Polytope of Type {2,18,18}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,18,18}*1944h
if this polytope has a name.
Group : SmallGroup(1944,951)
Rank : 4
Schlafli Type : {2,18,18}
Number of vertices, edges, etc : 2, 27, 243, 27
Order of s0s1s2s3 : 18
Order of s0s1s2s3s2s1 : 2
Special Properties :
Degenerate
Universal
Non-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 : {2,18,6}*648a, {2,6,18}*648c
9-fold quotients : {2,6,6}*216
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := (1,2);;
s1 := ( 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);;
s2 := ( 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)( 30,145)( 31,144)( 32,146)( 33,139)( 34,138)
( 35,140)( 36,142)( 37,141)( 38,143)( 39,163)( 40,162)( 41,164)( 42,157)
( 43,156)( 44,158)( 45,160)( 46,159)( 47,161)( 48,154)( 49,153)( 50,155)
( 51,148)( 52,147)( 53,149)( 54,151)( 55,150)( 56,152)( 57,115)( 58,114)
( 59,116)( 60,118)( 61,117)( 62,119)( 63,112)( 64,111)( 65,113)( 66,133)
( 67,132)( 68,134)( 69,136)( 70,135)( 71,137)( 72,130)( 73,129)( 74,131)
( 75,124)( 76,123)( 77,125)( 78,127)( 79,126)( 80,128)( 81,121)( 82,120)
( 83,122)(165,183)(166,185)(167,184)(168,186)(169,188)(170,187)(171,189)
(172,191)(173,190)(175,176)(178,179)(181,182)(192,244)(193,243)(194,245)
(195,238)(196,237)(197,239)(198,241)(199,240)(200,242)(201,235)(202,234)
(203,236)(204,229)(205,228)(206,230)(207,232)(208,231)(209,233)(210,226)
(211,225)(212,227)(213,220)(214,219)(215,221)(216,223)(217,222)(218,224);;
s3 := ( 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, 58)( 60, 64)( 61, 63)( 62, 65)( 66, 67)
( 69, 73)( 70, 72)( 71, 74)( 75, 76)( 78, 82)( 79, 81)( 80, 83)( 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)(138,139)(141,145)(142,144)(143,146)(147,148)(150,154)
(151,153)(152,155)(156,157)(159,163)(160,162)(161,164)(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)(219,220)(222,226)(223,225)(224,227)(228,229)(231,235)(232,234)
(233,236)(237,238)(240,244)(241,243)(242,245);;
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*s1*s0*s1,
s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3,
s3*s1*s2*s3*s2*s3*s1*s2*s3*s1*s2*s3*s2*s3*s1*s2,
s1*s2*s3*s2*s3*s1*s2*s1*s2*s3*s1*s2*s3*s2*s1*s2*s1*s2,
s1*s3*s2*s1*s3*s2*s1*s3*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s1*s2*s3*s2*s1*s2*s3*s2*s1*s2*s3*s2*s1*s2*s3*s2*s1*s2*s3*s2*s1*s2*s3*s2 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(245)!(1,2);
s1 := 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);
s2 := 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)( 30,145)( 31,144)( 32,146)( 33,139)
( 34,138)( 35,140)( 36,142)( 37,141)( 38,143)( 39,163)( 40,162)( 41,164)
( 42,157)( 43,156)( 44,158)( 45,160)( 46,159)( 47,161)( 48,154)( 49,153)
( 50,155)( 51,148)( 52,147)( 53,149)( 54,151)( 55,150)( 56,152)( 57,115)
( 58,114)( 59,116)( 60,118)( 61,117)( 62,119)( 63,112)( 64,111)( 65,113)
( 66,133)( 67,132)( 68,134)( 69,136)( 70,135)( 71,137)( 72,130)( 73,129)
( 74,131)( 75,124)( 76,123)( 77,125)( 78,127)( 79,126)( 80,128)( 81,121)
( 82,120)( 83,122)(165,183)(166,185)(167,184)(168,186)(169,188)(170,187)
(171,189)(172,191)(173,190)(175,176)(178,179)(181,182)(192,244)(193,243)
(194,245)(195,238)(196,237)(197,239)(198,241)(199,240)(200,242)(201,235)
(202,234)(203,236)(204,229)(205,228)(206,230)(207,232)(208,231)(209,233)
(210,226)(211,225)(212,227)(213,220)(214,219)(215,221)(216,223)(217,222)
(218,224);
s3 := 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, 58)( 60, 64)( 61, 63)( 62, 65)
( 66, 67)( 69, 73)( 70, 72)( 71, 74)( 75, 76)( 78, 82)( 79, 81)( 80, 83)
( 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)(138,139)(141,145)(142,144)(143,146)(147,148)
(150,154)(151,153)(152,155)(156,157)(159,163)(160,162)(161,164)(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)(219,220)(222,226)(223,225)(224,227)(228,229)(231,235)
(232,234)(233,236)(237,238)(240,244)(241,243)(242,245);
poly := sub<Sym(245)|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*s1*s0*s1, s0*s2*s0*s2, s0*s3*s0*s3,
s1*s3*s1*s3, s3*s1*s2*s3*s2*s3*s1*s2*s3*s1*s2*s3*s2*s3*s1*s2,
s1*s2*s3*s2*s3*s1*s2*s1*s2*s3*s1*s2*s3*s2*s1*s2*s1*s2,
s1*s3*s2*s1*s3*s2*s1*s3*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s1*s2*s3*s2*s1*s2*s3*s2*s1*s2*s3*s2*s1*s2*s3*s2*s1*s2*s3*s2*s1*s2*s3*s2 >;
to this polytope