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# Polytope of Type {18,9}

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

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

```
References : None.
to this polytope