# Polytope of Type {18,9}

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

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

```
References : None.
to this polytope