Polytope of Type {4,36,4}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,36,4}*1152d
if this polytope has a name.
Group : SmallGroup(1152,154208)
Rank : 4
Schlafli Type : {4,36,4}
Number of vertices, edges, etc : 4, 72, 72, 4
Order of s0s1s2s3 : 36
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 : {4,36,2}*576b, {4,18,4}*576c
3-fold quotients : {4,12,4}*384d
4-fold quotients : {4,18,2}*288b
6-fold quotients : {4,12,2}*192b, {4,6,4}*192c
8-fold quotients : {4,9,2}*144
12-fold quotients : {4,6,2}*96c
24-fold quotients : {4,3,2}*48
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := ( 1, 3)( 2, 4)( 5, 7)( 6, 8)( 9, 11)( 10, 12)( 13, 15)( 14, 16)
( 17, 19)( 18, 20)( 21, 23)( 22, 24)( 25, 27)( 26, 28)( 29, 31)( 30, 32)
( 33, 35)( 34, 36)( 37, 39)( 38, 40)( 41, 43)( 42, 44)( 45, 47)( 46, 48)
( 49, 51)( 50, 52)( 53, 55)( 54, 56)( 57, 59)( 58, 60)( 61, 63)( 62, 64)
( 65, 67)( 66, 68)( 69, 71)( 70, 72)( 73, 75)( 74, 76)( 77, 79)( 78, 80)
( 81, 83)( 82, 84)( 85, 87)( 86, 88)( 89, 91)( 90, 92)( 93, 95)( 94, 96)
( 97, 99)( 98,100)(101,103)(102,104)(105,107)(106,108)(109,111)(110,112)
(113,115)(114,116)(117,119)(118,120)(121,123)(122,124)(125,127)(126,128)
(129,131)(130,132)(133,135)(134,136)(137,139)(138,140)(141,143)(142,144)
(145,147)(146,148)(149,151)(150,152)(153,155)(154,156)(157,159)(158,160)
(161,163)(162,164)(165,167)(166,168)(169,171)(170,172)(173,175)(174,176)
(177,179)(178,180)(181,183)(182,184)(185,187)(186,188)(189,191)(190,192)
(193,195)(194,196)(197,199)(198,200)(201,203)(202,204)(205,207)(206,208)
(209,211)(210,212)(213,215)(214,216)(217,219)(218,220)(221,223)(222,224)
(225,227)(226,228)(229,231)(230,232)(233,235)(234,236)(237,239)(238,240)
(241,243)(242,244)(245,247)(246,248)(249,251)(250,252)(253,255)(254,256)
(257,259)(258,260)(261,263)(262,264)(265,267)(266,268)(269,271)(270,272)
(273,275)(274,276)(277,279)(278,280)(281,283)(282,284)(285,287)(286,288);;
s1 := ( 3, 4)( 5, 9)( 6, 10)( 7, 12)( 8, 11)( 13, 33)( 14, 34)( 15, 36)
( 16, 35)( 17, 29)( 18, 30)( 19, 32)( 20, 31)( 21, 25)( 22, 26)( 23, 28)
( 24, 27)( 39, 40)( 41, 45)( 42, 46)( 43, 48)( 44, 47)( 49, 69)( 50, 70)
( 51, 72)( 52, 71)( 53, 65)( 54, 66)( 55, 68)( 56, 67)( 57, 61)( 58, 62)
( 59, 64)( 60, 63)( 75, 76)( 77, 81)( 78, 82)( 79, 84)( 80, 83)( 85,105)
( 86,106)( 87,108)( 88,107)( 89,101)( 90,102)( 91,104)( 92,103)( 93, 97)
( 94, 98)( 95,100)( 96, 99)(111,112)(113,117)(114,118)(115,120)(116,119)
(121,141)(122,142)(123,144)(124,143)(125,137)(126,138)(127,140)(128,139)
(129,133)(130,134)(131,136)(132,135)(145,181)(146,182)(147,184)(148,183)
(149,189)(150,190)(151,192)(152,191)(153,185)(154,186)(155,188)(156,187)
(157,213)(158,214)(159,216)(160,215)(161,209)(162,210)(163,212)(164,211)
(165,205)(166,206)(167,208)(168,207)(169,201)(170,202)(171,204)(172,203)
(173,197)(174,198)(175,200)(176,199)(177,193)(178,194)(179,196)(180,195)
(217,253)(218,254)(219,256)(220,255)(221,261)(222,262)(223,264)(224,263)
(225,257)(226,258)(227,260)(228,259)(229,285)(230,286)(231,288)(232,287)
(233,281)(234,282)(235,284)(236,283)(237,277)(238,278)(239,280)(240,279)
(241,273)(242,274)(243,276)(244,275)(245,269)(246,270)(247,272)(248,271)
(249,265)(250,266)(251,268)(252,267);;
s2 := ( 1,157)( 2,160)( 3,159)( 4,158)( 5,165)( 6,168)( 7,167)( 8,166)
( 9,161)( 10,164)( 11,163)( 12,162)( 13,145)( 14,148)( 15,147)( 16,146)
( 17,153)( 18,156)( 19,155)( 20,154)( 21,149)( 22,152)( 23,151)( 24,150)
( 25,177)( 26,180)( 27,179)( 28,178)( 29,173)( 30,176)( 31,175)( 32,174)
( 33,169)( 34,172)( 35,171)( 36,170)( 37,193)( 38,196)( 39,195)( 40,194)
( 41,201)( 42,204)( 43,203)( 44,202)( 45,197)( 46,200)( 47,199)( 48,198)
( 49,181)( 50,184)( 51,183)( 52,182)( 53,189)( 54,192)( 55,191)( 56,190)
( 57,185)( 58,188)( 59,187)( 60,186)( 61,213)( 62,216)( 63,215)( 64,214)
( 65,209)( 66,212)( 67,211)( 68,210)( 69,205)( 70,208)( 71,207)( 72,206)
( 73,229)( 74,232)( 75,231)( 76,230)( 77,237)( 78,240)( 79,239)( 80,238)
( 81,233)( 82,236)( 83,235)( 84,234)( 85,217)( 86,220)( 87,219)( 88,218)
( 89,225)( 90,228)( 91,227)( 92,226)( 93,221)( 94,224)( 95,223)( 96,222)
( 97,249)( 98,252)( 99,251)(100,250)(101,245)(102,248)(103,247)(104,246)
(105,241)(106,244)(107,243)(108,242)(109,265)(110,268)(111,267)(112,266)
(113,273)(114,276)(115,275)(116,274)(117,269)(118,272)(119,271)(120,270)
(121,253)(122,256)(123,255)(124,254)(125,261)(126,264)(127,263)(128,262)
(129,257)(130,260)(131,259)(132,258)(133,285)(134,288)(135,287)(136,286)
(137,281)(138,284)(139,283)(140,282)(141,277)(142,280)(143,279)(144,278);;
s3 := (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);;
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*s0*s1*s0*s1*s0*s1,
s1*s2*s3*s2*s1*s2*s3*s2, s2*s3*s2*s3*s2*s3*s2*s3,
s0*s1*s2*s1*s0*s1*s2*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*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)!( 1, 3)( 2, 4)( 5, 7)( 6, 8)( 9, 11)( 10, 12)( 13, 15)
( 14, 16)( 17, 19)( 18, 20)( 21, 23)( 22, 24)( 25, 27)( 26, 28)( 29, 31)
( 30, 32)( 33, 35)( 34, 36)( 37, 39)( 38, 40)( 41, 43)( 42, 44)( 45, 47)
( 46, 48)( 49, 51)( 50, 52)( 53, 55)( 54, 56)( 57, 59)( 58, 60)( 61, 63)
( 62, 64)( 65, 67)( 66, 68)( 69, 71)( 70, 72)( 73, 75)( 74, 76)( 77, 79)
( 78, 80)( 81, 83)( 82, 84)( 85, 87)( 86, 88)( 89, 91)( 90, 92)( 93, 95)
( 94, 96)( 97, 99)( 98,100)(101,103)(102,104)(105,107)(106,108)(109,111)
(110,112)(113,115)(114,116)(117,119)(118,120)(121,123)(122,124)(125,127)
(126,128)(129,131)(130,132)(133,135)(134,136)(137,139)(138,140)(141,143)
(142,144)(145,147)(146,148)(149,151)(150,152)(153,155)(154,156)(157,159)
(158,160)(161,163)(162,164)(165,167)(166,168)(169,171)(170,172)(173,175)
(174,176)(177,179)(178,180)(181,183)(182,184)(185,187)(186,188)(189,191)
(190,192)(193,195)(194,196)(197,199)(198,200)(201,203)(202,204)(205,207)
(206,208)(209,211)(210,212)(213,215)(214,216)(217,219)(218,220)(221,223)
(222,224)(225,227)(226,228)(229,231)(230,232)(233,235)(234,236)(237,239)
(238,240)(241,243)(242,244)(245,247)(246,248)(249,251)(250,252)(253,255)
(254,256)(257,259)(258,260)(261,263)(262,264)(265,267)(266,268)(269,271)
(270,272)(273,275)(274,276)(277,279)(278,280)(281,283)(282,284)(285,287)
(286,288);
s1 := Sym(288)!( 3, 4)( 5, 9)( 6, 10)( 7, 12)( 8, 11)( 13, 33)( 14, 34)
( 15, 36)( 16, 35)( 17, 29)( 18, 30)( 19, 32)( 20, 31)( 21, 25)( 22, 26)
( 23, 28)( 24, 27)( 39, 40)( 41, 45)( 42, 46)( 43, 48)( 44, 47)( 49, 69)
( 50, 70)( 51, 72)( 52, 71)( 53, 65)( 54, 66)( 55, 68)( 56, 67)( 57, 61)
( 58, 62)( 59, 64)( 60, 63)( 75, 76)( 77, 81)( 78, 82)( 79, 84)( 80, 83)
( 85,105)( 86,106)( 87,108)( 88,107)( 89,101)( 90,102)( 91,104)( 92,103)
( 93, 97)( 94, 98)( 95,100)( 96, 99)(111,112)(113,117)(114,118)(115,120)
(116,119)(121,141)(122,142)(123,144)(124,143)(125,137)(126,138)(127,140)
(128,139)(129,133)(130,134)(131,136)(132,135)(145,181)(146,182)(147,184)
(148,183)(149,189)(150,190)(151,192)(152,191)(153,185)(154,186)(155,188)
(156,187)(157,213)(158,214)(159,216)(160,215)(161,209)(162,210)(163,212)
(164,211)(165,205)(166,206)(167,208)(168,207)(169,201)(170,202)(171,204)
(172,203)(173,197)(174,198)(175,200)(176,199)(177,193)(178,194)(179,196)
(180,195)(217,253)(218,254)(219,256)(220,255)(221,261)(222,262)(223,264)
(224,263)(225,257)(226,258)(227,260)(228,259)(229,285)(230,286)(231,288)
(232,287)(233,281)(234,282)(235,284)(236,283)(237,277)(238,278)(239,280)
(240,279)(241,273)(242,274)(243,276)(244,275)(245,269)(246,270)(247,272)
(248,271)(249,265)(250,266)(251,268)(252,267);
s2 := Sym(288)!( 1,157)( 2,160)( 3,159)( 4,158)( 5,165)( 6,168)( 7,167)
( 8,166)( 9,161)( 10,164)( 11,163)( 12,162)( 13,145)( 14,148)( 15,147)
( 16,146)( 17,153)( 18,156)( 19,155)( 20,154)( 21,149)( 22,152)( 23,151)
( 24,150)( 25,177)( 26,180)( 27,179)( 28,178)( 29,173)( 30,176)( 31,175)
( 32,174)( 33,169)( 34,172)( 35,171)( 36,170)( 37,193)( 38,196)( 39,195)
( 40,194)( 41,201)( 42,204)( 43,203)( 44,202)( 45,197)( 46,200)( 47,199)
( 48,198)( 49,181)( 50,184)( 51,183)( 52,182)( 53,189)( 54,192)( 55,191)
( 56,190)( 57,185)( 58,188)( 59,187)( 60,186)( 61,213)( 62,216)( 63,215)
( 64,214)( 65,209)( 66,212)( 67,211)( 68,210)( 69,205)( 70,208)( 71,207)
( 72,206)( 73,229)( 74,232)( 75,231)( 76,230)( 77,237)( 78,240)( 79,239)
( 80,238)( 81,233)( 82,236)( 83,235)( 84,234)( 85,217)( 86,220)( 87,219)
( 88,218)( 89,225)( 90,228)( 91,227)( 92,226)( 93,221)( 94,224)( 95,223)
( 96,222)( 97,249)( 98,252)( 99,251)(100,250)(101,245)(102,248)(103,247)
(104,246)(105,241)(106,244)(107,243)(108,242)(109,265)(110,268)(111,267)
(112,266)(113,273)(114,276)(115,275)(116,274)(117,269)(118,272)(119,271)
(120,270)(121,253)(122,256)(123,255)(124,254)(125,261)(126,264)(127,263)
(128,262)(129,257)(130,260)(131,259)(132,258)(133,285)(134,288)(135,287)
(136,286)(137,281)(138,284)(139,283)(140,282)(141,277)(142,280)(143,279)
(144,278);
s3 := Sym(288)!(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);
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*s0*s1*s0*s1*s0*s1, s1*s2*s3*s2*s1*s2*s3*s2,
s2*s3*s2*s3*s2*s3*s2*s3, s0*s1*s2*s1*s0*s1*s2*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*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