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# Polytope of Type {9,18}

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

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

```
References : None.
to this polytope