Polytope of Type {36,9} with group SmallGroup(1296,1782)

Overview

Group: SmallGroup(1296,1782)
Rank: 3
Schläfli Type: {36,9}
Vertices, edges, …: 72, 324, 18
Order of s₀s₁s₂: 18
Order of s₀s₁s₂s₁: 36
Also known as: if this polytope has a name.

Special Properties

Compact Hyperbolic Quotient
Locally Spherical
Orientable

Quotients maximal quotients in bold

3-fold

{12,9}*432

4-fold

{18,9}*324

9-fold

12-fold

{6,9}*108

18-fold

{4,9}*72

27-fold

{4,3}*48

36-fold

54-fold

{4,3}*24

108-fold

{2,3}*12

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=<s0*s2*s1*s0*s2*(s1*s0)^2*s2*s1*s2> of order 2

12 facets

6 of {18}*36
6 of {36}*72

36 vertex figures

36 of {9}*18

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

9 facets

9 of {36}*72

36 vertex figures

36 of {9}*18

Representations

Permutation Representation (GAP)

s0 := (  1,  3)(  2,  4)(  5,  7)(  6,  8)(  9, 11)( 10, 12)( 13, 27)( 14, 28)( 15, 25)( 16, 26)( 17, 31)( 18, 32)( 19, 29)( 20, 30)( 21, 35)( 22, 36)( 23, 33)( 24, 34)( 37, 99)( 38,100)( 39, 97)( 40, 98)( 41,103)( 42,104)( 43,101)( 44,102)( 45,107)( 46,108)( 47,105)( 48,106)( 49, 87)( 50, 88)( 51, 85)( 52, 86)( 53, 91)( 54, 92)( 55, 89)( 56, 90)( 57, 95)( 58, 96)( 59, 93)( 60, 94)( 61, 75)( 62, 76)( 63, 73)( 64, 74)( 65, 79)( 66, 80)( 67, 77)( 68, 78)( 69, 83)( 70, 84)( 71, 81)( 72, 82)(109,111)(110,112)(113,115)(114,116)(117,119)(118,120)(121,135)(122,136)(123,133)(124,134)(125,139)(126,140)(127,137)(128,138)(129,143)(130,144)(131,141)(132,142)(145,207)(146,208)(147,205)(148,206)(149,211)(150,212)(151,209)(152,210)(153,215)(154,216)(155,213)(156,214)(157,195)(158,196)(159,193)(160,194)(161,199)(162,200)(163,197)(164,198)(165,203)(166,204)(167,201)(168,202)(169,183)(170,184)(171,181)(172,182)(173,187)(174,188)(175,185)(176,186)(177,191)(178,192)(179,189)(180,190)(217,219)(218,220)(221,223)(222,224)(225,227)(226,228)(229,243)(230,244)(231,241)(232,242)(233,247)(234,248)(235,245)(236,246)(237,251)(238,252)(239,249)(240,250)(253,315)(254,316)(255,313)(256,314)(257,319)(258,320)(259,317)(260,318)(261,323)(262,324)(263,321)(264,322)(265,303)(266,304)(267,301)(268,302)(269,307)(270,308)(271,305)(272,306)(273,311)(274,312)(275,309)(276,310)(277,291)(278,292)(279,289)(280,290)(281,295)(282,296)(283,293)(284,294)(285,299)(286,300)(287,297)(288,298);;
s1 := (  1, 37)(  2, 38)(  3, 40)(  4, 39)(  5, 45)(  6, 46)(  7, 48)(  8, 47)(  9, 41)( 10, 42)( 11, 44)( 12, 43)( 13, 61)( 14, 62)( 15, 64)( 16, 63)( 17, 69)( 18, 70)( 19, 72)( 20, 71)( 21, 65)( 22, 66)( 23, 68)( 24, 67)( 25, 49)( 26, 50)( 27, 52)( 28, 51)( 29, 57)( 30, 58)( 31, 60)( 32, 59)( 33, 53)( 34, 54)( 35, 56)( 36, 55)( 73, 97)( 74, 98)( 75,100)( 76, 99)( 77,105)( 78,106)( 79,108)( 80,107)( 81,101)( 82,102)( 83,104)( 84,103)( 87, 88)( 89, 93)( 90, 94)( 91, 96)( 92, 95)(109,261)(110,262)(111,264)(112,263)(113,257)(114,258)(115,260)(116,259)(117,253)(118,254)(119,256)(120,255)(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)(145,225)(146,226)(147,228)(148,227)(149,221)(150,222)(151,224)(152,223)(153,217)(154,218)(155,220)(156,219)(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)(181,321)(182,322)(183,324)(184,323)(185,317)(186,318)(187,320)(188,319)(189,313)(190,314)(191,316)(192,315)(193,309)(194,310)(195,312)(196,311)(197,305)(198,306)(199,308)(200,307)(201,301)(202,302)(203,304)(204,303)(205,297)(206,298)(207,300)(208,299)(209,293)(210,294)(211,296)(212,295)(213,289)(214,290)(215,292)(216,291);;
s2 := (  1,109)(  2,112)(  3,111)(  4,110)(  5,117)(  6,120)(  7,119)(  8,118)(  9,113)( 10,116)( 11,115)( 12,114)( 13,133)( 14,136)( 15,135)( 16,134)( 17,141)( 18,144)( 19,143)( 20,142)( 21,137)( 22,140)( 23,139)( 24,138)( 25,121)( 26,124)( 27,123)( 28,122)( 29,129)( 30,132)( 31,131)( 32,130)( 33,125)( 34,128)( 35,127)( 36,126)( 37,205)( 38,208)( 39,207)( 40,206)( 41,213)( 42,216)( 43,215)( 44,214)( 45,209)( 46,212)( 47,211)( 48,210)( 49,193)( 50,196)( 51,195)( 52,194)( 53,201)( 54,204)( 55,203)( 56,202)( 57,197)( 58,200)( 59,199)( 60,198)( 61,181)( 62,184)( 63,183)( 64,182)( 65,189)( 66,192)( 67,191)( 68,190)( 69,185)( 70,188)( 71,187)( 72,186)( 73,169)( 74,172)( 75,171)( 76,170)( 77,177)( 78,180)( 79,179)( 80,178)( 81,173)( 82,176)( 83,175)( 84,174)( 85,157)( 86,160)( 87,159)( 88,158)( 89,165)( 90,168)( 91,167)( 92,166)( 93,161)( 94,164)( 95,163)( 96,162)( 97,145)( 98,148)( 99,147)(100,146)(101,153)(102,156)(103,155)(104,154)(105,149)(106,152)(107,151)(108,150)(217,225)(218,228)(219,227)(220,226)(222,224)(229,249)(230,252)(231,251)(232,250)(233,245)(234,248)(235,247)(236,246)(237,241)(238,244)(239,243)(240,242)(253,321)(254,324)(255,323)(256,322)(257,317)(258,320)(259,319)(260,318)(261,313)(262,316)(263,315)(264,314)(265,309)(266,312)(267,311)(268,310)(269,305)(270,308)(271,307)(272,306)(273,301)(274,304)(275,303)(276,302)(277,297)(278,300)(279,299)(280,298)(281,293)(282,296)(283,295)(284,294)(285,289)(286,292)(287,291)(288,290);;
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*s2*s0*s1*s0*s1*s0*s1*s0*s1*s2*s0*s1, 
s0*s1*s2*s0*s1*s0*s1*s2*s1*s0*s1*s2*s0*s1*s0*s1*s2*s1, 
s0*s1*s2*s1*s2*s1*s2*s0*s1*s0*s1*s2*s1*s2*s1*s2*s0*s1, 
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(324)!(  1,  3)(  2,  4)(  5,  7)(  6,  8)(  9, 11)( 10, 12)( 13, 27)( 14, 28)( 15, 25)( 16, 26)( 17, 31)( 18, 32)( 19, 29)( 20, 30)( 21, 35)( 22, 36)( 23, 33)( 24, 34)( 37, 99)( 38,100)( 39, 97)( 40, 98)( 41,103)( 42,104)( 43,101)( 44,102)( 45,107)( 46,108)( 47,105)( 48,106)( 49, 87)( 50, 88)( 51, 85)( 52, 86)( 53, 91)( 54, 92)( 55, 89)( 56, 90)( 57, 95)( 58, 96)( 59, 93)( 60, 94)( 61, 75)( 62, 76)( 63, 73)( 64, 74)( 65, 79)( 66, 80)( 67, 77)( 68, 78)( 69, 83)( 70, 84)( 71, 81)( 72, 82)(109,111)(110,112)(113,115)(114,116)(117,119)(118,120)(121,135)(122,136)(123,133)(124,134)(125,139)(126,140)(127,137)(128,138)(129,143)(130,144)(131,141)(132,142)(145,207)(146,208)(147,205)(148,206)(149,211)(150,212)(151,209)(152,210)(153,215)(154,216)(155,213)(156,214)(157,195)(158,196)(159,193)(160,194)(161,199)(162,200)(163,197)(164,198)(165,203)(166,204)(167,201)(168,202)(169,183)(170,184)(171,181)(172,182)(173,187)(174,188)(175,185)(176,186)(177,191)(178,192)(179,189)(180,190)(217,219)(218,220)(221,223)(222,224)(225,227)(226,228)(229,243)(230,244)(231,241)(232,242)(233,247)(234,248)(235,245)(236,246)(237,251)(238,252)(239,249)(240,250)(253,315)(254,316)(255,313)(256,314)(257,319)(258,320)(259,317)(260,318)(261,323)(262,324)(263,321)(264,322)(265,303)(266,304)(267,301)(268,302)(269,307)(270,308)(271,305)(272,306)(273,311)(274,312)(275,309)(276,310)(277,291)(278,292)(279,289)(280,290)(281,295)(282,296)(283,293)(284,294)(285,299)(286,300)(287,297)(288,298);
s1 := Sym(324)!(  1, 37)(  2, 38)(  3, 40)(  4, 39)(  5, 45)(  6, 46)(  7, 48)(  8, 47)(  9, 41)( 10, 42)( 11, 44)( 12, 43)( 13, 61)( 14, 62)( 15, 64)( 16, 63)( 17, 69)( 18, 70)( 19, 72)( 20, 71)( 21, 65)( 22, 66)( 23, 68)( 24, 67)( 25, 49)( 26, 50)( 27, 52)( 28, 51)( 29, 57)( 30, 58)( 31, 60)( 32, 59)( 33, 53)( 34, 54)( 35, 56)( 36, 55)( 73, 97)( 74, 98)( 75,100)( 76, 99)( 77,105)( 78,106)( 79,108)( 80,107)( 81,101)( 82,102)( 83,104)( 84,103)( 87, 88)( 89, 93)( 90, 94)( 91, 96)( 92, 95)(109,261)(110,262)(111,264)(112,263)(113,257)(114,258)(115,260)(116,259)(117,253)(118,254)(119,256)(120,255)(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)(145,225)(146,226)(147,228)(148,227)(149,221)(150,222)(151,224)(152,223)(153,217)(154,218)(155,220)(156,219)(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)(181,321)(182,322)(183,324)(184,323)(185,317)(186,318)(187,320)(188,319)(189,313)(190,314)(191,316)(192,315)(193,309)(194,310)(195,312)(196,311)(197,305)(198,306)(199,308)(200,307)(201,301)(202,302)(203,304)(204,303)(205,297)(206,298)(207,300)(208,299)(209,293)(210,294)(211,296)(212,295)(213,289)(214,290)(215,292)(216,291);
s2 := Sym(324)!(  1,109)(  2,112)(  3,111)(  4,110)(  5,117)(  6,120)(  7,119)(  8,118)(  9,113)( 10,116)( 11,115)( 12,114)( 13,133)( 14,136)( 15,135)( 16,134)( 17,141)( 18,144)( 19,143)( 20,142)( 21,137)( 22,140)( 23,139)( 24,138)( 25,121)( 26,124)( 27,123)( 28,122)( 29,129)( 30,132)( 31,131)( 32,130)( 33,125)( 34,128)( 35,127)( 36,126)( 37,205)( 38,208)( 39,207)( 40,206)( 41,213)( 42,216)( 43,215)( 44,214)( 45,209)( 46,212)( 47,211)( 48,210)( 49,193)( 50,196)( 51,195)( 52,194)( 53,201)( 54,204)( 55,203)( 56,202)( 57,197)( 58,200)( 59,199)( 60,198)( 61,181)( 62,184)( 63,183)( 64,182)( 65,189)( 66,192)( 67,191)( 68,190)( 69,185)( 70,188)( 71,187)( 72,186)( 73,169)( 74,172)( 75,171)( 76,170)( 77,177)( 78,180)( 79,179)( 80,178)( 81,173)( 82,176)( 83,175)( 84,174)( 85,157)( 86,160)( 87,159)( 88,158)( 89,165)( 90,168)( 91,167)( 92,166)( 93,161)( 94,164)( 95,163)( 96,162)( 97,145)( 98,148)( 99,147)(100,146)(101,153)(102,156)(103,155)(104,154)(105,149)(106,152)(107,151)(108,150)(217,225)(218,228)(219,227)(220,226)(222,224)(229,249)(230,252)(231,251)(232,250)(233,245)(234,248)(235,247)(236,246)(237,241)(238,244)(239,243)(240,242)(253,321)(254,324)(255,323)(256,322)(257,317)(258,320)(259,319)(260,318)(261,313)(262,316)(263,315)(264,314)(265,309)(266,312)(267,311)(268,310)(269,305)(270,308)(271,307)(272,306)(273,301)(274,304)(275,303)(276,302)(277,297)(278,300)(279,299)(280,298)(281,293)(282,296)(283,295)(284,294)(285,289)(286,292)(287,291)(288,290);
poly := sub<Sym(324)|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*s2*s0*s1*s0*s1*s0*s1*s0*s1*s2*s0*s1, 
s0*s1*s2*s0*s1*s0*s1*s2*s1*s0*s1*s2*s0*s1*s0*s1*s2*s1, 
s0*s1*s2*s1*s2*s1*s2*s0*s1*s0*s1*s2*s1*s2*s1*s2*s0*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 >;

References

None.

to this polytope.