Polytope of Type {6,27}

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

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

```
References : None.
to this polytope