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