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# Polytope of Type {18,18}

Atlas Canonical Name : {18,18}*972i
if this polytope has a name.
Group : SmallGroup(972,110)
Rank : 3
Schlafli Type : {18,18}
Number of vertices, edges, etc : 27, 243, 27
Order of s0s1s2 : 9
Order of s0s1s2s1 : 6
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Non-Orientable
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{18,18,2} of size 1944
Vertex Figure Of :
{2,18,18} of size 1944
Quotients (Maximal Quotients in Boldface) :
3-fold quotients : {6,18}*324a, {18,6}*324c
9-fold quotients : {6,6}*108
Covers (Minimal Covers in Boldface) :
2-fold covers : {18,18}*1944z
Permutation Representation (GAP) :
```s0 := (  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);;
s1 := (  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)( 55, 56)( 59, 60)( 61, 63)( 64, 74)( 65, 73)
( 66, 75)( 67, 76)( 68, 78)( 69, 77)( 70, 81)( 71, 80)( 72, 79)( 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)(136,236)(137,235)(138,237)
(139,238)(140,240)(141,239)(142,243)(143,242)(144,241)(145,227)(146,226)
(147,228)(148,229)(149,231)(150,230)(151,234)(152,233)(153,232)(154,218)
(155,217)(156,219)(157,220)(158,222)(159,221)(160,225)(161,224)(162,223);;
s2 := (  1, 82)(  2, 83)(  3, 84)(  4, 88)(  5, 89)(  6, 90)(  7, 85)(  8, 86)
(  9, 87)( 10,100)( 11,101)( 12,102)( 13,106)( 14,107)( 15,108)( 16,103)
( 17,104)( 18,105)( 19, 91)( 20, 92)( 21, 93)( 22, 97)( 23, 98)( 24, 99)
( 25, 94)( 26, 95)( 27, 96)( 28,112)( 29,113)( 30,114)( 31,109)( 32,110)
( 33,111)( 34,115)( 35,116)( 36,117)( 37,130)( 38,131)( 39,132)( 40,127)
( 41,128)( 42,129)( 43,133)( 44,134)( 45,135)( 46,121)( 47,122)( 48,123)
( 49,118)( 50,119)( 51,120)( 52,124)( 53,125)( 54,126)( 55,142)( 56,143)
( 57,144)( 58,139)( 59,140)( 60,141)( 61,136)( 62,137)( 63,138)( 64,160)
( 65,161)( 66,162)( 67,157)( 68,158)( 69,159)( 70,154)( 71,155)( 72,156)
( 73,151)( 74,152)( 75,153)( 76,148)( 77,149)( 78,150)( 79,145)( 80,146)
( 81,147)(163,181)(164,182)(165,183)(166,187)(167,188)(168,189)(169,184)
(170,185)(171,186)(175,178)(176,179)(177,180)(190,211)(191,212)(192,213)
(193,208)(194,209)(195,210)(196,214)(197,215)(198,216)(199,202)(200,203)
(201,204)(217,241)(218,242)(219,243)(220,238)(221,239)(222,240)(223,235)
(224,236)(225,237)(226,232)(227,233)(228,234);;
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*s0*s1*s0*s1*s2*s0*s1*s2*s0*s1*s0*s1,
s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s1*s2*s1*s0*s1*s0*s2*s1,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s2*s1*s2,
s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s2*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)( 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);
s1 := 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)( 55, 56)( 59, 60)( 61, 63)( 64, 74)
( 65, 73)( 66, 75)( 67, 76)( 68, 78)( 69, 77)( 70, 81)( 71, 80)( 72, 79)
( 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)(136,236)(137,235)
(138,237)(139,238)(140,240)(141,239)(142,243)(143,242)(144,241)(145,227)
(146,226)(147,228)(148,229)(149,231)(150,230)(151,234)(152,233)(153,232)
(154,218)(155,217)(156,219)(157,220)(158,222)(159,221)(160,225)(161,224)
(162,223);
s2 := Sym(243)!(  1, 82)(  2, 83)(  3, 84)(  4, 88)(  5, 89)(  6, 90)(  7, 85)
(  8, 86)(  9, 87)( 10,100)( 11,101)( 12,102)( 13,106)( 14,107)( 15,108)
( 16,103)( 17,104)( 18,105)( 19, 91)( 20, 92)( 21, 93)( 22, 97)( 23, 98)
( 24, 99)( 25, 94)( 26, 95)( 27, 96)( 28,112)( 29,113)( 30,114)( 31,109)
( 32,110)( 33,111)( 34,115)( 35,116)( 36,117)( 37,130)( 38,131)( 39,132)
( 40,127)( 41,128)( 42,129)( 43,133)( 44,134)( 45,135)( 46,121)( 47,122)
( 48,123)( 49,118)( 50,119)( 51,120)( 52,124)( 53,125)( 54,126)( 55,142)
( 56,143)( 57,144)( 58,139)( 59,140)( 60,141)( 61,136)( 62,137)( 63,138)
( 64,160)( 65,161)( 66,162)( 67,157)( 68,158)( 69,159)( 70,154)( 71,155)
( 72,156)( 73,151)( 74,152)( 75,153)( 76,148)( 77,149)( 78,150)( 79,145)
( 80,146)( 81,147)(163,181)(164,182)(165,183)(166,187)(167,188)(168,189)
(169,184)(170,185)(171,186)(175,178)(176,179)(177,180)(190,211)(191,212)
(192,213)(193,208)(194,209)(195,210)(196,214)(197,215)(198,216)(199,202)
(200,203)(201,204)(217,241)(218,242)(219,243)(220,238)(221,239)(222,240)
(223,235)(224,236)(225,237)(226,232)(227,233)(228,234);
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*s0*s1*s0*s1*s2*s0*s1*s2*s0*s1*s0*s1,
s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s1*s2*s1*s0*s1*s0*s2*s1,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s2*s1*s2,
s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s0*s1 >;

```
References : None.
to this polytope