# Polytope of Type {18,9}

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