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}*1944b
if this polytope has a name.
Group : SmallGroup(1944,952)
Rank : 5
Schlafli Type : {2,3,6,9}
Number of vertices, edges, etc : 2, 9, 27, 81, 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 : {2,3,6,9}*648, {2,3,6,3}*648b
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 := ( 6, 10)( 7, 11)( 8, 9)( 15, 19)( 16, 20)( 17, 18)( 24, 28)( 25, 29)
( 26, 27)( 30, 57)( 31, 58)( 32, 59)( 33, 64)( 34, 65)( 35, 63)( 36, 62)
( 37, 60)( 38, 61)( 39, 66)( 40, 67)( 41, 68)( 42, 73)( 43, 74)( 44, 72)
( 45, 71)( 46, 69)( 47, 70)( 48, 75)( 49, 76)( 50, 77)( 51, 82)( 52, 83)
( 53, 81)( 54, 80)( 55, 78)( 56, 79)( 87, 91)( 88, 92)( 89, 90)( 96,100)
( 97,101)( 98, 99)(105,109)(106,110)(107,108)(111,138)(112,139)(113,140)
(114,145)(115,146)(116,144)(117,143)(118,141)(119,142)(120,147)(121,148)
(122,149)(123,154)(124,155)(125,153)(126,152)(127,150)(128,151)(129,156)
(130,157)(131,158)(132,163)(133,164)(134,162)(135,161)(136,159)(137,160)
(168,172)(169,173)(170,171)(177,181)(178,182)(179,180)(186,190)(187,191)
(188,189)(192,219)(193,220)(194,221)(195,226)(196,227)(197,225)(198,224)
(199,222)(200,223)(201,228)(202,229)(203,230)(204,235)(205,236)(206,234)
(207,233)(208,231)(209,232)(210,237)(211,238)(212,239)(213,244)(214,245)
(215,243)(216,242)(217,240)(218,241);;
s2 := ( 3, 30)( 4, 31)( 5, 32)( 6, 36)( 7, 37)( 8, 38)( 9, 33)( 10, 34)
( 11, 35)( 12, 39)( 13, 40)( 14, 41)( 15, 45)( 16, 46)( 17, 47)( 18, 42)
( 19, 43)( 20, 44)( 21, 48)( 22, 49)( 23, 50)( 24, 54)( 25, 55)( 26, 56)
( 27, 51)( 28, 52)( 29, 53)( 60, 63)( 61, 64)( 62, 65)( 69, 72)( 70, 73)
( 71, 74)( 78, 81)( 79, 82)( 80, 83)( 84,111)( 85,112)( 86,113)( 87,117)
( 88,118)( 89,119)( 90,114)( 91,115)( 92,116)( 93,120)( 94,121)( 95,122)
( 96,126)( 97,127)( 98,128)( 99,123)(100,124)(101,125)(102,129)(103,130)
(104,131)(105,135)(106,136)(107,137)(108,132)(109,133)(110,134)(141,144)
(142,145)(143,146)(150,153)(151,154)(152,155)(159,162)(160,163)(161,164)
(165,192)(166,193)(167,194)(168,198)(169,199)(170,200)(171,195)(172,196)
(173,197)(174,201)(175,202)(176,203)(177,207)(178,208)(179,209)(180,204)
(181,205)(182,206)(183,210)(184,211)(185,212)(186,216)(187,217)(188,218)
(189,213)(190,214)(191,215)(222,225)(223,226)(224,227)(231,234)(232,235)
(233,236)(240,243)(241,244)(242,245);;
s3 := ( 3, 84)( 4, 86)( 5, 85)( 6, 90)( 7, 92)( 8, 91)( 9, 87)( 10, 89)
( 11, 88)( 12,102)( 13,104)( 14,103)( 15,108)( 16,110)( 17,109)( 18,105)
( 19,107)( 20,106)( 21, 93)( 22, 95)( 23, 94)( 24, 99)( 25,101)( 26,100)
( 27, 96)( 28, 98)( 29, 97)( 30,114)( 31,116)( 32,115)( 33,111)( 34,113)
( 35,112)( 36,117)( 37,119)( 38,118)( 39,132)( 40,134)( 41,133)( 42,129)
( 43,131)( 44,130)( 45,135)( 46,137)( 47,136)( 48,123)( 49,125)( 50,124)
( 51,120)( 52,122)( 53,121)( 54,126)( 55,128)( 56,127)( 57,145)( 58,144)
( 59,146)( 60,142)( 61,141)( 62,143)( 63,139)( 64,138)( 65,140)( 66,163)
( 67,162)( 68,164)( 69,160)( 70,159)( 71,161)( 72,157)( 73,156)( 74,158)
( 75,154)( 76,153)( 77,155)( 78,151)( 79,150)( 80,152)( 81,148)( 82,147)
( 83,149)(165,183)(166,185)(167,184)(168,189)(169,191)(170,190)(171,186)
(172,188)(173,187)(175,176)(177,180)(178,182)(179,181)(192,213)(193,215)
(194,214)(195,210)(196,212)(197,211)(198,216)(199,218)(200,217)(201,204)
(202,206)(203,205)(208,209)(219,244)(220,243)(221,245)(222,241)(223,240)
(224,242)(225,238)(226,237)(227,239)(228,235)(229,234)(230,236)(231,232);;
s4 := ( 4, 5)( 6, 9)( 7, 11)( 8, 10)( 12, 21)( 13, 23)( 14, 22)( 15, 27)
( 16, 29)( 17, 28)( 18, 24)( 19, 26)( 20, 25)( 31, 32)( 33, 36)( 34, 38)
( 35, 37)( 39, 48)( 40, 50)( 41, 49)( 42, 54)( 43, 56)( 44, 55)( 45, 51)
( 46, 53)( 47, 52)( 58, 59)( 60, 63)( 61, 65)( 62, 64)( 66, 75)( 67, 77)
( 68, 76)( 69, 81)( 70, 83)( 71, 82)( 72, 78)( 73, 80)( 74, 79)( 84,183)
( 85,185)( 86,184)( 87,189)( 88,191)( 89,190)( 90,186)( 91,188)( 92,187)
( 93,174)( 94,176)( 95,175)( 96,180)( 97,182)( 98,181)( 99,177)(100,179)
(101,178)(102,165)(103,167)(104,166)(105,171)(106,173)(107,172)(108,168)
(109,170)(110,169)(111,210)(112,212)(113,211)(114,216)(115,218)(116,217)
(117,213)(118,215)(119,214)(120,201)(121,203)(122,202)(123,207)(124,209)
(125,208)(126,204)(127,206)(128,205)(129,192)(130,194)(131,193)(132,198)
(133,200)(134,199)(135,195)(136,197)(137,196)(138,237)(139,239)(140,238)
(141,243)(142,245)(143,244)(144,240)(145,242)(146,241)(147,228)(148,230)
(149,229)(150,234)(151,236)(152,235)(153,231)(154,233)(155,232)(156,219)
(157,221)(158,220)(159,225)(160,227)(161,226)(162,222)(163,224)(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, s4*s2*s3*s2*s3*s4*s2*s3*s2*s3,
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*s3*s4 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(245)!(1,2);
s1 := Sym(245)!( 6, 10)( 7, 11)( 8, 9)( 15, 19)( 16, 20)( 17, 18)( 24, 28)
( 25, 29)( 26, 27)( 30, 57)( 31, 58)( 32, 59)( 33, 64)( 34, 65)( 35, 63)
( 36, 62)( 37, 60)( 38, 61)( 39, 66)( 40, 67)( 41, 68)( 42, 73)( 43, 74)
( 44, 72)( 45, 71)( 46, 69)( 47, 70)( 48, 75)( 49, 76)( 50, 77)( 51, 82)
( 52, 83)( 53, 81)( 54, 80)( 55, 78)( 56, 79)( 87, 91)( 88, 92)( 89, 90)
( 96,100)( 97,101)( 98, 99)(105,109)(106,110)(107,108)(111,138)(112,139)
(113,140)(114,145)(115,146)(116,144)(117,143)(118,141)(119,142)(120,147)
(121,148)(122,149)(123,154)(124,155)(125,153)(126,152)(127,150)(128,151)
(129,156)(130,157)(131,158)(132,163)(133,164)(134,162)(135,161)(136,159)
(137,160)(168,172)(169,173)(170,171)(177,181)(178,182)(179,180)(186,190)
(187,191)(188,189)(192,219)(193,220)(194,221)(195,226)(196,227)(197,225)
(198,224)(199,222)(200,223)(201,228)(202,229)(203,230)(204,235)(205,236)
(206,234)(207,233)(208,231)(209,232)(210,237)(211,238)(212,239)(213,244)
(214,245)(215,243)(216,242)(217,240)(218,241);
s2 := Sym(245)!( 3, 30)( 4, 31)( 5, 32)( 6, 36)( 7, 37)( 8, 38)( 9, 33)
( 10, 34)( 11, 35)( 12, 39)( 13, 40)( 14, 41)( 15, 45)( 16, 46)( 17, 47)
( 18, 42)( 19, 43)( 20, 44)( 21, 48)( 22, 49)( 23, 50)( 24, 54)( 25, 55)
( 26, 56)( 27, 51)( 28, 52)( 29, 53)( 60, 63)( 61, 64)( 62, 65)( 69, 72)
( 70, 73)( 71, 74)( 78, 81)( 79, 82)( 80, 83)( 84,111)( 85,112)( 86,113)
( 87,117)( 88,118)( 89,119)( 90,114)( 91,115)( 92,116)( 93,120)( 94,121)
( 95,122)( 96,126)( 97,127)( 98,128)( 99,123)(100,124)(101,125)(102,129)
(103,130)(104,131)(105,135)(106,136)(107,137)(108,132)(109,133)(110,134)
(141,144)(142,145)(143,146)(150,153)(151,154)(152,155)(159,162)(160,163)
(161,164)(165,192)(166,193)(167,194)(168,198)(169,199)(170,200)(171,195)
(172,196)(173,197)(174,201)(175,202)(176,203)(177,207)(178,208)(179,209)
(180,204)(181,205)(182,206)(183,210)(184,211)(185,212)(186,216)(187,217)
(188,218)(189,213)(190,214)(191,215)(222,225)(223,226)(224,227)(231,234)
(232,235)(233,236)(240,243)(241,244)(242,245);
s3 := Sym(245)!( 3, 84)( 4, 86)( 5, 85)( 6, 90)( 7, 92)( 8, 91)( 9, 87)
( 10, 89)( 11, 88)( 12,102)( 13,104)( 14,103)( 15,108)( 16,110)( 17,109)
( 18,105)( 19,107)( 20,106)( 21, 93)( 22, 95)( 23, 94)( 24, 99)( 25,101)
( 26,100)( 27, 96)( 28, 98)( 29, 97)( 30,114)( 31,116)( 32,115)( 33,111)
( 34,113)( 35,112)( 36,117)( 37,119)( 38,118)( 39,132)( 40,134)( 41,133)
( 42,129)( 43,131)( 44,130)( 45,135)( 46,137)( 47,136)( 48,123)( 49,125)
( 50,124)( 51,120)( 52,122)( 53,121)( 54,126)( 55,128)( 56,127)( 57,145)
( 58,144)( 59,146)( 60,142)( 61,141)( 62,143)( 63,139)( 64,138)( 65,140)
( 66,163)( 67,162)( 68,164)( 69,160)( 70,159)( 71,161)( 72,157)( 73,156)
( 74,158)( 75,154)( 76,153)( 77,155)( 78,151)( 79,150)( 80,152)( 81,148)
( 82,147)( 83,149)(165,183)(166,185)(167,184)(168,189)(169,191)(170,190)
(171,186)(172,188)(173,187)(175,176)(177,180)(178,182)(179,181)(192,213)
(193,215)(194,214)(195,210)(196,212)(197,211)(198,216)(199,218)(200,217)
(201,204)(202,206)(203,205)(208,209)(219,244)(220,243)(221,245)(222,241)
(223,240)(224,242)(225,238)(226,237)(227,239)(228,235)(229,234)(230,236)
(231,232);
s4 := Sym(245)!( 4, 5)( 6, 9)( 7, 11)( 8, 10)( 12, 21)( 13, 23)( 14, 22)
( 15, 27)( 16, 29)( 17, 28)( 18, 24)( 19, 26)( 20, 25)( 31, 32)( 33, 36)
( 34, 38)( 35, 37)( 39, 48)( 40, 50)( 41, 49)( 42, 54)( 43, 56)( 44, 55)
( 45, 51)( 46, 53)( 47, 52)( 58, 59)( 60, 63)( 61, 65)( 62, 64)( 66, 75)
( 67, 77)( 68, 76)( 69, 81)( 70, 83)( 71, 82)( 72, 78)( 73, 80)( 74, 79)
( 84,183)( 85,185)( 86,184)( 87,189)( 88,191)( 89,190)( 90,186)( 91,188)
( 92,187)( 93,174)( 94,176)( 95,175)( 96,180)( 97,182)( 98,181)( 99,177)
(100,179)(101,178)(102,165)(103,167)(104,166)(105,171)(106,173)(107,172)
(108,168)(109,170)(110,169)(111,210)(112,212)(113,211)(114,216)(115,218)
(116,217)(117,213)(118,215)(119,214)(120,201)(121,203)(122,202)(123,207)
(124,209)(125,208)(126,204)(127,206)(128,205)(129,192)(130,194)(131,193)
(132,198)(133,200)(134,199)(135,195)(136,197)(137,196)(138,237)(139,239)
(140,238)(141,243)(142,245)(143,244)(144,240)(145,242)(146,241)(147,228)
(148,230)(149,229)(150,234)(151,236)(152,235)(153,231)(154,233)(155,232)
(156,219)(157,221)(158,220)(159,225)(160,227)(161,226)(162,222)(163,224)
(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,
s4*s2*s3*s2*s3*s4*s2*s3*s2*s3, 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*s3*s4 >;
to this polytope