Polytope of Type {72,4} with group SmallGroup(1152,154353)

Overview

Group: SmallGroup(1152,154353)
Rank: 3
Schläfli Type: {72,4}
Vertices, edges, …: 144, 288, 8
Order of s₀s₁s₂: 72
Order of s₀s₁s₂s₁: 4
Also known as: if this polytope has a name.

Special Properties

Compact Hyperbolic Quotient
Locally Spherical
Orientable
Self-Petrie

Quotients maximal quotients in bold

2-fold

{36,4}*576b

3-fold

{24,4}*384d

4-fold

6-fold

{12,4}*192b

8-fold

12-fold

16-fold

24-fold

32-fold

{9,2}*36

48-fold

72-fold

{4,2}*16

96-fold

{3,2}*12

144-fold

{2,2}*8

Covers minimal covers in bold

None in this atlas.

Irregular Quotients of which this is a minimal cover

Click an entry to reveal its facets and vertex figures.

P/N, where N=<s1*s0*s1*s2*s1*s0*s2*s1> of order 2

4 facets

4 of {72}*144

96 vertex figures

48 of {4}*8
48 of {2}*4

P/N, where N=<s0*s1*s2*s1*s0*(s1*s2)^2> of order 2

4 facets

4 of {72}*144

72 vertex figures

72 of {4}*8

Representations

Permutation Representation (GAP)

s0 := (  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)( 73,109)( 74,110)( 75,112)( 76,111)( 77,117)( 78,118)( 79,120)( 80,119)( 81,113)( 82,114)( 83,116)( 84,115)( 85,141)( 86,142)( 87,144)( 88,143)( 89,137)( 90,138)( 91,140)( 92,139)( 93,133)( 94,134)( 95,136)( 96,135)( 97,129)( 98,130)( 99,132)(100,131)(101,125)(102,126)(103,128)(104,127)(105,121)(106,122)(107,124)(108,123)(145,217)(146,218)(147,220)(148,219)(149,225)(150,226)(151,228)(152,227)(153,221)(154,222)(155,224)(156,223)(157,249)(158,250)(159,252)(160,251)(161,245)(162,246)(163,248)(164,247)(165,241)(166,242)(167,244)(168,243)(169,237)(170,238)(171,240)(172,239)(173,233)(174,234)(175,236)(176,235)(177,229)(178,230)(179,232)(180,231)(181,253)(182,254)(183,256)(184,255)(185,261)(186,262)(187,264)(188,263)(189,257)(190,258)(191,260)(192,259)(193,285)(194,286)(195,288)(196,287)(197,281)(198,282)(199,284)(200,283)(201,277)(202,278)(203,280)(204,279)(205,273)(206,274)(207,276)(208,275)(209,269)(210,270)(211,272)(212,271)(213,265)(214,266)(215,268)(216,267);;
s1 := (  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)( 73,265)( 74,268)( 75,267)( 76,266)( 77,273)( 78,276)( 79,275)( 80,274)( 81,269)( 82,272)( 83,271)( 84,270)( 85,253)( 86,256)( 87,255)( 88,254)( 89,261)( 90,264)( 91,263)( 92,262)( 93,257)( 94,260)( 95,259)( 96,258)( 97,285)( 98,288)( 99,287)(100,286)(101,281)(102,284)(103,283)(104,282)(105,277)(106,280)(107,279)(108,278)(109,229)(110,232)(111,231)(112,230)(113,237)(114,240)(115,239)(116,238)(117,233)(118,236)(119,235)(120,234)(121,217)(122,220)(123,219)(124,218)(125,225)(126,228)(127,227)(128,226)(129,221)(130,224)(131,223)(132,222)(133,249)(134,252)(135,251)(136,250)(137,245)(138,248)(139,247)(140,246)(141,241)(142,244)(143,243)(144,242);;
s2 := (  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)(145,182)(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]);;

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, s1*s2*s1*s2*s1*s2*s1*s2, 
s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1, 
s0*s1*s0*s2*s1*s2*s1*s0*s1*s0*s1*s0*s2*s1*s2*s1*s0*s1, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s2*s1*s0*s1*s0*s1*s0*s2*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 ];;
poly := F / rels;;

Permutation Representation (Magma)

s0 := 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)( 73,109)( 74,110)( 75,112)( 76,111)( 77,117)( 78,118)( 79,120)( 80,119)( 81,113)( 82,114)( 83,116)( 84,115)( 85,141)( 86,142)( 87,144)( 88,143)( 89,137)( 90,138)( 91,140)( 92,139)( 93,133)( 94,134)( 95,136)( 96,135)( 97,129)( 98,130)( 99,132)(100,131)(101,125)(102,126)(103,128)(104,127)(105,121)(106,122)(107,124)(108,123)(145,217)(146,218)(147,220)(148,219)(149,225)(150,226)(151,228)(152,227)(153,221)(154,222)(155,224)(156,223)(157,249)(158,250)(159,252)(160,251)(161,245)(162,246)(163,248)(164,247)(165,241)(166,242)(167,244)(168,243)(169,237)(170,238)(171,240)(172,239)(173,233)(174,234)(175,236)(176,235)(177,229)(178,230)(179,232)(180,231)(181,253)(182,254)(183,256)(184,255)(185,261)(186,262)(187,264)(188,263)(189,257)(190,258)(191,260)(192,259)(193,285)(194,286)(195,288)(196,287)(197,281)(198,282)(199,284)(200,283)(201,277)(202,278)(203,280)(204,279)(205,273)(206,274)(207,276)(208,275)(209,269)(210,270)(211,272)(212,271)(213,265)(214,266)(215,268)(216,267);
s1 := 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)( 73,265)( 74,268)( 75,267)( 76,266)( 77,273)( 78,276)( 79,275)( 80,274)( 81,269)( 82,272)( 83,271)( 84,270)( 85,253)( 86,256)( 87,255)( 88,254)( 89,261)( 90,264)( 91,263)( 92,262)( 93,257)( 94,260)( 95,259)( 96,258)( 97,285)( 98,288)( 99,287)(100,286)(101,281)(102,284)(103,283)(104,282)(105,277)(106,280)(107,279)(108,278)(109,229)(110,232)(111,231)(112,230)(113,237)(114,240)(115,239)(116,238)(117,233)(118,236)(119,235)(120,234)(121,217)(122,220)(123,219)(124,218)(125,225)(126,228)(127,227)(128,226)(129,221)(130,224)(131,223)(132,222)(133,249)(134,252)(135,251)(136,250)(137,245)(138,248)(139,247)(140,246)(141,241)(142,244)(143,243)(144,242);
s2 := 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)(145,182)(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>;

Finitely Presented Group Representation (Magma)

poly<s0,s1,s2> := Group< s0,s1,s2 | s0*s0, s1*s1, s2*s2, 
s0*s2*s0*s2, s1*s2*s1*s2*s1*s2*s1*s2, 
s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1, 
s0*s1*s0*s2*s1*s2*s1*s0*s1*s0*s1*s0*s2*s1*s2*s1*s0*s1, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s2*s1*s0*s1*s0*s1*s0*s2*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 >;

References

None.

to this polytope.