Polytope of Type {27,6,6}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {27,6,6}*1944b
if this polytope has a name.
Group : SmallGroup(1944,2343)
Rank : 4
Schlafli Type : {27,6,6}
Number of vertices, edges, etc : 27, 81, 18, 6
Order of s0s1s2s3 : 54
Order of s0s1s2s3s2s1 : 2
Special Properties :
Universal
Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
None in this Atlas
Vertex Figure Of :
None in this Atlas
Quotients (Maximal Quotients in Boldface) :
3-fold quotients : {27,2,6}*648, {27,6,2}*648, {9,6,6}*648b
6-fold quotients : {27,2,3}*324
9-fold quotients : {27,2,2}*216, {9,2,6}*216, {9,6,2}*216, {3,6,6}*216b
18-fold quotients : {9,2,3}*108
27-fold quotients : {9,2,2}*72, {3,2,6}*72, {3,6,2}*72
54-fold quotients : {3,2,3}*36
81-fold quotients : {3,2,2}*24
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := ( 2, 3)( 4, 8)( 5, 7)( 6, 9)( 10, 23)( 11, 22)( 12, 24)( 13, 20)
( 14, 19)( 15, 21)( 16, 27)( 17, 26)( 18, 25)( 29, 30)( 31, 35)( 32, 34)
( 33, 36)( 37, 50)( 38, 49)( 39, 51)( 40, 47)( 41, 46)( 42, 48)( 43, 54)
( 44, 53)( 45, 52)( 56, 57)( 58, 62)( 59, 61)( 60, 63)( 64, 77)( 65, 76)
( 66, 78)( 67, 74)( 68, 73)( 69, 75)( 70, 81)( 71, 80)( 72, 79)( 82,163)
( 83,165)( 84,164)( 85,170)( 86,169)( 87,171)( 88,167)( 89,166)( 90,168)
( 91,185)( 92,184)( 93,186)( 94,182)( 95,181)( 96,183)( 97,189)( 98,188)
( 99,187)(100,176)(101,175)(102,177)(103,173)(104,172)(105,174)(106,180)
(107,179)(108,178)(109,190)(110,192)(111,191)(112,197)(113,196)(114,198)
(115,194)(116,193)(117,195)(118,212)(119,211)(120,213)(121,209)(122,208)
(123,210)(124,216)(125,215)(126,214)(127,203)(128,202)(129,204)(130,200)
(131,199)(132,201)(133,207)(134,206)(135,205)(136,217)(137,219)(138,218)
(139,224)(140,223)(141,225)(142,221)(143,220)(144,222)(145,239)(146,238)
(147,240)(148,236)(149,235)(150,237)(151,243)(152,242)(153,241)(154,230)
(155,229)(156,231)(157,227)(158,226)(159,228)(160,234)(161,233)(162,232);;
s1 := ( 1, 91)( 2, 93)( 3, 92)( 4, 98)( 5, 97)( 6, 99)( 7, 95)( 8, 94)
( 9, 96)( 10, 82)( 11, 84)( 12, 83)( 13, 89)( 14, 88)( 15, 90)( 16, 86)
( 17, 85)( 18, 87)( 19,104)( 20,103)( 21,105)( 22,101)( 23,100)( 24,102)
( 25,108)( 26,107)( 27,106)( 28,118)( 29,120)( 30,119)( 31,125)( 32,124)
( 33,126)( 34,122)( 35,121)( 36,123)( 37,109)( 38,111)( 39,110)( 40,116)
( 41,115)( 42,117)( 43,113)( 44,112)( 45,114)( 46,131)( 47,130)( 48,132)
( 49,128)( 50,127)( 51,129)( 52,135)( 53,134)( 54,133)( 55,145)( 56,147)
( 57,146)( 58,152)( 59,151)( 60,153)( 61,149)( 62,148)( 63,150)( 64,136)
( 65,138)( 66,137)( 67,143)( 68,142)( 69,144)( 70,140)( 71,139)( 72,141)
( 73,158)( 74,157)( 75,159)( 76,155)( 77,154)( 78,156)( 79,162)( 80,161)
( 81,160)(163,172)(164,174)(165,173)(166,179)(167,178)(168,180)(169,176)
(170,175)(171,177)(181,185)(182,184)(183,186)(187,189)(190,199)(191,201)
(192,200)(193,206)(194,205)(195,207)(196,203)(197,202)(198,204)(208,212)
(209,211)(210,213)(214,216)(217,226)(218,228)(219,227)(220,233)(221,232)
(222,234)(223,230)(224,229)(225,231)(235,239)(236,238)(237,240)(241,243);;
s2 := ( 28, 55)( 29, 56)( 30, 57)( 31, 58)( 32, 59)( 33, 60)( 34, 61)( 35, 62)
( 36, 63)( 37, 64)( 38, 65)( 39, 66)( 40, 67)( 41, 68)( 42, 69)( 43, 70)
( 44, 71)( 45, 72)( 46, 73)( 47, 74)( 48, 75)( 49, 76)( 50, 77)( 51, 78)
( 52, 79)( 53, 80)( 54, 81)( 82,163)( 83,164)( 84,165)( 85,166)( 86,167)
( 87,168)( 88,169)( 89,170)( 90,171)( 91,172)( 92,173)( 93,174)( 94,175)
( 95,176)( 96,177)( 97,178)( 98,179)( 99,180)(100,181)(101,182)(102,183)
(103,184)(104,185)(105,186)(106,187)(107,188)(108,189)(109,217)(110,218)
(111,219)(112,220)(113,221)(114,222)(115,223)(116,224)(117,225)(118,226)
(119,227)(120,228)(121,229)(122,230)(123,231)(124,232)(125,233)(126,234)
(127,235)(128,236)(129,237)(130,238)(131,239)(132,240)(133,241)(134,242)
(135,243)(136,190)(137,191)(138,192)(139,193)(140,194)(141,195)(142,196)
(143,197)(144,198)(145,199)(146,200)(147,201)(148,202)(149,203)(150,204)
(151,205)(152,206)(153,207)(154,208)(155,209)(156,210)(157,211)(158,212)
(159,213)(160,214)(161,215)(162,216);;
s3 := ( 1, 28)( 2, 29)( 3, 30)( 4, 31)( 5, 32)( 6, 33)( 7, 34)( 8, 35)
( 9, 36)( 10, 37)( 11, 38)( 12, 39)( 13, 40)( 14, 41)( 15, 42)( 16, 43)
( 17, 44)( 18, 45)( 19, 46)( 20, 47)( 21, 48)( 22, 49)( 23, 50)( 24, 51)
( 25, 52)( 26, 53)( 27, 54)( 82,109)( 83,110)( 84,111)( 85,112)( 86,113)
( 87,114)( 88,115)( 89,116)( 90,117)( 91,118)( 92,119)( 93,120)( 94,121)
( 95,122)( 96,123)( 97,124)( 98,125)( 99,126)(100,127)(101,128)(102,129)
(103,130)(104,131)(105,132)(106,133)(107,134)(108,135)(163,190)(164,191)
(165,192)(166,193)(167,194)(168,195)(169,196)(170,197)(171,198)(172,199)
(173,200)(174,201)(175,202)(176,203)(177,204)(178,205)(179,206)(180,207)
(181,208)(182,209)(183,210)(184,211)(185,212)(186,213)(187,214)(188,215)
(189,216);;
poly := Group([s0,s1,s2,s3]);;
Finitely Presented Group Representation (GAP) :
F := FreeGroup("s0","s1","s2","s3");;
s0 := F.1;; s1 := F.2;; s2 := F.3;; s3 := F.4;;
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s0*s2*s0*s2,
s0*s3*s0*s3, s1*s3*s1*s3, s1*s2*s3*s2*s1*s2*s3*s2,
s2*s0*s1*s2*s1*s2*s0*s1*s2*s1, s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(243)!( 2, 3)( 4, 8)( 5, 7)( 6, 9)( 10, 23)( 11, 22)( 12, 24)
( 13, 20)( 14, 19)( 15, 21)( 16, 27)( 17, 26)( 18, 25)( 29, 30)( 31, 35)
( 32, 34)( 33, 36)( 37, 50)( 38, 49)( 39, 51)( 40, 47)( 41, 46)( 42, 48)
( 43, 54)( 44, 53)( 45, 52)( 56, 57)( 58, 62)( 59, 61)( 60, 63)( 64, 77)
( 65, 76)( 66, 78)( 67, 74)( 68, 73)( 69, 75)( 70, 81)( 71, 80)( 72, 79)
( 82,163)( 83,165)( 84,164)( 85,170)( 86,169)( 87,171)( 88,167)( 89,166)
( 90,168)( 91,185)( 92,184)( 93,186)( 94,182)( 95,181)( 96,183)( 97,189)
( 98,188)( 99,187)(100,176)(101,175)(102,177)(103,173)(104,172)(105,174)
(106,180)(107,179)(108,178)(109,190)(110,192)(111,191)(112,197)(113,196)
(114,198)(115,194)(116,193)(117,195)(118,212)(119,211)(120,213)(121,209)
(122,208)(123,210)(124,216)(125,215)(126,214)(127,203)(128,202)(129,204)
(130,200)(131,199)(132,201)(133,207)(134,206)(135,205)(136,217)(137,219)
(138,218)(139,224)(140,223)(141,225)(142,221)(143,220)(144,222)(145,239)
(146,238)(147,240)(148,236)(149,235)(150,237)(151,243)(152,242)(153,241)
(154,230)(155,229)(156,231)(157,227)(158,226)(159,228)(160,234)(161,233)
(162,232);
s1 := Sym(243)!( 1, 91)( 2, 93)( 3, 92)( 4, 98)( 5, 97)( 6, 99)( 7, 95)
( 8, 94)( 9, 96)( 10, 82)( 11, 84)( 12, 83)( 13, 89)( 14, 88)( 15, 90)
( 16, 86)( 17, 85)( 18, 87)( 19,104)( 20,103)( 21,105)( 22,101)( 23,100)
( 24,102)( 25,108)( 26,107)( 27,106)( 28,118)( 29,120)( 30,119)( 31,125)
( 32,124)( 33,126)( 34,122)( 35,121)( 36,123)( 37,109)( 38,111)( 39,110)
( 40,116)( 41,115)( 42,117)( 43,113)( 44,112)( 45,114)( 46,131)( 47,130)
( 48,132)( 49,128)( 50,127)( 51,129)( 52,135)( 53,134)( 54,133)( 55,145)
( 56,147)( 57,146)( 58,152)( 59,151)( 60,153)( 61,149)( 62,148)( 63,150)
( 64,136)( 65,138)( 66,137)( 67,143)( 68,142)( 69,144)( 70,140)( 71,139)
( 72,141)( 73,158)( 74,157)( 75,159)( 76,155)( 77,154)( 78,156)( 79,162)
( 80,161)( 81,160)(163,172)(164,174)(165,173)(166,179)(167,178)(168,180)
(169,176)(170,175)(171,177)(181,185)(182,184)(183,186)(187,189)(190,199)
(191,201)(192,200)(193,206)(194,205)(195,207)(196,203)(197,202)(198,204)
(208,212)(209,211)(210,213)(214,216)(217,226)(218,228)(219,227)(220,233)
(221,232)(222,234)(223,230)(224,229)(225,231)(235,239)(236,238)(237,240)
(241,243);
s2 := Sym(243)!( 28, 55)( 29, 56)( 30, 57)( 31, 58)( 32, 59)( 33, 60)( 34, 61)
( 35, 62)( 36, 63)( 37, 64)( 38, 65)( 39, 66)( 40, 67)( 41, 68)( 42, 69)
( 43, 70)( 44, 71)( 45, 72)( 46, 73)( 47, 74)( 48, 75)( 49, 76)( 50, 77)
( 51, 78)( 52, 79)( 53, 80)( 54, 81)( 82,163)( 83,164)( 84,165)( 85,166)
( 86,167)( 87,168)( 88,169)( 89,170)( 90,171)( 91,172)( 92,173)( 93,174)
( 94,175)( 95,176)( 96,177)( 97,178)( 98,179)( 99,180)(100,181)(101,182)
(102,183)(103,184)(104,185)(105,186)(106,187)(107,188)(108,189)(109,217)
(110,218)(111,219)(112,220)(113,221)(114,222)(115,223)(116,224)(117,225)
(118,226)(119,227)(120,228)(121,229)(122,230)(123,231)(124,232)(125,233)
(126,234)(127,235)(128,236)(129,237)(130,238)(131,239)(132,240)(133,241)
(134,242)(135,243)(136,190)(137,191)(138,192)(139,193)(140,194)(141,195)
(142,196)(143,197)(144,198)(145,199)(146,200)(147,201)(148,202)(149,203)
(150,204)(151,205)(152,206)(153,207)(154,208)(155,209)(156,210)(157,211)
(158,212)(159,213)(160,214)(161,215)(162,216);
s3 := Sym(243)!( 1, 28)( 2, 29)( 3, 30)( 4, 31)( 5, 32)( 6, 33)( 7, 34)
( 8, 35)( 9, 36)( 10, 37)( 11, 38)( 12, 39)( 13, 40)( 14, 41)( 15, 42)
( 16, 43)( 17, 44)( 18, 45)( 19, 46)( 20, 47)( 21, 48)( 22, 49)( 23, 50)
( 24, 51)( 25, 52)( 26, 53)( 27, 54)( 82,109)( 83,110)( 84,111)( 85,112)
( 86,113)( 87,114)( 88,115)( 89,116)( 90,117)( 91,118)( 92,119)( 93,120)
( 94,121)( 95,122)( 96,123)( 97,124)( 98,125)( 99,126)(100,127)(101,128)
(102,129)(103,130)(104,131)(105,132)(106,133)(107,134)(108,135)(163,190)
(164,191)(165,192)(166,193)(167,194)(168,195)(169,196)(170,197)(171,198)
(172,199)(173,200)(174,201)(175,202)(176,203)(177,204)(178,205)(179,206)
(180,207)(181,208)(182,209)(183,210)(184,211)(185,212)(186,213)(187,214)
(188,215)(189,216);
poly := sub<Sym(243)|s0,s1,s2,s3>;
Finitely Presented Group Representation (Magma) :
poly<s0,s1,s2,s3> := Group< s0,s1,s2,s3 | s0*s0, s1*s1, s2*s2,
s3*s3, s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3,
s1*s2*s3*s2*s1*s2*s3*s2, s2*s0*s1*s2*s1*s2*s0*s1*s2*s1,
s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 >;
References : None.
to this polytope