# Polytope of Type {18,18}

Atlas Canonical Name : {18,18}*972c
if this polytope has a name.
Group : SmallGroup(972,104)
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
Petrial
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 : {18,6}*324b, {6,18}*324c
9-fold quotients : {6,6}*108
Covers (Minimal Covers in Boldface) :
2-fold covers : {18,18}*1944k
Permutation Representation (GAP) :
```s0 := (  4,  7)(  5,  8)(  6,  9)( 10, 20)( 11, 21)( 12, 19)( 13, 26)( 14, 27)
( 15, 25)( 16, 23)( 17, 24)( 18, 22)( 31, 34)( 32, 35)( 33, 36)( 37, 47)
( 38, 48)( 39, 46)( 40, 53)( 41, 54)( 42, 52)( 43, 50)( 44, 51)( 45, 49)
( 58, 61)( 59, 62)( 60, 63)( 64, 74)( 65, 75)( 66, 73)( 67, 80)( 68, 81)
( 69, 79)( 70, 77)( 71, 78)( 72, 76)( 82,166)( 83,167)( 84,168)( 85,163)
( 86,164)( 87,165)( 88,169)( 89,170)( 90,171)( 91,185)( 92,186)( 93,184)
( 94,182)( 95,183)( 96,181)( 97,188)( 98,189)( 99,187)(100,177)(101,175)
(102,176)(103,174)(104,172)(105,173)(106,180)(107,178)(108,179)(109,193)
(110,194)(111,195)(112,190)(113,191)(114,192)(115,196)(116,197)(117,198)
(118,212)(119,213)(120,211)(121,209)(122,210)(123,208)(124,215)(125,216)
(126,214)(127,204)(128,202)(129,203)(130,201)(131,199)(132,200)(133,207)
(134,205)(135,206)(136,220)(137,221)(138,222)(139,217)(140,218)(141,219)
(142,223)(143,224)(144,225)(145,239)(146,240)(147,238)(148,236)(149,237)
(150,235)(151,242)(152,243)(153,241)(154,231)(155,229)(156,230)(157,228)
(158,226)(159,227)(160,234)(161,232)(162,233);;
s1 := (  1, 82)(  2, 84)(  3, 83)(  4, 88)(  5, 90)(  6, 89)(  7, 85)(  8, 87)
(  9, 86)( 10, 97)( 11, 99)( 12, 98)( 13, 94)( 14, 96)( 15, 95)( 16, 91)
( 17, 93)( 18, 92)( 19,103)( 20,105)( 21,104)( 22,100)( 23,102)( 24,101)
( 25,106)( 26,108)( 27,107)( 28,158)( 29,157)( 30,159)( 31,155)( 32,154)
( 33,156)( 34,161)( 35,160)( 36,162)( 37,137)( 38,136)( 39,138)( 40,143)
( 41,142)( 42,144)( 43,140)( 44,139)( 45,141)( 46,152)( 47,151)( 48,153)
( 49,149)( 50,148)( 51,150)( 52,146)( 53,145)( 54,147)( 55,119)( 56,118)
( 57,120)( 58,125)( 59,124)( 60,126)( 61,122)( 62,121)( 63,123)( 64,134)
( 65,133)( 66,135)( 67,131)( 68,130)( 69,132)( 70,128)( 71,127)( 72,129)
( 73,113)( 74,112)( 75,114)( 76,110)( 77,109)( 78,111)( 79,116)( 80,115)
( 81,117)(163,166)(164,168)(165,167)(170,171)(173,174)(175,178)(176,180)
(177,179)(181,187)(182,189)(183,188)(185,186)(190,242)(191,241)(192,243)
(193,239)(194,238)(195,240)(196,236)(197,235)(198,237)(199,221)(200,220)
(201,222)(202,218)(203,217)(204,219)(205,224)(206,223)(207,225)(208,227)
(209,226)(210,228)(211,233)(212,232)(213,234)(214,230)(215,229)(216,231);;
s2 := (  1, 28)(  2, 30)(  3, 29)(  4, 31)(  5, 33)(  6, 32)(  7, 34)(  8, 36)
(  9, 35)( 10, 46)( 11, 48)( 12, 47)( 13, 49)( 14, 51)( 15, 50)( 16, 52)
( 17, 54)( 18, 53)( 19, 37)( 20, 39)( 21, 38)( 22, 40)( 23, 42)( 24, 41)
( 25, 43)( 26, 45)( 27, 44)( 55, 56)( 58, 59)( 61, 62)( 64, 74)( 65, 73)
( 66, 75)( 67, 77)( 68, 76)( 69, 78)( 70, 80)( 71, 79)( 72, 81)( 82,109)
( 83,111)( 84,110)( 85,112)( 86,114)( 87,113)( 88,115)( 89,117)( 90,116)
( 91,127)( 92,129)( 93,128)( 94,130)( 95,132)( 96,131)( 97,133)( 98,135)
( 99,134)(100,118)(101,120)(102,119)(103,121)(104,123)(105,122)(106,124)
(107,126)(108,125)(136,137)(139,140)(142,143)(145,155)(146,154)(147,156)
(148,158)(149,157)(150,159)(151,161)(152,160)(153,162)(163,190)(164,192)
(165,191)(166,193)(167,195)(168,194)(169,196)(170,198)(171,197)(172,208)
(173,210)(174,209)(175,211)(176,213)(177,212)(178,214)(179,216)(180,215)
(181,199)(182,201)(183,200)(184,202)(185,204)(186,203)(187,205)(188,207)
(189,206)(217,218)(220,221)(223,224)(226,236)(227,235)(228,237)(229,239)
(230,238)(231,240)(232,242)(233,241)(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, s0*s1*s0*s1*s0*s1*s0*s1*s2*s1*s0*s2*s1*s2*s1,
s0*s1*s0*s1*s0*s1*s2*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s2,
s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1 ];;
poly := F / rels;;

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

```
References : None.
to this polytope