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# Polytope of Type {18,18}

Atlas Canonical Name : {18,18}*1944af
if this polytope has a name.
Group : SmallGroup(1944,2339)
Rank : 3
Schlafli Type : {18,18}
Number of vertices, edges, etc : 54, 486, 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 :
None in this Atlas
Vertex Figure Of :
None in this Atlas
Quotients (Maximal Quotients in Boldface) :
3-fold quotients : {18,18}*648c, {18,6}*648i
6-fold quotients : {9,18}*324
9-fold quotients : {18,6}*216a, {18,6}*216b, {6,6}*216d
18-fold quotients : {9,6}*108
27-fold quotients : {18,2}*72, {6,6}*72a, {6,6}*72b, {6,6}*72c
54-fold quotients : {9,2}*36, {3,6}*36, {6,3}*36
81-fold quotients : {2,6}*24, {6,2}*24
162-fold quotients : {2,3}*12, {3,2}*12
243-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
```s0 := (  2,  3)(  4,  7)(  5,  9)(  6,  8)( 10, 22)( 11, 24)( 12, 23)( 13, 19)
( 14, 21)( 15, 20)( 16, 25)( 17, 27)( 18, 26)( 29, 30)( 31, 34)( 32, 36)
( 33, 35)( 37, 49)( 38, 51)( 39, 50)( 40, 46)( 41, 48)( 42, 47)( 43, 52)
( 44, 54)( 45, 53)( 56, 57)( 58, 61)( 59, 63)( 60, 62)( 64, 76)( 65, 78)
( 66, 77)( 67, 73)( 68, 75)( 69, 74)( 70, 79)( 71, 81)( 72, 80)( 82,164)
( 83,163)( 84,165)( 85,170)( 86,169)( 87,171)( 88,167)( 89,166)( 90,168)
( 91,185)( 92,184)( 93,186)( 94,182)( 95,181)( 96,183)( 97,188)( 98,187)
( 99,189)(100,176)(101,175)(102,177)(103,173)(104,172)(105,174)(106,179)
(107,178)(108,180)(109,191)(110,190)(111,192)(112,197)(113,196)(114,198)
(115,194)(116,193)(117,195)(118,212)(119,211)(120,213)(121,209)(122,208)
(123,210)(124,215)(125,214)(126,216)(127,203)(128,202)(129,204)(130,200)
(131,199)(132,201)(133,206)(134,205)(135,207)(136,218)(137,217)(138,219)
(139,224)(140,223)(141,225)(142,221)(143,220)(144,222)(145,239)(146,238)
(147,240)(148,236)(149,235)(150,237)(151,242)(152,241)(153,243)(154,230)
(155,229)(156,231)(157,227)(158,226)(159,228)(160,233)(161,232)(162,234);;
s1 := (  1, 91)(  2, 93)(  3, 92)(  4, 97)(  5, 99)(  6, 98)(  7, 94)(  8, 96)
(  9, 95)( 10, 82)( 11, 84)( 12, 83)( 13, 88)( 14, 90)( 15, 89)( 16, 85)
( 17, 87)( 18, 86)( 19,103)( 20,105)( 21,104)( 22,100)( 23,102)( 24,101)
( 25,106)( 26,108)( 27,107)( 28,145)( 29,147)( 30,146)( 31,151)( 32,153)
( 33,152)( 34,148)( 35,150)( 36,149)( 37,136)( 38,138)( 39,137)( 40,142)
( 41,144)( 42,143)( 43,139)( 44,141)( 45,140)( 46,157)( 47,159)( 48,158)
( 49,154)( 50,156)( 51,155)( 52,160)( 53,162)( 54,161)( 55,118)( 56,120)
( 57,119)( 58,124)( 59,126)( 60,125)( 61,121)( 62,123)( 63,122)( 64,109)
( 65,111)( 66,110)( 67,115)( 68,117)( 69,116)( 70,112)( 71,114)( 72,113)
( 73,130)( 74,132)( 75,131)( 76,127)( 77,129)( 78,128)( 79,133)( 80,135)
( 81,134)(163,173)(164,172)(165,174)(166,179)(167,178)(168,180)(169,176)
(170,175)(171,177)(181,185)(182,184)(183,186)(187,188)(190,227)(191,226)
(192,228)(193,233)(194,232)(195,234)(196,230)(197,229)(198,231)(199,218)
(200,217)(201,219)(202,224)(203,223)(204,225)(205,221)(206,220)(207,222)
(208,239)(209,238)(210,240)(211,236)(212,235)(213,237)(214,242)(215,241)
(216,243);;
s2 := (  1, 28)(  2, 29)(  3, 30)(  4, 34)(  5, 35)(  6, 36)(  7, 31)(  8, 32)
(  9, 33)( 10, 49)( 11, 50)( 12, 51)( 13, 46)( 14, 47)( 15, 48)( 16, 52)
( 17, 53)( 18, 54)( 19, 40)( 20, 41)( 21, 42)( 22, 37)( 23, 38)( 24, 39)
( 25, 43)( 26, 44)( 27, 45)( 58, 61)( 59, 62)( 60, 63)( 64, 76)( 65, 77)
( 66, 78)( 67, 73)( 68, 74)( 69, 75)( 70, 79)( 71, 80)( 72, 81)( 82,109)
( 83,110)( 84,111)( 85,115)( 86,116)( 87,117)( 88,112)( 89,113)( 90,114)
( 91,130)( 92,131)( 93,132)( 94,127)( 95,128)( 96,129)( 97,133)( 98,134)
( 99,135)(100,121)(101,122)(102,123)(103,118)(104,119)(105,120)(106,124)
(107,125)(108,126)(139,142)(140,143)(141,144)(145,157)(146,158)(147,159)
(148,154)(149,155)(150,156)(151,160)(152,161)(153,162)(163,190)(164,191)
(165,192)(166,196)(167,197)(168,198)(169,193)(170,194)(171,195)(172,211)
(173,212)(174,213)(175,208)(176,209)(177,210)(178,214)(179,215)(180,216)
(181,202)(182,203)(183,204)(184,199)(185,200)(186,201)(187,205)(188,206)
(189,207)(220,223)(221,224)(222,225)(226,238)(227,239)(228,240)(229,235)
(230,236)(231,237)(232,241)(233,242)(234,243);;
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*s0*s1*s2*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,
s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s0*s1*s2*s1*s2*s1*s2*s1*s2*s1*s0*s1,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 ];;
poly := F / rels;;

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

```
References : None.
to this polytope