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# Polytope of Type {6,9}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,9}*972e
if this polytope has a name.
Group : SmallGroup(972,111)
Rank : 3
Schlafli Type : {6,9}
Number of vertices, edges, etc : 54, 243, 81
Order of s0s1s2 : 18
Order of s0s1s2s1 : 6
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Related Polytopes :
Facet
Vertex Figure
Dual
Petrial
Facet Of :
{6,9,2} of size 1944
Vertex Figure Of :
{2,6,9} of size 1944
Quotients (Maximal Quotients in Boldface) :
3-fold quotients : {6,9}*324a, {6,9}*324c, {6,9}*324d, {6,3}*324
9-fold quotients : {6,9}*108, {6,3}*108
27-fold quotients : {2,9}*36, {6,3}*36
81-fold quotients : {2,3}*12
Covers (Minimal Covers in Boldface) :
2-fold covers : {6,18}*1944i
Permutation Representation (GAP) :
```s0 := (  4,  8)(  5,  9)(  6,  7)( 13, 17)( 14, 18)( 15, 16)( 22, 26)( 23, 27)
( 24, 25)( 28, 55)( 29, 56)( 30, 57)( 31, 62)( 32, 63)( 33, 61)( 34, 60)
( 35, 58)( 36, 59)( 37, 64)( 38, 65)( 39, 66)( 40, 71)( 41, 72)( 42, 70)
( 43, 69)( 44, 67)( 45, 68)( 46, 73)( 47, 74)( 48, 75)( 49, 80)( 50, 81)
( 51, 79)( 52, 78)( 53, 76)( 54, 77)( 85, 89)( 86, 90)( 87, 88)( 94, 98)
( 95, 99)( 96, 97)(103,107)(104,108)(105,106)(109,136)(110,137)(111,138)
(112,143)(113,144)(114,142)(115,141)(116,139)(117,140)(118,145)(119,146)
(120,147)(121,152)(122,153)(123,151)(124,150)(125,148)(126,149)(127,154)
(128,155)(129,156)(130,161)(131,162)(132,160)(133,159)(134,157)(135,158)
(166,170)(167,171)(168,169)(175,179)(176,180)(177,178)(184,188)(185,189)
(186,187)(190,217)(191,218)(192,219)(193,224)(194,225)(195,223)(196,222)
(197,220)(198,221)(199,226)(200,227)(201,228)(202,233)(203,234)(204,232)
(205,231)(206,229)(207,230)(208,235)(209,236)(210,237)(211,242)(212,243)
(213,241)(214,240)(215,238)(216,239);;
s1 := (  1, 28)(  2, 30)(  3, 29)(  4, 31)(  5, 33)(  6, 32)(  7, 34)(  8, 36)
(  9, 35)( 10, 46)( 11, 48)( 12, 47)( 13, 49)( 14, 51)( 15, 50)( 16, 52)
( 17, 54)( 18, 53)( 19, 37)( 20, 39)( 21, 38)( 22, 40)( 23, 42)( 24, 41)
( 25, 43)( 26, 45)( 27, 44)( 56, 57)( 59, 60)( 62, 63)( 64, 73)( 65, 75)
( 66, 74)( 67, 76)( 68, 78)( 69, 77)( 70, 79)( 71, 81)( 72, 80)( 82,208)
( 83,210)( 84,209)( 85,211)( 86,213)( 87,212)( 88,214)( 89,216)( 90,215)
( 91,199)( 92,201)( 93,200)( 94,202)( 95,204)( 96,203)( 97,205)( 98,207)
( 99,206)(100,190)(101,192)(102,191)(103,193)(104,195)(105,194)(106,196)
(107,198)(108,197)(109,181)(110,183)(111,182)(112,184)(113,186)(114,185)
(115,187)(116,189)(117,188)(118,172)(119,174)(120,173)(121,175)(122,177)
(123,176)(124,178)(125,180)(126,179)(127,163)(128,165)(129,164)(130,166)
(131,168)(132,167)(133,169)(134,171)(135,170)(136,235)(137,237)(138,236)
(139,238)(140,240)(141,239)(142,241)(143,243)(144,242)(145,226)(146,228)
(147,227)(148,229)(149,231)(150,230)(151,232)(152,234)(153,233)(154,217)
(155,219)(156,218)(157,220)(158,222)(159,221)(160,223)(161,225)(162,224);;
s2 := (  1, 82)(  2, 84)(  3, 83)(  4, 87)(  5, 86)(  6, 85)(  7, 89)(  8, 88)
(  9, 90)( 10,100)( 11,102)( 12,101)( 13,105)( 14,104)( 15,103)( 16,107)
( 17,106)( 18,108)( 19, 91)( 20, 93)( 21, 92)( 22, 96)( 23, 95)( 24, 94)
( 25, 98)( 26, 97)( 27, 99)( 28,143)( 29,142)( 30,144)( 31,136)( 32,138)
( 33,137)( 34,141)( 35,140)( 36,139)( 37,161)( 38,160)( 39,162)( 40,154)
( 41,156)( 42,155)( 43,159)( 44,158)( 45,157)( 46,152)( 47,151)( 48,153)
( 49,145)( 50,147)( 51,146)( 52,150)( 53,149)( 54,148)( 55,112)( 56,114)
( 57,113)( 58,117)( 59,116)( 60,115)( 61,110)( 62,109)( 63,111)( 64,130)
( 65,132)( 66,131)( 67,135)( 68,134)( 69,133)( 70,128)( 71,127)( 72,129)
( 73,121)( 74,123)( 75,122)( 76,126)( 77,125)( 78,124)( 79,119)( 80,118)
( 81,120)(163,181)(164,183)(165,182)(166,186)(167,185)(168,184)(169,188)
(170,187)(171,189)(173,174)(175,177)(178,179)(190,242)(191,241)(192,243)
(193,235)(194,237)(195,236)(196,240)(197,239)(198,238)(199,233)(200,232)
(201,234)(202,226)(203,228)(204,227)(205,231)(206,230)(207,229)(208,224)
(209,223)(210,225)(211,217)(212,219)(213,218)(214,222)(215,221)(216,220);;
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*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,
s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s0*s1 ];;
poly := F / rels;;

```
Permutation Representation (Magma) :
```s0 := Sym(243)!(  4,  8)(  5,  9)(  6,  7)( 13, 17)( 14, 18)( 15, 16)( 22, 26)
( 23, 27)( 24, 25)( 28, 55)( 29, 56)( 30, 57)( 31, 62)( 32, 63)( 33, 61)
( 34, 60)( 35, 58)( 36, 59)( 37, 64)( 38, 65)( 39, 66)( 40, 71)( 41, 72)
( 42, 70)( 43, 69)( 44, 67)( 45, 68)( 46, 73)( 47, 74)( 48, 75)( 49, 80)
( 50, 81)( 51, 79)( 52, 78)( 53, 76)( 54, 77)( 85, 89)( 86, 90)( 87, 88)
( 94, 98)( 95, 99)( 96, 97)(103,107)(104,108)(105,106)(109,136)(110,137)
(111,138)(112,143)(113,144)(114,142)(115,141)(116,139)(117,140)(118,145)
(119,146)(120,147)(121,152)(122,153)(123,151)(124,150)(125,148)(126,149)
(127,154)(128,155)(129,156)(130,161)(131,162)(132,160)(133,159)(134,157)
(135,158)(166,170)(167,171)(168,169)(175,179)(176,180)(177,178)(184,188)
(185,189)(186,187)(190,217)(191,218)(192,219)(193,224)(194,225)(195,223)
(196,222)(197,220)(198,221)(199,226)(200,227)(201,228)(202,233)(203,234)
(204,232)(205,231)(206,229)(207,230)(208,235)(209,236)(210,237)(211,242)
(212,243)(213,241)(214,240)(215,238)(216,239);
s1 := Sym(243)!(  1, 28)(  2, 30)(  3, 29)(  4, 31)(  5, 33)(  6, 32)(  7, 34)
(  8, 36)(  9, 35)( 10, 46)( 11, 48)( 12, 47)( 13, 49)( 14, 51)( 15, 50)
( 16, 52)( 17, 54)( 18, 53)( 19, 37)( 20, 39)( 21, 38)( 22, 40)( 23, 42)
( 24, 41)( 25, 43)( 26, 45)( 27, 44)( 56, 57)( 59, 60)( 62, 63)( 64, 73)
( 65, 75)( 66, 74)( 67, 76)( 68, 78)( 69, 77)( 70, 79)( 71, 81)( 72, 80)
( 82,208)( 83,210)( 84,209)( 85,211)( 86,213)( 87,212)( 88,214)( 89,216)
( 90,215)( 91,199)( 92,201)( 93,200)( 94,202)( 95,204)( 96,203)( 97,205)
( 98,207)( 99,206)(100,190)(101,192)(102,191)(103,193)(104,195)(105,194)
(106,196)(107,198)(108,197)(109,181)(110,183)(111,182)(112,184)(113,186)
(114,185)(115,187)(116,189)(117,188)(118,172)(119,174)(120,173)(121,175)
(122,177)(123,176)(124,178)(125,180)(126,179)(127,163)(128,165)(129,164)
(130,166)(131,168)(132,167)(133,169)(134,171)(135,170)(136,235)(137,237)
(138,236)(139,238)(140,240)(141,239)(142,241)(143,243)(144,242)(145,226)
(146,228)(147,227)(148,229)(149,231)(150,230)(151,232)(152,234)(153,233)
(154,217)(155,219)(156,218)(157,220)(158,222)(159,221)(160,223)(161,225)
(162,224);
s2 := Sym(243)!(  1, 82)(  2, 84)(  3, 83)(  4, 87)(  5, 86)(  6, 85)(  7, 89)
(  8, 88)(  9, 90)( 10,100)( 11,102)( 12,101)( 13,105)( 14,104)( 15,103)
( 16,107)( 17,106)( 18,108)( 19, 91)( 20, 93)( 21, 92)( 22, 96)( 23, 95)
( 24, 94)( 25, 98)( 26, 97)( 27, 99)( 28,143)( 29,142)( 30,144)( 31,136)
( 32,138)( 33,137)( 34,141)( 35,140)( 36,139)( 37,161)( 38,160)( 39,162)
( 40,154)( 41,156)( 42,155)( 43,159)( 44,158)( 45,157)( 46,152)( 47,151)
( 48,153)( 49,145)( 50,147)( 51,146)( 52,150)( 53,149)( 54,148)( 55,112)
( 56,114)( 57,113)( 58,117)( 59,116)( 60,115)( 61,110)( 62,109)( 63,111)
( 64,130)( 65,132)( 66,131)( 67,135)( 68,134)( 69,133)( 70,128)( 71,127)
( 72,129)( 73,121)( 74,123)( 75,122)( 76,126)( 77,125)( 78,124)( 79,119)
( 80,118)( 81,120)(163,181)(164,183)(165,182)(166,186)(167,185)(168,184)
(169,188)(170,187)(171,189)(173,174)(175,177)(178,179)(190,242)(191,241)
(192,243)(193,235)(194,237)(195,236)(196,240)(197,239)(198,238)(199,233)
(200,232)(201,234)(202,226)(203,228)(204,227)(205,231)(206,230)(207,229)
(208,224)(209,223)(210,225)(211,217)(212,219)(213,218)(214,222)(215,221)
(216,220);
poly := sub<Sym(243)|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*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,
s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s0*s1 >;

```
References : None.
to this polytope