# Polytope of Type {18,6}

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