Polytope of Type {4,36}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,36}*576b
if this polytope has a name.
Group : SmallGroup(576,4968)
Rank : 3
Schlafli Type : {4,36}
Number of vertices, edges, etc : 8, 144, 72
Order of s0s1s2 : 36
Order of s0s1s2s1 : 4
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{4,36,2} of size 1152
Vertex Figure Of :
{2,4,36} of size 1152
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {4,36}*288b, {4,36}*288c, {4,18}*288
3-fold quotients : {4,12}*192b
4-fold quotients : {2,36}*144, {4,9}*144, {4,18}*144b, {4,18}*144c
6-fold quotients : {4,12}*96b, {4,12}*96c, {4,6}*96
8-fold quotients : {4,9}*72, {2,18}*72
12-fold quotients : {2,12}*48, {4,3}*48, {4,6}*48b, {4,6}*48c
16-fold quotients : {2,9}*36
24-fold quotients : {4,3}*24, {2,6}*24
36-fold quotients : {2,4}*16
48-fold quotients : {2,3}*12
72-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
2-fold covers : {4,36}*1152d, {8,36}*1152e, {8,36}*1152f, {4,72}*1152c, {4,72}*1152d
3-fold covers : {4,108}*1728b, {12,36}*1728e, {12,36}*1728f
Permutation Representation (GAP) :
s0 := ( 1,147)( 2,148)( 3,145)( 4,146)( 5,151)( 6,152)( 7,149)( 8,150)
( 9,155)( 10,156)( 11,153)( 12,154)( 13,159)( 14,160)( 15,157)( 16,158)
( 17,163)( 18,164)( 19,161)( 20,162)( 21,167)( 22,168)( 23,165)( 24,166)
( 25,171)( 26,172)( 27,169)( 28,170)( 29,175)( 30,176)( 31,173)( 32,174)
( 33,179)( 34,180)( 35,177)( 36,178)( 37,183)( 38,184)( 39,181)( 40,182)
( 41,187)( 42,188)( 43,185)( 44,186)( 45,191)( 46,192)( 47,189)( 48,190)
( 49,195)( 50,196)( 51,193)( 52,194)( 53,199)( 54,200)( 55,197)( 56,198)
( 57,203)( 58,204)( 59,201)( 60,202)( 61,207)( 62,208)( 63,205)( 64,206)
( 65,211)( 66,212)( 67,209)( 68,210)( 69,215)( 70,216)( 71,213)( 72,214)
( 73,219)( 74,220)( 75,217)( 76,218)( 77,223)( 78,224)( 79,221)( 80,222)
( 81,227)( 82,228)( 83,225)( 84,226)( 85,231)( 86,232)( 87,229)( 88,230)
( 89,235)( 90,236)( 91,233)( 92,234)( 93,239)( 94,240)( 95,237)( 96,238)
( 97,243)( 98,244)( 99,241)(100,242)(101,247)(102,248)(103,245)(104,246)
(105,251)(106,252)(107,249)(108,250)(109,255)(110,256)(111,253)(112,254)
(113,259)(114,260)(115,257)(116,258)(117,263)(118,264)(119,261)(120,262)
(121,267)(122,268)(123,265)(124,266)(125,271)(126,272)(127,269)(128,270)
(129,275)(130,276)(131,273)(132,274)(133,279)(134,280)(135,277)(136,278)
(137,283)(138,284)(139,281)(140,282)(141,287)(142,288)(143,285)(144,286);;
s1 := ( 2, 3)( 5, 9)( 6, 11)( 7, 10)( 8, 12)( 13, 33)( 14, 35)( 15, 34)
( 16, 36)( 17, 29)( 18, 31)( 19, 30)( 20, 32)( 21, 25)( 22, 27)( 23, 26)
( 24, 28)( 38, 39)( 41, 45)( 42, 47)( 43, 46)( 44, 48)( 49, 69)( 50, 71)
( 51, 70)( 52, 72)( 53, 65)( 54, 67)( 55, 66)( 56, 68)( 57, 61)( 58, 63)
( 59, 62)( 60, 64)( 73,109)( 74,111)( 75,110)( 76,112)( 77,117)( 78,119)
( 79,118)( 80,120)( 81,113)( 82,115)( 83,114)( 84,116)( 85,141)( 86,143)
( 87,142)( 88,144)( 89,137)( 90,139)( 91,138)( 92,140)( 93,133)( 94,135)
( 95,134)( 96,136)( 97,129)( 98,131)( 99,130)(100,132)(101,125)(102,127)
(103,126)(104,128)(105,121)(106,123)(107,122)(108,124)(146,147)(149,153)
(150,155)(151,154)(152,156)(157,177)(158,179)(159,178)(160,180)(161,173)
(162,175)(163,174)(164,176)(165,169)(166,171)(167,170)(168,172)(182,183)
(185,189)(186,191)(187,190)(188,192)(193,213)(194,215)(195,214)(196,216)
(197,209)(198,211)(199,210)(200,212)(201,205)(202,207)(203,206)(204,208)
(217,253)(218,255)(219,254)(220,256)(221,261)(222,263)(223,262)(224,264)
(225,257)(226,259)(227,258)(228,260)(229,285)(230,287)(231,286)(232,288)
(233,281)(234,283)(235,282)(236,284)(237,277)(238,279)(239,278)(240,280)
(241,273)(242,275)(243,274)(244,276)(245,269)(246,271)(247,270)(248,272)
(249,265)(250,267)(251,266)(252,268);;
s2 := ( 1, 97)( 2,100)( 3, 99)( 4, 98)( 5,105)( 6,108)( 7,107)( 8,106)
( 9,101)( 10,104)( 11,103)( 12,102)( 13, 85)( 14, 88)( 15, 87)( 16, 86)
( 17, 93)( 18, 96)( 19, 95)( 20, 94)( 21, 89)( 22, 92)( 23, 91)( 24, 90)
( 25, 73)( 26, 76)( 27, 75)( 28, 74)( 29, 81)( 30, 84)( 31, 83)( 32, 82)
( 33, 77)( 34, 80)( 35, 79)( 36, 78)( 37,133)( 38,136)( 39,135)( 40,134)
( 41,141)( 42,144)( 43,143)( 44,142)( 45,137)( 46,140)( 47,139)( 48,138)
( 49,121)( 50,124)( 51,123)( 52,122)( 53,129)( 54,132)( 55,131)( 56,130)
( 57,125)( 58,128)( 59,127)( 60,126)( 61,109)( 62,112)( 63,111)( 64,110)
( 65,117)( 66,120)( 67,119)( 68,118)( 69,113)( 70,116)( 71,115)( 72,114)
(145,241)(146,244)(147,243)(148,242)(149,249)(150,252)(151,251)(152,250)
(153,245)(154,248)(155,247)(156,246)(157,229)(158,232)(159,231)(160,230)
(161,237)(162,240)(163,239)(164,238)(165,233)(166,236)(167,235)(168,234)
(169,217)(170,220)(171,219)(172,218)(173,225)(174,228)(175,227)(176,226)
(177,221)(178,224)(179,223)(180,222)(181,277)(182,280)(183,279)(184,278)
(185,285)(186,288)(187,287)(188,286)(189,281)(190,284)(191,283)(192,282)
(193,265)(194,268)(195,267)(196,266)(197,273)(198,276)(199,275)(200,274)
(201,269)(202,272)(203,271)(204,270)(205,253)(206,256)(207,255)(208,254)
(209,261)(210,264)(211,263)(212,262)(213,257)(214,260)(215,259)(216,258);;
poly := Group([s0,s1,s2]);;
Finitely Presented Group Representation (GAP) :
F := FreeGroup("s0","s1","s2");;
s0 := F.1;; s1 := F.2;; s2 := F.3;;
rels := [ s0*s0, s1*s1, s2*s2, s0*s2*s0*s2, s0*s1*s0*s1*s0*s1*s0*s1,
s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*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)!( 1,147)( 2,148)( 3,145)( 4,146)( 5,151)( 6,152)( 7,149)
( 8,150)( 9,155)( 10,156)( 11,153)( 12,154)( 13,159)( 14,160)( 15,157)
( 16,158)( 17,163)( 18,164)( 19,161)( 20,162)( 21,167)( 22,168)( 23,165)
( 24,166)( 25,171)( 26,172)( 27,169)( 28,170)( 29,175)( 30,176)( 31,173)
( 32,174)( 33,179)( 34,180)( 35,177)( 36,178)( 37,183)( 38,184)( 39,181)
( 40,182)( 41,187)( 42,188)( 43,185)( 44,186)( 45,191)( 46,192)( 47,189)
( 48,190)( 49,195)( 50,196)( 51,193)( 52,194)( 53,199)( 54,200)( 55,197)
( 56,198)( 57,203)( 58,204)( 59,201)( 60,202)( 61,207)( 62,208)( 63,205)
( 64,206)( 65,211)( 66,212)( 67,209)( 68,210)( 69,215)( 70,216)( 71,213)
( 72,214)( 73,219)( 74,220)( 75,217)( 76,218)( 77,223)( 78,224)( 79,221)
( 80,222)( 81,227)( 82,228)( 83,225)( 84,226)( 85,231)( 86,232)( 87,229)
( 88,230)( 89,235)( 90,236)( 91,233)( 92,234)( 93,239)( 94,240)( 95,237)
( 96,238)( 97,243)( 98,244)( 99,241)(100,242)(101,247)(102,248)(103,245)
(104,246)(105,251)(106,252)(107,249)(108,250)(109,255)(110,256)(111,253)
(112,254)(113,259)(114,260)(115,257)(116,258)(117,263)(118,264)(119,261)
(120,262)(121,267)(122,268)(123,265)(124,266)(125,271)(126,272)(127,269)
(128,270)(129,275)(130,276)(131,273)(132,274)(133,279)(134,280)(135,277)
(136,278)(137,283)(138,284)(139,281)(140,282)(141,287)(142,288)(143,285)
(144,286);
s1 := Sym(288)!( 2, 3)( 5, 9)( 6, 11)( 7, 10)( 8, 12)( 13, 33)( 14, 35)
( 15, 34)( 16, 36)( 17, 29)( 18, 31)( 19, 30)( 20, 32)( 21, 25)( 22, 27)
( 23, 26)( 24, 28)( 38, 39)( 41, 45)( 42, 47)( 43, 46)( 44, 48)( 49, 69)
( 50, 71)( 51, 70)( 52, 72)( 53, 65)( 54, 67)( 55, 66)( 56, 68)( 57, 61)
( 58, 63)( 59, 62)( 60, 64)( 73,109)( 74,111)( 75,110)( 76,112)( 77,117)
( 78,119)( 79,118)( 80,120)( 81,113)( 82,115)( 83,114)( 84,116)( 85,141)
( 86,143)( 87,142)( 88,144)( 89,137)( 90,139)( 91,138)( 92,140)( 93,133)
( 94,135)( 95,134)( 96,136)( 97,129)( 98,131)( 99,130)(100,132)(101,125)
(102,127)(103,126)(104,128)(105,121)(106,123)(107,122)(108,124)(146,147)
(149,153)(150,155)(151,154)(152,156)(157,177)(158,179)(159,178)(160,180)
(161,173)(162,175)(163,174)(164,176)(165,169)(166,171)(167,170)(168,172)
(182,183)(185,189)(186,191)(187,190)(188,192)(193,213)(194,215)(195,214)
(196,216)(197,209)(198,211)(199,210)(200,212)(201,205)(202,207)(203,206)
(204,208)(217,253)(218,255)(219,254)(220,256)(221,261)(222,263)(223,262)
(224,264)(225,257)(226,259)(227,258)(228,260)(229,285)(230,287)(231,286)
(232,288)(233,281)(234,283)(235,282)(236,284)(237,277)(238,279)(239,278)
(240,280)(241,273)(242,275)(243,274)(244,276)(245,269)(246,271)(247,270)
(248,272)(249,265)(250,267)(251,266)(252,268);
s2 := Sym(288)!( 1, 97)( 2,100)( 3, 99)( 4, 98)( 5,105)( 6,108)( 7,107)
( 8,106)( 9,101)( 10,104)( 11,103)( 12,102)( 13, 85)( 14, 88)( 15, 87)
( 16, 86)( 17, 93)( 18, 96)( 19, 95)( 20, 94)( 21, 89)( 22, 92)( 23, 91)
( 24, 90)( 25, 73)( 26, 76)( 27, 75)( 28, 74)( 29, 81)( 30, 84)( 31, 83)
( 32, 82)( 33, 77)( 34, 80)( 35, 79)( 36, 78)( 37,133)( 38,136)( 39,135)
( 40,134)( 41,141)( 42,144)( 43,143)( 44,142)( 45,137)( 46,140)( 47,139)
( 48,138)( 49,121)( 50,124)( 51,123)( 52,122)( 53,129)( 54,132)( 55,131)
( 56,130)( 57,125)( 58,128)( 59,127)( 60,126)( 61,109)( 62,112)( 63,111)
( 64,110)( 65,117)( 66,120)( 67,119)( 68,118)( 69,113)( 70,116)( 71,115)
( 72,114)(145,241)(146,244)(147,243)(148,242)(149,249)(150,252)(151,251)
(152,250)(153,245)(154,248)(155,247)(156,246)(157,229)(158,232)(159,231)
(160,230)(161,237)(162,240)(163,239)(164,238)(165,233)(166,236)(167,235)
(168,234)(169,217)(170,220)(171,219)(172,218)(173,225)(174,228)(175,227)
(176,226)(177,221)(178,224)(179,223)(180,222)(181,277)(182,280)(183,279)
(184,278)(185,285)(186,288)(187,287)(188,286)(189,281)(190,284)(191,283)
(192,282)(193,265)(194,268)(195,267)(196,266)(197,273)(198,276)(199,275)
(200,274)(201,269)(202,272)(203,271)(204,270)(205,253)(206,256)(207,255)
(208,254)(209,261)(210,264)(211,263)(212,262)(213,257)(214,260)(215,259)
(216,258);
poly := sub<Sym(288)|s0,s1,s2>;
Finitely Presented Group Representation (Magma) :
poly<s0,s1,s2> := Group< s0,s1,s2 | s0*s0, s1*s1, s2*s2,
s0*s2*s0*s2, s0*s1*s0*s1*s0*s1*s0*s1,
s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*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