# Polytope of Type {18,18}

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