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# Polytope of Type {18,18}

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 : 6
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Self-Dual
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}*648a, {6,18}*648i, {18,6}*648i
9-fold quotients : {6,18}*216a, {6,18}*216b, {18,6}*216a, {18,6}*216b, {6,6}*216d
18-fold quotients : {6,9}*108, {9,6}*108
27-fold quotients : {2,18}*72, {18,2}*72, {6,6}*72a, {6,6}*72b, {6,6}*72c
54-fold quotients : {2,9}*36, {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 := (  4,  7)(  5,  8)(  6,  9)( 10, 22)( 11, 23)( 12, 24)( 13, 19)( 14, 20)
( 15, 21)( 16, 25)( 17, 26)( 18, 27)( 28, 55)( 29, 56)( 30, 57)( 31, 61)
( 32, 62)( 33, 63)( 34, 58)( 35, 59)( 36, 60)( 37, 76)( 38, 77)( 39, 78)
( 40, 73)( 41, 74)( 42, 75)( 43, 79)( 44, 80)( 45, 81)( 46, 67)( 47, 68)
( 48, 69)( 49, 64)( 50, 65)( 51, 66)( 52, 70)( 53, 71)( 54, 72)( 85, 88)
( 86, 89)( 87, 90)( 91,103)( 92,104)( 93,105)( 94,100)( 95,101)( 96,102)
( 97,106)( 98,107)( 99,108)(109,136)(110,137)(111,138)(112,142)(113,143)
(114,144)(115,139)(116,140)(117,141)(118,157)(119,158)(120,159)(121,154)
(122,155)(123,156)(124,160)(125,161)(126,162)(127,148)(128,149)(129,150)
(130,145)(131,146)(132,147)(133,151)(134,152)(135,153)(166,169)(167,170)
(168,171)(172,184)(173,185)(174,186)(175,181)(176,182)(177,183)(178,187)
(179,188)(180,189)(190,217)(191,218)(192,219)(193,223)(194,224)(195,225)
(196,220)(197,221)(198,222)(199,238)(200,239)(201,240)(202,235)(203,236)
(204,237)(205,241)(206,242)(207,243)(208,229)(209,230)(210,231)(211,226)
(212,227)(213,228)(214,232)(215,233)(216,234);;
s1 := (  1, 37)(  2, 39)(  3, 38)(  4, 43)(  5, 45)(  6, 44)(  7, 40)(  8, 42)
(  9, 41)( 10, 28)( 11, 30)( 12, 29)( 13, 34)( 14, 36)( 15, 35)( 16, 31)
( 17, 33)( 18, 32)( 19, 49)( 20, 51)( 21, 50)( 22, 46)( 23, 48)( 24, 47)
( 25, 52)( 26, 54)( 27, 53)( 55, 64)( 56, 66)( 57, 65)( 58, 70)( 59, 72)
( 60, 71)( 61, 67)( 62, 69)( 63, 68)( 73, 76)( 74, 78)( 75, 77)( 80, 81)
( 82,200)( 83,199)( 84,201)( 85,206)( 86,205)( 87,207)( 88,203)( 89,202)
( 90,204)( 91,191)( 92,190)( 93,192)( 94,197)( 95,196)( 96,198)( 97,194)
( 98,193)( 99,195)(100,212)(101,211)(102,213)(103,209)(104,208)(105,210)
(106,215)(107,214)(108,216)(109,173)(110,172)(111,174)(112,179)(113,178)
(114,180)(115,176)(116,175)(117,177)(118,164)(119,163)(120,165)(121,170)
(122,169)(123,171)(124,167)(125,166)(126,168)(127,185)(128,184)(129,186)
(130,182)(131,181)(132,183)(133,188)(134,187)(135,189)(136,227)(137,226)
(138,228)(139,233)(140,232)(141,234)(142,230)(143,229)(144,231)(145,218)
(146,217)(147,219)(148,224)(149,223)(150,225)(151,221)(152,220)(153,222)
(154,239)(155,238)(156,240)(157,236)(158,235)(159,237)(160,242)(161,241)
(162,243);;
s2 := (  1, 82)(  2, 84)(  3, 83)(  4, 85)(  5, 87)(  6, 86)(  7, 88)(  8, 90)
(  9, 89)( 10, 91)( 11, 93)( 12, 92)( 13, 94)( 14, 96)( 15, 95)( 16, 97)
( 17, 99)( 18, 98)( 19,100)( 20,102)( 21,101)( 22,103)( 23,105)( 24,104)
( 25,106)( 26,108)( 27,107)( 28,136)( 29,138)( 30,137)( 31,139)( 32,141)
( 33,140)( 34,142)( 35,144)( 36,143)( 37,145)( 38,147)( 39,146)( 40,148)
( 41,150)( 42,149)( 43,151)( 44,153)( 45,152)( 46,154)( 47,156)( 48,155)
( 49,157)( 50,159)( 51,158)( 52,160)( 53,162)( 54,161)( 55,109)( 56,111)
( 57,110)( 58,112)( 59,114)( 60,113)( 61,115)( 62,117)( 63,116)( 64,118)
( 65,120)( 66,119)( 67,121)( 68,123)( 69,122)( 70,124)( 71,126)( 72,125)
( 73,127)( 74,129)( 75,128)( 76,130)( 77,132)( 78,131)( 79,133)( 80,135)
( 81,134)(163,164)(166,167)(169,170)(172,173)(175,176)(178,179)(181,182)
(184,185)(187,188)(190,218)(191,217)(192,219)(193,221)(194,220)(195,222)
(196,224)(197,223)(198,225)(199,227)(200,226)(201,228)(202,230)(203,229)
(204,231)(205,233)(206,232)(207,234)(208,236)(209,235)(210,237)(211,239)
(212,238)(213,240)(214,242)(215,241)(216,243);;
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,
s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*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*s0*s1*s0*s1*s0*s1,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*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)!(  4,  7)(  5,  8)(  6,  9)( 10, 22)( 11, 23)( 12, 24)( 13, 19)
( 14, 20)( 15, 21)( 16, 25)( 17, 26)( 18, 27)( 28, 55)( 29, 56)( 30, 57)
( 31, 61)( 32, 62)( 33, 63)( 34, 58)( 35, 59)( 36, 60)( 37, 76)( 38, 77)
( 39, 78)( 40, 73)( 41, 74)( 42, 75)( 43, 79)( 44, 80)( 45, 81)( 46, 67)
( 47, 68)( 48, 69)( 49, 64)( 50, 65)( 51, 66)( 52, 70)( 53, 71)( 54, 72)
( 85, 88)( 86, 89)( 87, 90)( 91,103)( 92,104)( 93,105)( 94,100)( 95,101)
( 96,102)( 97,106)( 98,107)( 99,108)(109,136)(110,137)(111,138)(112,142)
(113,143)(114,144)(115,139)(116,140)(117,141)(118,157)(119,158)(120,159)
(121,154)(122,155)(123,156)(124,160)(125,161)(126,162)(127,148)(128,149)
(129,150)(130,145)(131,146)(132,147)(133,151)(134,152)(135,153)(166,169)
(167,170)(168,171)(172,184)(173,185)(174,186)(175,181)(176,182)(177,183)
(178,187)(179,188)(180,189)(190,217)(191,218)(192,219)(193,223)(194,224)
(195,225)(196,220)(197,221)(198,222)(199,238)(200,239)(201,240)(202,235)
(203,236)(204,237)(205,241)(206,242)(207,243)(208,229)(209,230)(210,231)
(211,226)(212,227)(213,228)(214,232)(215,233)(216,234);
s1 := Sym(243)!(  1, 37)(  2, 39)(  3, 38)(  4, 43)(  5, 45)(  6, 44)(  7, 40)
(  8, 42)(  9, 41)( 10, 28)( 11, 30)( 12, 29)( 13, 34)( 14, 36)( 15, 35)
( 16, 31)( 17, 33)( 18, 32)( 19, 49)( 20, 51)( 21, 50)( 22, 46)( 23, 48)
( 24, 47)( 25, 52)( 26, 54)( 27, 53)( 55, 64)( 56, 66)( 57, 65)( 58, 70)
( 59, 72)( 60, 71)( 61, 67)( 62, 69)( 63, 68)( 73, 76)( 74, 78)( 75, 77)
( 80, 81)( 82,200)( 83,199)( 84,201)( 85,206)( 86,205)( 87,207)( 88,203)
( 89,202)( 90,204)( 91,191)( 92,190)( 93,192)( 94,197)( 95,196)( 96,198)
( 97,194)( 98,193)( 99,195)(100,212)(101,211)(102,213)(103,209)(104,208)
(105,210)(106,215)(107,214)(108,216)(109,173)(110,172)(111,174)(112,179)
(113,178)(114,180)(115,176)(116,175)(117,177)(118,164)(119,163)(120,165)
(121,170)(122,169)(123,171)(124,167)(125,166)(126,168)(127,185)(128,184)
(129,186)(130,182)(131,181)(132,183)(133,188)(134,187)(135,189)(136,227)
(137,226)(138,228)(139,233)(140,232)(141,234)(142,230)(143,229)(144,231)
(145,218)(146,217)(147,219)(148,224)(149,223)(150,225)(151,221)(152,220)
(153,222)(154,239)(155,238)(156,240)(157,236)(158,235)(159,237)(160,242)
(161,241)(162,243);
s2 := Sym(243)!(  1, 82)(  2, 84)(  3, 83)(  4, 85)(  5, 87)(  6, 86)(  7, 88)
(  8, 90)(  9, 89)( 10, 91)( 11, 93)( 12, 92)( 13, 94)( 14, 96)( 15, 95)
( 16, 97)( 17, 99)( 18, 98)( 19,100)( 20,102)( 21,101)( 22,103)( 23,105)
( 24,104)( 25,106)( 26,108)( 27,107)( 28,136)( 29,138)( 30,137)( 31,139)
( 32,141)( 33,140)( 34,142)( 35,144)( 36,143)( 37,145)( 38,147)( 39,146)
( 40,148)( 41,150)( 42,149)( 43,151)( 44,153)( 45,152)( 46,154)( 47,156)
( 48,155)( 49,157)( 50,159)( 51,158)( 52,160)( 53,162)( 54,161)( 55,109)
( 56,111)( 57,110)( 58,112)( 59,114)( 60,113)( 61,115)( 62,117)( 63,116)
( 64,118)( 65,120)( 66,119)( 67,121)( 68,123)( 69,122)( 70,124)( 71,126)
( 72,125)( 73,127)( 74,129)( 75,128)( 76,130)( 77,132)( 78,131)( 79,133)
( 80,135)( 81,134)(163,164)(166,167)(169,170)(172,173)(175,176)(178,179)
(181,182)(184,185)(187,188)(190,218)(191,217)(192,219)(193,221)(194,220)
(195,222)(196,224)(197,223)(198,225)(199,227)(200,226)(201,228)(202,230)
(203,229)(204,231)(205,233)(206,232)(207,234)(208,236)(209,235)(210,237)
(211,239)(212,238)(213,240)(214,242)(215,241)(216,243);
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,
s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*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*s0*s1*s0*s1*s0*s1,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 >;

```
References : None.
to this polytope