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# Polytope of Type {9,18}

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

```
References : None.
to this polytope