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# Polytope of Type {54,6}

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

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

```
References : None.
to this polytope