( 45,136)( 46,135)( 47,137)( 48,121)( 49,120)( 50,122)
( 51,124)( 52,123)( 53,125)( 54,127)( 55,126)( 56,128)( 57,140)( 58,139)
( 59,138)( 60,143)( 61,142)( 62,141)( 63,146)( 64,145)( 65,144)( 66,158)
( 67,157)( 68,156)( 69,161)( 70,160)( 71,159)( 72,164)( 73,163)( 74,162)
( 75,149)( 76,148)( 77,147)( 78,152)( 79,151)( 80,150)( 81,155)( 82,154)
( 83,153)(165,183)(166,185)(167,184)(168,186)(169,188)(170,187)(171,189)
(172,191)(173,190)(175,176)(178,179)(181,182)(192,211)(193,210)(194,212)
(195,214)(196,213)(197,215)(198,217)(199,216)(200,218)(201,202)(204,205)
(207,208)(219,239)(220,238)(221,237)(222,242)(223,241)(224,240)(225,245)
(226,244)(227,243)(228,230)(231,233)(234,236);;
s4 := ( 4, 5)( 7, 8)( 10, 11)( 12, 21)( 13, 23)( 14, 22)( 15, 24)( 16, 26)
( 17, 25)( 18, 27)( 19, 29)( 20, 28)( 31, 32)( 34, 35)( 37, 38)( 39, 48)
( 40, 50)( 41, 49)( 42, 51)( 43, 53)( 44, 52)( 45, 54)( 46, 56)( 47, 55)
( 58, 59)( 61, 62)( 64, 65)( 66, 75)( 67, 77)( 68, 76)( 69, 78)( 70, 80)
( 71, 79)( 72, 81)( 73, 83)( 74, 82)( 84,183)( 85,185)( 86,184)( 87,186)
( 88,188)( 89,187)( 90,189)( 91,191)( 92,190)( 93,174)( 94,176)( 95,175)
( 96,177)( 97,179)( 98,178)( 99,180)(100,182)(101,181)(102,165)(103,167)
(104,166)(105,168)(106,170)(107,169)(108,171)(109,173)(110,172)(111,210)
(112,212)(113,211)(114,213)(115,215)(116,214)(117,216)(118,218)(119,217)
(120,201)(121,203)(122,202)(123,204)(124,206)(125,205)(126,207)(127,209)
(128,208)(129,192)(130,194)(131,193)(132,195)(133,197)(134,196)(135,198)
(136,200)(137,199)(138,237)(139,239)(140,238)(141,240)(142,242)(143,241)
(144,243)(145,245)(146,244)(147,228)(148,230)(149,229)(150,231)(151,233)
(152,232)(153,234)(154,236)(155,235)(156,219)(157,221)(158,220)(159,222)
(160,224)(161,223)(162,225)(163,227)(164,226);;
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*s1*s2*s3*s2*s3*s1*s2*s3*s2, s4*s2*s3*s2*s3*s4*s2*s3*s2*s3,
s1*s2*s3*s2*s1*s2*s1*s2*s3*s2*s1*s2,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
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, 63)( 31, 65)( 32, 64)
( 33, 60)( 34, 62)( 35, 61)( 36, 57)( 37, 59)( 38, 58)( 39, 72)( 40, 74)
( 41, 73)( 42, 69)( 43, 71)( 44, 70)( 45, 66)( 46, 68)( 47, 67)( 48, 81)
( 49, 83)( 50, 82)( 51, 78)( 52, 80)( 53, 79)( 54, 75)( 55, 77)( 56, 76)
( 85, 86)( 87, 90)( 88, 92)( 89, 91)( 94, 95)( 96, 99)( 97,101)( 98,100)
(103,104)(105,108)(106,110)(107,109)(111,144)(112,146)(113,145)(114,141)
(115,143)(116,142)(117,138)(118,140)(119,139)(120,153)(121,155)(122,154)
(123,150)(124,152)(125,151)(126,147)(127,149)(128,148)(129,162)(130,164)
(131,163)(132,159)(133,161)(134,160)(135,156)(136,158)(137,157)(166,167)
(168,171)(169,173)(170,172)(175,176)(177,180)(178,182)(179,181)(184,185)
(186,189)(187,191)(188,190)(192,225)(193,227)(194,226)(195,222)(196,224)
(197,223)(198,219)(199,221)(200,220)(201,234)(202,236)(203,235)(204,231)
(205,233)(206,232)(207,228)(208,230)(209,229)(210,243)(211,245)(212,244)
(213,240)(214,242)(215,241)(216,237)(217,239)(218,238);
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)( 57, 63)( 58, 65)( 59, 64)( 61, 62)
( 66, 72)( 67, 74)( 68, 73)( 70, 71)( 75, 81)( 76, 83)( 77, 82)( 79, 80)
( 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)(138,144)(139,146)(140,145)(142,143)(147,153)
(148,155)(149,154)(151,152)(156,162)(157,164)(158,163)(160,161)(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)(219,225)(220,227)(221,226)(223,224)(228,234)(229,236)
(230,235)(232,233)(237,243)(238,245)(239,244)(241,242);
s3 := Sym(245)!( 3, 84)( 4, 86)( 5, 85)( 6, 87)( 7, 89)( 8, 88)( 9, 90)
( 10, 92)( 11, 91)( 12,102)( 13,104)( 14,103)( 15,105)( 16,107)( 17,106)
( 18,108)( 19,110)( 20,109)( 21, 93)( 22, 95)( 23, 94)( 24, 96)( 25, 98)
( 26, 97)( 27, 99)( 28,101)( 29,100)( 30,112)( 31,111)( 32,113)( 33,115)
( 34,114)( 35,116)( 36,118)( 37,117)( 38,119)( 39,130)( 40,129)( 41,131)
( 42,133)( 43,132)( 44,134)( 45,136)( 46,135)( 47,137)( 48,121)( 49,120)
( 50,122)( 51,124)( 52,123)( 53,125)( 54,127)( 55,126)( 56,128)( 57,140)
( 58,139)( 59,138)( 60,143)( 61,142)( 62,141)( 63,146)( 64,145)( 65,144)
( 66,158)( 67,157)( 68,156)( 69,161)( 70,160)( 71,159)( 72,164)( 73,163)
( 74,162)( 75,149)( 76,148)( 77,147)( 78,152)( 79,151)( 80,150)( 81,155)
( 82,154)( 83,153)(165,183)(166,185)(167,184)(168,186)(169,188)(170,187)
(171,189)(172,191)(173,190)(175,176)(178,179)(181,182)(192,211)(193,210)
(194,212)(195,214)(196,213)(197,215)(198,217)(199,216)(200,218)(201,202)
(204,205)(207,208)(219,239)(220,238)(221,237)(222,242)(223,241)(224,240)
(225,245)(226,244)(227,243)(228,230)(231,233)(234,236);
s4 := Sym(245)!( 4, 5)( 7, 8)( 10, 11)( 12, 21)( 13, 23)( 14, 22)( 15, 24)
( 16, 26)( 17, 25)( 18, 27)( 19, 29)( 20, 28)( 31, 32)( 34, 35)( 37, 38)
( 39, 48)( 40, 50)( 41, 49)( 42, 51)( 43, 53)( 44, 52)( 45, 54)( 46, 56)
( 47, 55)( 58, 59)( 61, 62)( 64, 65)( 66, 75)( 67, 77)( 68, 76)( 69, 78)
( 70, 80)( 71, 79)( 72, 81)( 73, 83)( 74, 82)( 84,183)( 85,185)( 86,184)
( 87,186)( 88,188)( 89,187)( 90,189)( 91,191)( 92,190)( 93,174)( 94,176)
( 95,175)( 96,177)( 97,179)( 98,178)( 99,180)(100,182)(101,181)(102,165)
(103,167)(104,166)(105,168)(106,170)(107,169)(108,171)(109,173)(110,172)
(111,210)(112,212)(113,211)(114,213)(115,215)(116,214)(117,216)(118,218)
(119,217)(120,201)(121,203)(122,202)(123,204)(124,206)(125,205)(126,207)
(127,209)(128,208)(129,192)(130,194)(131,193)(132,195)(133,197)(134,196)
(135,198)(136,200)(137,199)(138,237)(139,239)(140,238)(141,240)(142,242)
(143,241)(144,243)(145,245)(146,244)(147,228)(148,230)(149,229)(150,231)
(151,233)(152,232)(153,234)(154,236)(155,235)(156,219)(157,221)(158,220)
(159,222)(160,224)(161,223)(162,225)(163,227)(164,226);
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, s3*s1*s2*s3*s2*s3*s1*s2*s3*s2,
s4*s2*s3*s2*s3*s4*s2*s3*s2*s3, s1*s2*s3*s2*s1*s2*s1*s2*s3*s2*s1*s2,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4 >;
to this polytope