Polytope of Type {8,18,4}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {8,18,4}*1152b
if this polytope has a name.
Group : SmallGroup(1152,154366)
Rank : 4
Schlafli Type : {8,18,4}
Number of vertices, edges, etc : 8, 72, 36, 4
Order of s0s1s2s3 : 72
Order of s0s1s2s3s2s1 : 2
Special Properties :
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) :
2-fold quotients : {4,18,4}*576b
3-fold quotients : {8,6,4}*384b
4-fold quotients : {2,18,4}*288b
6-fold quotients : {4,6,4}*192b
8-fold quotients : {2,9,4}*144
12-fold quotients : {2,6,4}*96c
24-fold quotients : {2,3,4}*48
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := ( 73,109)( 74,110)( 75,111)( 76,112)( 77,113)( 78,114)( 79,115)( 80,116)
( 81,117)( 82,118)( 83,119)( 84,120)( 85,121)( 86,122)( 87,123)( 88,124)
( 89,125)( 90,126)( 91,127)( 92,128)( 93,129)( 94,130)( 95,131)( 96,132)
( 97,133)( 98,134)( 99,135)(100,136)(101,137)(102,138)(103,139)(104,140)
(105,141)(106,142)(107,143)(108,144)(145,217)(146,218)(147,219)(148,220)
(149,221)(150,222)(151,223)(152,224)(153,225)(154,226)(155,227)(156,228)
(157,229)(158,230)(159,231)(160,232)(161,233)(162,234)(163,235)(164,236)
(165,237)(166,238)(167,239)(168,240)(169,241)(170,242)(171,243)(172,244)
(173,245)(174,246)(175,247)(176,248)(177,249)(178,250)(179,251)(180,252)
(181,253)(182,254)(183,255)(184,256)(185,257)(186,258)(187,259)(188,260)
(189,261)(190,262)(191,263)(192,264)(193,265)(194,266)(195,267)(196,268)
(197,269)(198,270)(199,271)(200,272)(201,273)(202,274)(203,275)(204,276)
(205,277)(206,278)(207,279)(208,280)(209,281)(210,282)(211,283)(212,284)
(213,285)(214,286)(215,287)(216,288);;
s1 := ( 1,145)( 2,146)( 3,148)( 4,147)( 5,153)( 6,154)( 7,156)( 8,155)
( 9,149)( 10,150)( 11,152)( 12,151)( 13,177)( 14,178)( 15,180)( 16,179)
( 17,173)( 18,174)( 19,176)( 20,175)( 21,169)( 22,170)( 23,172)( 24,171)
( 25,165)( 26,166)( 27,168)( 28,167)( 29,161)( 30,162)( 31,164)( 32,163)
( 33,157)( 34,158)( 35,160)( 36,159)( 37,181)( 38,182)( 39,184)( 40,183)
( 41,189)( 42,190)( 43,192)( 44,191)( 45,185)( 46,186)( 47,188)( 48,187)
( 49,213)( 50,214)( 51,216)( 52,215)( 53,209)( 54,210)( 55,212)( 56,211)
( 57,205)( 58,206)( 59,208)( 60,207)( 61,201)( 62,202)( 63,204)( 64,203)
( 65,197)( 66,198)( 67,200)( 68,199)( 69,193)( 70,194)( 71,196)( 72,195)
( 73,253)( 74,254)( 75,256)( 76,255)( 77,261)( 78,262)( 79,264)( 80,263)
( 81,257)( 82,258)( 83,260)( 84,259)( 85,285)( 86,286)( 87,288)( 88,287)
( 89,281)( 90,282)( 91,284)( 92,283)( 93,277)( 94,278)( 95,280)( 96,279)
( 97,273)( 98,274)( 99,276)(100,275)(101,269)(102,270)(103,272)(104,271)
(105,265)(106,266)(107,268)(108,267)(109,217)(110,218)(111,220)(112,219)
(113,225)(114,226)(115,228)(116,227)(117,221)(118,222)(119,224)(120,223)
(121,249)(122,250)(123,252)(124,251)(125,245)(126,246)(127,248)(128,247)
(129,241)(130,242)(131,244)(132,243)(133,237)(134,238)(135,240)(136,239)
(137,233)(138,234)(139,236)(140,235)(141,229)(142,230)(143,232)(144,231);;
s2 := ( 1, 13)( 2, 16)( 3, 15)( 4, 14)( 5, 21)( 6, 24)( 7, 23)( 8, 22)
( 9, 17)( 10, 20)( 11, 19)( 12, 18)( 25, 33)( 26, 36)( 27, 35)( 28, 34)
( 30, 32)( 37, 49)( 38, 52)( 39, 51)( 40, 50)( 41, 57)( 42, 60)( 43, 59)
( 44, 58)( 45, 53)( 46, 56)( 47, 55)( 48, 54)( 61, 69)( 62, 72)( 63, 71)
( 64, 70)( 66, 68)( 73, 85)( 74, 88)( 75, 87)( 76, 86)( 77, 93)( 78, 96)
( 79, 95)( 80, 94)( 81, 89)( 82, 92)( 83, 91)( 84, 90)( 97,105)( 98,108)
( 99,107)(100,106)(102,104)(109,121)(110,124)(111,123)(112,122)(113,129)
(114,132)(115,131)(116,130)(117,125)(118,128)(119,127)(120,126)(133,141)
(134,144)(135,143)(136,142)(138,140)(145,157)(146,160)(147,159)(148,158)
(149,165)(150,168)(151,167)(152,166)(153,161)(154,164)(155,163)(156,162)
(169,177)(170,180)(171,179)(172,178)(174,176)(181,193)(182,196)(183,195)
(184,194)(185,201)(186,204)(187,203)(188,202)(189,197)(190,200)(191,199)
(192,198)(205,213)(206,216)(207,215)(208,214)(210,212)(217,229)(218,232)
(219,231)(220,230)(221,237)(222,240)(223,239)(224,238)(225,233)(226,236)
(227,235)(228,234)(241,249)(242,252)(243,251)(244,250)(246,248)(253,265)
(254,268)(255,267)(256,266)(257,273)(258,276)(259,275)(260,274)(261,269)
(262,272)(263,271)(264,270)(277,285)(278,288)(279,287)(280,286)(282,284);;
s3 := ( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9, 10)( 11, 12)( 13, 14)( 15, 16)
( 17, 18)( 19, 20)( 21, 22)( 23, 24)( 25, 26)( 27, 28)( 29, 30)( 31, 32)
( 33, 34)( 35, 36)( 37, 38)( 39, 40)( 41, 42)( 43, 44)( 45, 46)( 47, 48)
( 49, 50)( 51, 52)( 53, 54)( 55, 56)( 57, 58)( 59, 60)( 61, 62)( 63, 64)
( 65, 66)( 67, 68)( 69, 70)( 71, 72)( 73, 74)( 75, 76)( 77, 78)( 79, 80)
( 81, 82)( 83, 84)( 85, 86)( 87, 88)( 89, 90)( 91, 92)( 93, 94)( 95, 96)
( 97, 98)( 99,100)(101,102)(103,104)(105,106)(107,108)(109,110)(111,112)
(113,114)(115,116)(117,118)(119,120)(121,122)(123,124)(125,126)(127,128)
(129,130)(131,132)(133,134)(135,136)(137,138)(139,140)(141,142)(143,144)
(145,146)(147,148)(149,150)(151,152)(153,154)(155,156)(157,158)(159,160)
(161,162)(163,164)(165,166)(167,168)(169,170)(171,172)(173,174)(175,176)
(177,178)(179,180)(181,182)(183,184)(185,186)(187,188)(189,190)(191,192)
(193,194)(195,196)(197,198)(199,200)(201,202)(203,204)(205,206)(207,208)
(209,210)(211,212)(213,214)(215,216)(217,218)(219,220)(221,222)(223,224)
(225,226)(227,228)(229,230)(231,232)(233,234)(235,236)(237,238)(239,240)
(241,242)(243,244)(245,246)(247,248)(249,250)(251,252)(253,254)(255,256)
(257,258)(259,260)(261,262)(263,264)(265,266)(267,268)(269,270)(271,272)
(273,274)(275,276)(277,278)(279,280)(281,282)(283,284)(285,286)(287,288);;
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*s2*s0*s2,
s0*s3*s0*s3, s1*s3*s1*s3, s0*s1*s2*s1*s0*s1*s2*s1,
s2*s3*s2*s3*s2*s3*s2*s3, s3*s2*s1*s3*s2*s3*s2*s1*s2,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(288)!( 73,109)( 74,110)( 75,111)( 76,112)( 77,113)( 78,114)( 79,115)
( 80,116)( 81,117)( 82,118)( 83,119)( 84,120)( 85,121)( 86,122)( 87,123)
( 88,124)( 89,125)( 90,126)( 91,127)( 92,128)( 93,129)( 94,130)( 95,131)
( 96,132)( 97,133)( 98,134)( 99,135)(100,136)(101,137)(102,138)(103,139)
(104,140)(105,141)(106,142)(107,143)(108,144)(145,217)(146,218)(147,219)
(148,220)(149,221)(150,222)(151,223)(152,224)(153,225)(154,226)(155,227)
(156,228)(157,229)(158,230)(159,231)(160,232)(161,233)(162,234)(163,235)
(164,236)(165,237)(166,238)(167,239)(168,240)(169,241)(170,242)(171,243)
(172,244)(173,245)(174,246)(175,247)(176,248)(177,249)(178,250)(179,251)
(180,252)(181,253)(182,254)(183,255)(184,256)(185,257)(186,258)(187,259)
(188,260)(189,261)(190,262)(191,263)(192,264)(193,265)(194,266)(195,267)
(196,268)(197,269)(198,270)(199,271)(200,272)(201,273)(202,274)(203,275)
(204,276)(205,277)(206,278)(207,279)(208,280)(209,281)(210,282)(211,283)
(212,284)(213,285)(214,286)(215,287)(216,288);
s1 := Sym(288)!( 1,145)( 2,146)( 3,148)( 4,147)( 5,153)( 6,154)( 7,156)
( 8,155)( 9,149)( 10,150)( 11,152)( 12,151)( 13,177)( 14,178)( 15,180)
( 16,179)( 17,173)( 18,174)( 19,176)( 20,175)( 21,169)( 22,170)( 23,172)
( 24,171)( 25,165)( 26,166)( 27,168)( 28,167)( 29,161)( 30,162)( 31,164)
( 32,163)( 33,157)( 34,158)( 35,160)( 36,159)( 37,181)( 38,182)( 39,184)
( 40,183)( 41,189)( 42,190)( 43,192)( 44,191)( 45,185)( 46,186)( 47,188)
( 48,187)( 49,213)( 50,214)( 51,216)( 52,215)( 53,209)( 54,210)( 55,212)
( 56,211)( 57,205)( 58,206)( 59,208)( 60,207)( 61,201)( 62,202)( 63,204)
( 64,203)( 65,197)( 66,198)( 67,200)( 68,199)( 69,193)( 70,194)( 71,196)
( 72,195)( 73,253)( 74,254)( 75,256)( 76,255)( 77,261)( 78,262)( 79,264)
( 80,263)( 81,257)( 82,258)( 83,260)( 84,259)( 85,285)( 86,286)( 87,288)
( 88,287)( 89,281)( 90,282)( 91,284)( 92,283)( 93,277)( 94,278)( 95,280)
( 96,279)( 97,273)( 98,274)( 99,276)(100,275)(101,269)(102,270)(103,272)
(104,271)(105,265)(106,266)(107,268)(108,267)(109,217)(110,218)(111,220)
(112,219)(113,225)(114,226)(115,228)(116,227)(117,221)(118,222)(119,224)
(120,223)(121,249)(122,250)(123,252)(124,251)(125,245)(126,246)(127,248)
(128,247)(129,241)(130,242)(131,244)(132,243)(133,237)(134,238)(135,240)
(136,239)(137,233)(138,234)(139,236)(140,235)(141,229)(142,230)(143,232)
(144,231);
s2 := Sym(288)!( 1, 13)( 2, 16)( 3, 15)( 4, 14)( 5, 21)( 6, 24)( 7, 23)
( 8, 22)( 9, 17)( 10, 20)( 11, 19)( 12, 18)( 25, 33)( 26, 36)( 27, 35)
( 28, 34)( 30, 32)( 37, 49)( 38, 52)( 39, 51)( 40, 50)( 41, 57)( 42, 60)
( 43, 59)( 44, 58)( 45, 53)( 46, 56)( 47, 55)( 48, 54)( 61, 69)( 62, 72)
( 63, 71)( 64, 70)( 66, 68)( 73, 85)( 74, 88)( 75, 87)( 76, 86)( 77, 93)
( 78, 96)( 79, 95)( 80, 94)( 81, 89)( 82, 92)( 83, 91)( 84, 90)( 97,105)
( 98,108)( 99,107)(100,106)(102,104)(109,121)(110,124)(111,123)(112,122)
(113,129)(114,132)(115,131)(116,130)(117,125)(118,128)(119,127)(120,126)
(133,141)(134,144)(135,143)(136,142)(138,140)(145,157)(146,160)(147,159)
(148,158)(149,165)(150,168)(151,167)(152,166)(153,161)(154,164)(155,163)
(156,162)(169,177)(170,180)(171,179)(172,178)(174,176)(181,193)(182,196)
(183,195)(184,194)(185,201)(186,204)(187,203)(188,202)(189,197)(190,200)
(191,199)(192,198)(205,213)(206,216)(207,215)(208,214)(210,212)(217,229)
(218,232)(219,231)(220,230)(221,237)(222,240)(223,239)(224,238)(225,233)
(226,236)(227,235)(228,234)(241,249)(242,252)(243,251)(244,250)(246,248)
(253,265)(254,268)(255,267)(256,266)(257,273)(258,276)(259,275)(260,274)
(261,269)(262,272)(263,271)(264,270)(277,285)(278,288)(279,287)(280,286)
(282,284);
s3 := Sym(288)!( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9, 10)( 11, 12)( 13, 14)
( 15, 16)( 17, 18)( 19, 20)( 21, 22)( 23, 24)( 25, 26)( 27, 28)( 29, 30)
( 31, 32)( 33, 34)( 35, 36)( 37, 38)( 39, 40)( 41, 42)( 43, 44)( 45, 46)
( 47, 48)( 49, 50)( 51, 52)( 53, 54)( 55, 56)( 57, 58)( 59, 60)( 61, 62)
( 63, 64)( 65, 66)( 67, 68)( 69, 70)( 71, 72)( 73, 74)( 75, 76)( 77, 78)
( 79, 80)( 81, 82)( 83, 84)( 85, 86)( 87, 88)( 89, 90)( 91, 92)( 93, 94)
( 95, 96)( 97, 98)( 99,100)(101,102)(103,104)(105,106)(107,108)(109,110)
(111,112)(113,114)(115,116)(117,118)(119,120)(121,122)(123,124)(125,126)
(127,128)(129,130)(131,132)(133,134)(135,136)(137,138)(139,140)(141,142)
(143,144)(145,146)(147,148)(149,150)(151,152)(153,154)(155,156)(157,158)
(159,160)(161,162)(163,164)(165,166)(167,168)(169,170)(171,172)(173,174)
(175,176)(177,178)(179,180)(181,182)(183,184)(185,186)(187,188)(189,190)
(191,192)(193,194)(195,196)(197,198)(199,200)(201,202)(203,204)(205,206)
(207,208)(209,210)(211,212)(213,214)(215,216)(217,218)(219,220)(221,222)
(223,224)(225,226)(227,228)(229,230)(231,232)(233,234)(235,236)(237,238)
(239,240)(241,242)(243,244)(245,246)(247,248)(249,250)(251,252)(253,254)
(255,256)(257,258)(259,260)(261,262)(263,264)(265,266)(267,268)(269,270)
(271,272)(273,274)(275,276)(277,278)(279,280)(281,282)(283,284)(285,286)
(287,288);
poly := sub<Sym(288)|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*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3,
s0*s1*s2*s1*s0*s1*s2*s1, s2*s3*s2*s3*s2*s3*s2*s3,
s3*s2*s1*s3*s2*s3*s2*s1*s2, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 >;
References : None.
to this polytope