# Polytope of Type {9,18}

Atlas Canonical Name : {9,18}*972d
if this polytope has a name.
Group : SmallGroup(972,104)
Rank : 3
Schlafli Type : {9,18}
Number of vertices, edges, etc : 27, 243, 54
Order of s0s1s2 : 18
Order of s0s1s2s1 : 18
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Related Polytopes :
Facet
Vertex Figure
Dual
Petrial
Facet Of :
{9,18,2} of size 1944
Vertex Figure Of :
{2,9,18} of size 1944
Quotients (Maximal Quotients in Boldface) :
3-fold quotients : {9,6}*324d, {3,18}*324
9-fold quotients : {3,6}*108
27-fold quotients : {3,6}*36
81-fold quotients : {3,2}*12
Covers (Minimal Covers in Boldface) :
2-fold covers : {18,18}*1944l
Permutation Representation (GAP) :
```s0 := (  2,  3)(  4,  7)(  5,  9)(  6,  8)( 10, 18)( 11, 17)( 12, 16)( 13, 15)
( 19, 23)( 20, 22)( 21, 24)( 25, 26)( 28, 56)( 29, 55)( 30, 57)( 31, 62)
( 32, 61)( 33, 63)( 34, 59)( 35, 58)( 36, 60)( 37, 70)( 38, 72)( 39, 71)
( 40, 67)( 41, 69)( 42, 68)( 43, 64)( 44, 66)( 45, 65)( 46, 78)( 47, 77)
( 48, 76)( 49, 75)( 50, 74)( 51, 73)( 52, 81)( 53, 80)( 54, 79)( 82,166)
( 83,168)( 84,167)( 85,163)( 86,165)( 87,164)( 88,169)( 89,171)( 90,170)
( 91,174)( 92,173)( 93,172)( 94,180)( 95,179)( 96,178)( 97,177)( 98,176)
( 99,175)(100,188)(101,187)(102,189)(103,185)(104,184)(105,186)(106,182)
(107,181)(108,183)(109,221)(110,220)(111,222)(112,218)(113,217)(114,219)
(115,224)(116,223)(117,225)(118,226)(119,228)(120,227)(121,232)(122,234)
(123,233)(124,229)(125,231)(126,230)(127,243)(128,242)(129,241)(130,240)
(131,239)(132,238)(133,237)(134,236)(135,235)(136,194)(137,193)(138,195)
(139,191)(140,190)(141,192)(142,197)(143,196)(144,198)(145,199)(146,201)
(147,200)(148,205)(149,207)(150,206)(151,202)(152,204)(153,203)(154,216)
(155,215)(156,214)(157,213)(158,212)(159,211)(160,210)(161,209)(162,208);;
s1 := (  1,199)(  2,201)(  3,200)(  4,205)(  5,207)(  6,206)(  7,202)(  8,204)
(  9,203)( 10,209)( 11,208)( 12,210)( 13,215)( 14,214)( 15,216)( 16,212)
( 17,211)( 18,213)( 19,192)( 20,191)( 21,190)( 22,198)( 23,197)( 24,196)
( 25,195)( 26,194)( 27,193)( 28,183)( 29,182)( 30,181)( 31,189)( 32,188)
( 33,187)( 34,186)( 35,185)( 36,184)( 37,163)( 38,165)( 39,164)( 40,169)
( 41,171)( 42,170)( 43,166)( 44,168)( 45,167)( 46,173)( 47,172)( 48,174)
( 49,179)( 50,178)( 51,180)( 52,176)( 53,175)( 54,177)( 55,225)( 56,224)
( 57,223)( 58,222)( 59,221)( 60,220)( 61,219)( 62,218)( 63,217)( 64,232)
( 65,234)( 66,233)( 67,229)( 68,231)( 69,230)( 70,226)( 71,228)( 72,227)
( 73,242)( 74,241)( 75,243)( 76,239)( 77,238)( 78,240)( 79,236)( 80,235)
( 81,237)( 82,118)( 83,120)( 84,119)( 85,124)( 86,126)( 87,125)( 88,121)
( 89,123)( 90,122)( 91,128)( 92,127)( 93,129)( 94,134)( 95,133)( 96,135)
( 97,131)( 98,130)( 99,132)(100,111)(101,110)(102,109)(103,117)(104,116)
(105,115)(106,114)(107,113)(108,112)(136,144)(137,143)(138,142)(139,141)
(145,151)(146,153)(147,152)(149,150)(154,161)(155,160)(156,162)(157,158);;
s2 := (  2,  3)(  5,  6)(  8,  9)( 10, 22)( 11, 24)( 12, 23)( 13, 25)( 14, 27)
( 15, 26)( 16, 19)( 17, 21)( 18, 20)( 28, 56)( 29, 55)( 30, 57)( 31, 59)
( 32, 58)( 33, 60)( 34, 62)( 35, 61)( 36, 63)( 37, 77)( 38, 76)( 39, 78)
( 40, 80)( 41, 79)( 42, 81)( 43, 74)( 44, 73)( 45, 75)( 46, 71)( 47, 70)
( 48, 72)( 49, 65)( 50, 64)( 51, 66)( 52, 68)( 53, 67)( 54, 69)( 83, 84)
( 86, 87)( 89, 90)( 91,103)( 92,105)( 93,104)( 94,106)( 95,108)( 96,107)
( 97,100)( 98,102)( 99,101)(109,137)(110,136)(111,138)(112,140)(113,139)
(114,141)(115,143)(116,142)(117,144)(118,158)(119,157)(120,159)(121,161)
(122,160)(123,162)(124,155)(125,154)(126,156)(127,152)(128,151)(129,153)
(130,146)(131,145)(132,147)(133,149)(134,148)(135,150)(164,165)(167,168)
(170,171)(172,184)(173,186)(174,185)(175,187)(176,189)(177,188)(178,181)
(179,183)(180,182)(190,218)(191,217)(192,219)(193,221)(194,220)(195,222)
(196,224)(197,223)(198,225)(199,239)(200,238)(201,240)(202,242)(203,241)
(204,243)(205,236)(206,235)(207,237)(208,233)(209,232)(210,234)(211,227)
(212,226)(213,228)(214,230)(215,229)(216,231);;
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*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1,
s1*s2*s1*s0*s2*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s1*s0,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s2*s0*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s0*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1 ];;
poly := F / rels;;

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

```
References : None.
to this polytope