Polytope of Type {2,18,6}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,18,6}*1944d
if this polytope has a name.
Group : SmallGroup(1944,952)
Rank : 4
Schlafli Type : {2,18,6}
Number of vertices, edges, etc : 2, 81, 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, {2,6,6}*648a, {2,18,6}*648b, {2,18,6}*648c
9-fold quotients : {2,6,6}*216
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := (1,2);;
s1 := ( 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);;
s2 := ( 3, 84)( 4, 86)( 5, 85)( 6, 89)( 7, 88)( 8, 87)( 9, 91)( 10, 90)
( 11, 92)( 12,102)( 13,104)( 14,103)( 15,107)( 16,106)( 17,105)( 18,109)
( 19,108)( 20,110)( 21, 93)( 22, 95)( 23, 94)( 24, 98)( 25, 97)( 26, 96)
( 27,100)( 28, 99)( 29,101)( 30,145)( 31,144)( 32,146)( 33,138)( 34,140)
( 35,139)( 36,143)( 37,142)( 38,141)( 39,163)( 40,162)( 41,164)( 42,156)
( 43,158)( 44,157)( 45,161)( 46,160)( 47,159)( 48,154)( 49,153)( 50,155)
( 51,147)( 52,149)( 53,148)( 54,152)( 55,151)( 56,150)( 57,114)( 58,116)
( 59,115)( 60,119)( 61,118)( 62,117)( 63,112)( 64,111)( 65,113)( 66,132)
( 67,134)( 68,133)( 69,137)( 70,136)( 71,135)( 72,130)( 73,129)( 74,131)
( 75,123)( 76,125)( 77,124)( 78,128)( 79,127)( 80,126)( 81,121)( 82,120)
( 83,122)(165,183)(166,185)(167,184)(168,188)(169,187)(170,186)(171,190)
(172,189)(173,191)(175,176)(177,179)(180,181)(192,244)(193,243)(194,245)
(195,237)(196,239)(197,238)(198,242)(199,241)(200,240)(201,235)(202,234)
(203,236)(204,228)(205,230)(206,229)(207,233)(208,232)(209,231)(210,226)
(211,225)(212,227)(213,219)(214,221)(215,220)(216,224)(217,223)(218,222);;
s3 := ( 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);;
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,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3,
s3*s1*s2*s3*s2*s3*s1*s2*s3*s1*s2*s3*s2*s3*s1*s2,
s1*s2*s1*s2*s1*s2*s3*s1*s2*s1*s2*s1*s2*s3*s1*s2*s1*s2*s1*s2*s3,
s1*s2*s3*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s3*s2*s1*s2*s1*s2*s1*s2*s1*s2 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(245)!(1,2);
s1 := 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);
s2 := Sym(245)!( 3, 84)( 4, 86)( 5, 85)( 6, 89)( 7, 88)( 8, 87)( 9, 91)
( 10, 90)( 11, 92)( 12,102)( 13,104)( 14,103)( 15,107)( 16,106)( 17,105)
( 18,109)( 19,108)( 20,110)( 21, 93)( 22, 95)( 23, 94)( 24, 98)( 25, 97)
( 26, 96)( 27,100)( 28, 99)( 29,101)( 30,145)( 31,144)( 32,146)( 33,138)
( 34,140)( 35,139)( 36,143)( 37,142)( 38,141)( 39,163)( 40,162)( 41,164)
( 42,156)( 43,158)( 44,157)( 45,161)( 46,160)( 47,159)( 48,154)( 49,153)
( 50,155)( 51,147)( 52,149)( 53,148)( 54,152)( 55,151)( 56,150)( 57,114)
( 58,116)( 59,115)( 60,119)( 61,118)( 62,117)( 63,112)( 64,111)( 65,113)
( 66,132)( 67,134)( 68,133)( 69,137)( 70,136)( 71,135)( 72,130)( 73,129)
( 74,131)( 75,123)( 76,125)( 77,124)( 78,128)( 79,127)( 80,126)( 81,121)
( 82,120)( 83,122)(165,183)(166,185)(167,184)(168,188)(169,187)(170,186)
(171,190)(172,189)(173,191)(175,176)(177,179)(180,181)(192,244)(193,243)
(194,245)(195,237)(196,239)(197,238)(198,242)(199,241)(200,240)(201,235)
(202,234)(203,236)(204,228)(205,230)(206,229)(207,233)(208,232)(209,231)
(210,226)(211,225)(212,227)(213,219)(214,221)(215,220)(216,224)(217,223)
(218,222);
s3 := 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);
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, s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3,
s3*s1*s2*s3*s2*s3*s1*s2*s3*s1*s2*s3*s2*s3*s1*s2,
s1*s2*s1*s2*s1*s2*s3*s1*s2*s1*s2*s1*s2*s3*s1*s2*s1*s2*s1*s2*s3,
s1*s2*s3*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s3*s2*s1*s2*s1*s2*s1*s2*s1*s2 >;
to this polytope