# Polytope of Type {18,9}

Atlas Canonical Name : {18,9}*972d
if this polytope has a name.
Group : SmallGroup(972,104)
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
Related Polytopes :
Facet
Vertex Figure
Dual
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, {18,3}*324
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}*1944j
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, 56)( 29, 55)( 30, 57)( 31, 59)
( 32, 58)( 33, 60)( 34, 62)( 35, 61)( 36, 63)( 37, 77)( 38, 76)( 39, 78)
( 40, 80)( 41, 79)( 42, 81)( 43, 74)( 44, 73)( 45, 75)( 46, 71)( 47, 70)
( 48, 72)( 49, 65)( 50, 64)( 51, 66)( 52, 68)( 53, 67)( 54, 69)( 83, 84)
( 86, 87)( 89, 90)( 91,103)( 92,105)( 93,104)( 94,106)( 95,108)( 96,107)
( 97,100)( 98,102)( 99,101)(109,137)(110,136)(111,138)(112,140)(113,139)
(114,141)(115,143)(116,142)(117,144)(118,158)(119,157)(120,159)(121,161)
(122,160)(123,162)(124,155)(125,154)(126,156)(127,152)(128,151)(129,153)
(130,146)(131,145)(132,147)(133,149)(134,148)(135,150)(164,165)(167,168)
(170,171)(172,184)(173,186)(174,185)(175,187)(176,189)(177,188)(178,181)
(179,183)(180,182)(190,218)(191,217)(192,219)(193,221)(194,220)(195,222)
(196,224)(197,223)(198,225)(199,239)(200,238)(201,240)(202,242)(203,241)
(204,243)(205,236)(206,235)(207,237)(208,233)(209,232)(210,234)(211,227)
(212,226)(213,228)(214,230)(215,229)(216,231);;
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, 56)( 58, 62)( 59, 61)( 60, 63)( 65, 66)
( 67, 70)( 68, 72)( 69, 71)( 73, 75)( 76, 81)( 77, 80)( 78, 79)( 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)(136,221)(137,220)(138,222)
(139,218)(140,217)(141,219)(142,224)(143,223)(144,225)(145,229)(146,231)
(147,230)(148,226)(149,228)(150,227)(151,232)(152,234)(153,233)(154,240)
(155,239)(156,238)(157,237)(158,236)(159,235)(160,243)(161,242)(162,241);;
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)( 28,158)( 29,157)( 30,159)( 31,155)( 32,154)
( 33,156)( 34,161)( 35,160)( 36,162)( 37,137)( 38,136)( 39,138)( 40,143)
( 41,142)( 42,144)( 43,140)( 44,139)( 45,141)( 46,152)( 47,151)( 48,153)
( 49,149)( 50,148)( 51,150)( 52,146)( 53,145)( 54,147)( 55,119)( 56,118)
( 57,120)( 58,125)( 59,124)( 60,126)( 61,122)( 62,121)( 63,123)( 64,134)
( 65,133)( 66,135)( 67,131)( 68,130)( 69,132)( 70,128)( 71,127)( 72,129)
( 73,113)( 74,112)( 75,114)( 76,110)( 77,109)( 78,111)( 79,116)( 80,115)
( 81,117)(163,166)(164,168)(165,167)(170,171)(173,174)(175,178)(176,180)
(177,179)(181,187)(182,189)(183,188)(185,186)(190,242)(191,241)(192,243)
(193,239)(194,238)(195,240)(196,236)(197,235)(198,237)(199,221)(200,220)
(201,222)(202,218)(203,217)(204,219)(205,224)(206,223)(207,225)(208,227)
(209,226)(210,228)(211,233)(212,232)(213,234)(214,230)(215,229)(216,231);;
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*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1,
s2*s0*s1*s0*s1*s2*s1*s0*s1*s2*s0*s1*s2*s0*s1*s2*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*s0*s1*s0*s1*s0*s1*s0*s1*s2*s0*s1*s0*s1*s0*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, 56)( 29, 55)( 30, 57)
( 31, 59)( 32, 58)( 33, 60)( 34, 62)( 35, 61)( 36, 63)( 37, 77)( 38, 76)
( 39, 78)( 40, 80)( 41, 79)( 42, 81)( 43, 74)( 44, 73)( 45, 75)( 46, 71)
( 47, 70)( 48, 72)( 49, 65)( 50, 64)( 51, 66)( 52, 68)( 53, 67)( 54, 69)
( 83, 84)( 86, 87)( 89, 90)( 91,103)( 92,105)( 93,104)( 94,106)( 95,108)
( 96,107)( 97,100)( 98,102)( 99,101)(109,137)(110,136)(111,138)(112,140)
(113,139)(114,141)(115,143)(116,142)(117,144)(118,158)(119,157)(120,159)
(121,161)(122,160)(123,162)(124,155)(125,154)(126,156)(127,152)(128,151)
(129,153)(130,146)(131,145)(132,147)(133,149)(134,148)(135,150)(164,165)
(167,168)(170,171)(172,184)(173,186)(174,185)(175,187)(176,189)(177,188)
(178,181)(179,183)(180,182)(190,218)(191,217)(192,219)(193,221)(194,220)
(195,222)(196,224)(197,223)(198,225)(199,239)(200,238)(201,240)(202,242)
(203,241)(204,243)(205,236)(206,235)(207,237)(208,233)(209,232)(210,234)
(211,227)(212,226)(213,228)(214,230)(215,229)(216,231);
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, 56)( 58, 62)( 59, 61)( 60, 63)
( 65, 66)( 67, 70)( 68, 72)( 69, 71)( 73, 75)( 76, 81)( 77, 80)( 78, 79)
( 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)(136,221)(137,220)
(138,222)(139,218)(140,217)(141,219)(142,224)(143,223)(144,225)(145,229)
(146,231)(147,230)(148,226)(149,228)(150,227)(151,232)(152,234)(153,233)
(154,240)(155,239)(156,238)(157,237)(158,236)(159,235)(160,243)(161,242)
(162,241);
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)( 28,158)( 29,157)( 30,159)( 31,155)
( 32,154)( 33,156)( 34,161)( 35,160)( 36,162)( 37,137)( 38,136)( 39,138)
( 40,143)( 41,142)( 42,144)( 43,140)( 44,139)( 45,141)( 46,152)( 47,151)
( 48,153)( 49,149)( 50,148)( 51,150)( 52,146)( 53,145)( 54,147)( 55,119)
( 56,118)( 57,120)( 58,125)( 59,124)( 60,126)( 61,122)( 62,121)( 63,123)
( 64,134)( 65,133)( 66,135)( 67,131)( 68,130)( 69,132)( 70,128)( 71,127)
( 72,129)( 73,113)( 74,112)( 75,114)( 76,110)( 77,109)( 78,111)( 79,116)
( 80,115)( 81,117)(163,166)(164,168)(165,167)(170,171)(173,174)(175,178)
(176,180)(177,179)(181,187)(182,189)(183,188)(185,186)(190,242)(191,241)
(192,243)(193,239)(194,238)(195,240)(196,236)(197,235)(198,237)(199,221)
(200,220)(201,222)(202,218)(203,217)(204,219)(205,224)(206,223)(207,225)
(208,227)(209,226)(210,228)(211,233)(212,232)(213,234)(214,230)(215,229)
(216,231);
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*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1,
s2*s0*s1*s0*s1*s2*s1*s0*s1*s2*s0*s1*s2*s0*s1*s2*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*s0*s1*s0*s1*s0*s1*s0*s1*s2*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 >;

```
References : None.
to this polytope