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# Polytope of Type {18,9}

Atlas Canonical Name : {18,9}*972f
if this polytope has a name.
Group : SmallGroup(972,108)
Rank : 3
Schlafli Type : {18,9}
Number of vertices, edges, etc : 54, 243, 27
Order of s0s1s2 : 6
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}*324c
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}*1944p
Permutation Representation (GAP) :
```s0 := (  4,  8)(  5,  9)(  6,  7)( 10, 19)( 11, 20)( 12, 21)( 13, 26)( 14, 27)
( 15, 25)( 16, 24)( 17, 22)( 18, 23)( 31, 35)( 32, 36)( 33, 34)( 37, 46)
( 38, 47)( 39, 48)( 40, 53)( 41, 54)( 42, 52)( 43, 51)( 44, 49)( 45, 50)
( 58, 62)( 59, 63)( 60, 61)( 64, 73)( 65, 74)( 66, 75)( 67, 80)( 68, 81)
( 69, 79)( 70, 78)( 71, 76)( 72, 77)( 82,181)( 83,182)( 84,183)( 85,188)
( 86,189)( 87,187)( 88,186)( 89,184)( 90,185)( 91,172)( 92,173)( 93,174)
( 94,179)( 95,180)( 96,178)( 97,177)( 98,175)( 99,176)(100,163)(101,164)
(102,165)(103,170)(104,171)(105,169)(106,168)(107,166)(108,167)(109,208)
(110,209)(111,210)(112,215)(113,216)(114,214)(115,213)(116,211)(117,212)
(118,199)(119,200)(120,201)(121,206)(122,207)(123,205)(124,204)(125,202)
(126,203)(127,190)(128,191)(129,192)(130,197)(131,198)(132,196)(133,195)
(134,193)(135,194)(136,235)(137,236)(138,237)(139,242)(140,243)(141,241)
(142,240)(143,238)(144,239)(145,226)(146,227)(147,228)(148,233)(149,234)
(150,232)(151,231)(152,229)(153,230)(154,217)(155,218)(156,219)(157,224)
(158,225)(159,223)(160,222)(161,220)(162,221);;
s1 := (  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)( 28,142)( 29,144)( 30,143)( 31,136)( 32,138)
( 33,137)( 34,139)( 35,141)( 36,140)( 37,160)( 38,162)( 39,161)( 40,154)
( 41,156)( 42,155)( 43,157)( 44,159)( 45,158)( 46,151)( 47,153)( 48,152)
( 49,145)( 50,147)( 51,146)( 52,148)( 53,150)( 54,149)( 55,112)( 56,114)
( 57,113)( 58,115)( 59,117)( 60,116)( 61,109)( 62,111)( 63,110)( 64,130)
( 65,132)( 66,131)( 67,133)( 68,135)( 69,134)( 70,127)( 71,129)( 72,128)
( 73,121)( 74,123)( 75,122)( 76,124)( 77,126)( 78,125)( 79,118)( 80,120)
( 81,119)(163,181)(164,183)(165,182)(166,184)(167,186)(168,185)(169,187)
(170,189)(171,188)(173,174)(176,177)(179,180)(190,241)(191,243)(192,242)
(193,235)(194,237)(195,236)(196,238)(197,240)(198,239)(199,232)(200,234)
(201,233)(202,226)(203,228)(204,227)(205,229)(206,231)(207,230)(208,223)
(209,225)(210,224)(211,217)(212,219)(213,218)(214,220)(215,222)(216,221);;
s2 := (  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)( 56, 57)( 58, 60)( 61, 62)( 64, 73)( 65, 75)
( 66, 74)( 67, 78)( 68, 77)( 69, 76)( 70, 80)( 71, 79)( 72, 81)( 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)(136,235)(137,237)(138,236)
(139,240)(140,239)(141,238)(142,242)(143,241)(144,243)(145,226)(146,228)
(147,227)(148,231)(149,230)(150,229)(151,233)(152,232)(153,234)(154,217)
(155,219)(156,218)(157,222)(158,221)(159,220)(160,224)(161,223)(162,225);;
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,
s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1 ];;
poly := F / rels;;

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

```
References : None.
to this polytope