Polytope of Type {6,6,12} with group SmallGroup(864,2455)

Overview

Group: SmallGroup(864,2455)
Rank: 4
Schläfli Type: {6,6,12}
Vertices, edges, …: 6, 18, 36, 12
Order of s₀s₁s₂s₃: 12
Order of s₀s₁s₂s₃s₂s₁: 6
Also known as: if this polytope has a name.

Special Properties

Universal
Orientable
Flat

Quotients maximal quotients in bold

2-fold

3-fold

{6,2,12}*288

4-fold

6-fold

8-fold

{3,6,3}*108

9-fold

12-fold

18-fold

24-fold

{3,2,3}*36

27-fold

{2,2,4}*32

36-fold

54-fold

{2,2,2}*16

Covers minimal covers in bold

2-fold

Irregular Quotients of which this is a minimal cover

None.

Representations

Permutation Representation (GAP)

s0 := (  2,  3)(  4,  7)(  5,  9)(  6,  8)( 11, 12)( 13, 16)( 14, 18)( 15, 17)( 20, 21)( 22, 25)( 23, 27)( 24, 26)( 29, 30)( 31, 34)( 32, 36)( 33, 35)( 38, 39)( 40, 43)( 41, 45)( 42, 44)( 47, 48)( 49, 52)( 50, 54)( 51, 53)( 56, 57)( 58, 61)( 59, 63)( 60, 62)( 65, 66)( 67, 70)( 68, 72)( 69, 71)( 74, 75)( 76, 79)( 77, 81)( 78, 80)( 83, 84)( 85, 88)( 86, 90)( 87, 89)( 92, 93)( 94, 97)( 95, 99)( 96, 98)(101,102)(103,106)(104,108)(105,107)(110,111)(112,115)(113,117)(114,116)(119,120)(121,124)(122,126)(123,125)(128,129)(130,133)(131,135)(132,134)(137,138)(139,142)(140,144)(141,143)(146,147)(148,151)(149,153)(150,152)(155,156)(157,160)(158,162)(159,161)(164,165)(166,169)(167,171)(168,170)(173,174)(175,178)(176,180)(177,179)(182,183)(184,187)(185,189)(186,188)(191,192)(193,196)(194,198)(195,197)(200,201)(202,205)(203,207)(204,206)(209,210)(211,214)(212,216)(213,215);;
s1 := (  1,112)(  2,114)(  3,113)(  4,109)(  5,111)(  6,110)(  7,115)(  8,117)(  9,116)( 10,121)( 11,123)( 12,122)( 13,118)( 14,120)( 15,119)( 16,124)( 17,126)( 18,125)( 19,130)( 20,132)( 21,131)( 22,127)( 23,129)( 24,128)( 25,133)( 26,135)( 27,134)( 28,139)( 29,141)( 30,140)( 31,136)( 32,138)( 33,137)( 34,142)( 35,144)( 36,143)( 37,148)( 38,150)( 39,149)( 40,145)( 41,147)( 42,146)( 43,151)( 44,153)( 45,152)( 46,157)( 47,159)( 48,158)( 49,154)( 50,156)( 51,155)( 52,160)( 53,162)( 54,161)( 55,166)( 56,168)( 57,167)( 58,163)( 59,165)( 60,164)( 61,169)( 62,171)( 63,170)( 64,175)( 65,177)( 66,176)( 67,172)( 68,174)( 69,173)( 70,178)( 71,180)( 72,179)( 73,184)( 74,186)( 75,185)( 76,181)( 77,183)( 78,182)( 79,187)( 80,189)( 81,188)( 82,193)( 83,195)( 84,194)( 85,190)( 86,192)( 87,191)( 88,196)( 89,198)( 90,197)( 91,202)( 92,204)( 93,203)( 94,199)( 95,201)( 96,200)( 97,205)( 98,207)( 99,206)(100,211)(101,213)(102,212)(103,208)(104,210)(105,209)(106,214)(107,216)(108,215);;
s2 := (  1, 10)(  2, 12)(  3, 11)(  4, 14)(  5, 13)(  6, 15)(  7, 18)(  8, 17)(  9, 16)( 20, 21)( 22, 23)( 25, 27)( 28, 37)( 29, 39)( 30, 38)( 31, 41)( 32, 40)( 33, 42)( 34, 45)( 35, 44)( 36, 43)( 47, 48)( 49, 50)( 52, 54)( 55, 91)( 56, 93)( 57, 92)( 58, 95)( 59, 94)( 60, 96)( 61, 99)( 62, 98)( 63, 97)( 64, 82)( 65, 84)( 66, 83)( 67, 86)( 68, 85)( 69, 87)( 70, 90)( 71, 89)( 72, 88)( 73,100)( 74,102)( 75,101)( 76,104)( 77,103)( 78,105)( 79,108)( 80,107)( 81,106)(109,118)(110,120)(111,119)(112,122)(113,121)(114,123)(115,126)(116,125)(117,124)(128,129)(130,131)(133,135)(136,145)(137,147)(138,146)(139,149)(140,148)(141,150)(142,153)(143,152)(144,151)(155,156)(157,158)(160,162)(163,199)(164,201)(165,200)(166,203)(167,202)(168,204)(169,207)(170,206)(171,205)(172,190)(173,192)(174,191)(175,194)(176,193)(177,195)(178,198)(179,197)(180,196)(181,208)(182,210)(183,209)(184,212)(185,211)(186,213)(187,216)(188,215)(189,214);;
s3 := (  1, 55)(  2, 57)(  3, 56)(  4, 58)(  5, 60)(  6, 59)(  7, 61)(  8, 63)(  9, 62)( 10, 73)( 11, 75)( 12, 74)( 13, 76)( 14, 78)( 15, 77)( 16, 79)( 17, 81)( 18, 80)( 19, 64)( 20, 66)( 21, 65)( 22, 67)( 23, 69)( 24, 68)( 25, 70)( 26, 72)( 27, 71)( 28, 82)( 29, 84)( 30, 83)( 31, 85)( 32, 87)( 33, 86)( 34, 88)( 35, 90)( 36, 89)( 37,100)( 38,102)( 39,101)( 40,103)( 41,105)( 42,104)( 43,106)( 44,108)( 45,107)( 46, 91)( 47, 93)( 48, 92)( 49, 94)( 50, 96)( 51, 95)( 52, 97)( 53, 99)( 54, 98)(109,163)(110,165)(111,164)(112,166)(113,168)(114,167)(115,169)(116,171)(117,170)(118,181)(119,183)(120,182)(121,184)(122,186)(123,185)(124,187)(125,189)(126,188)(127,172)(128,174)(129,173)(130,175)(131,177)(132,176)(133,178)(134,180)(135,179)(136,190)(137,192)(138,191)(139,193)(140,195)(141,194)(142,196)(143,198)(144,197)(145,208)(146,210)(147,209)(148,211)(149,213)(150,212)(151,214)(152,216)(153,215)(154,199)(155,201)(156,200)(157,202)(158,204)(159,203)(160,205)(161,207)(162,206);;
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, s2*s0*s1*s2*s1*s2*s0*s1*s2*s1, 
s1*s2*s3*s1*s2*s1*s2*s3*s1*s2, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2*s1, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 ];;
poly := F / rels;;

Permutation Representation (Magma)

s0 := Sym(216)!(  2,  3)(  4,  7)(  5,  9)(  6,  8)( 11, 12)( 13, 16)( 14, 18)( 15, 17)( 20, 21)( 22, 25)( 23, 27)( 24, 26)( 29, 30)( 31, 34)( 32, 36)( 33, 35)( 38, 39)( 40, 43)( 41, 45)( 42, 44)( 47, 48)( 49, 52)( 50, 54)( 51, 53)( 56, 57)( 58, 61)( 59, 63)( 60, 62)( 65, 66)( 67, 70)( 68, 72)( 69, 71)( 74, 75)( 76, 79)( 77, 81)( 78, 80)( 83, 84)( 85, 88)( 86, 90)( 87, 89)( 92, 93)( 94, 97)( 95, 99)( 96, 98)(101,102)(103,106)(104,108)(105,107)(110,111)(112,115)(113,117)(114,116)(119,120)(121,124)(122,126)(123,125)(128,129)(130,133)(131,135)(132,134)(137,138)(139,142)(140,144)(141,143)(146,147)(148,151)(149,153)(150,152)(155,156)(157,160)(158,162)(159,161)(164,165)(166,169)(167,171)(168,170)(173,174)(175,178)(176,180)(177,179)(182,183)(184,187)(185,189)(186,188)(191,192)(193,196)(194,198)(195,197)(200,201)(202,205)(203,207)(204,206)(209,210)(211,214)(212,216)(213,215);
s1 := Sym(216)!(  1,112)(  2,114)(  3,113)(  4,109)(  5,111)(  6,110)(  7,115)(  8,117)(  9,116)( 10,121)( 11,123)( 12,122)( 13,118)( 14,120)( 15,119)( 16,124)( 17,126)( 18,125)( 19,130)( 20,132)( 21,131)( 22,127)( 23,129)( 24,128)( 25,133)( 26,135)( 27,134)( 28,139)( 29,141)( 30,140)( 31,136)( 32,138)( 33,137)( 34,142)( 35,144)( 36,143)( 37,148)( 38,150)( 39,149)( 40,145)( 41,147)( 42,146)( 43,151)( 44,153)( 45,152)( 46,157)( 47,159)( 48,158)( 49,154)( 50,156)( 51,155)( 52,160)( 53,162)( 54,161)( 55,166)( 56,168)( 57,167)( 58,163)( 59,165)( 60,164)( 61,169)( 62,171)( 63,170)( 64,175)( 65,177)( 66,176)( 67,172)( 68,174)( 69,173)( 70,178)( 71,180)( 72,179)( 73,184)( 74,186)( 75,185)( 76,181)( 77,183)( 78,182)( 79,187)( 80,189)( 81,188)( 82,193)( 83,195)( 84,194)( 85,190)( 86,192)( 87,191)( 88,196)( 89,198)( 90,197)( 91,202)( 92,204)( 93,203)( 94,199)( 95,201)( 96,200)( 97,205)( 98,207)( 99,206)(100,211)(101,213)(102,212)(103,208)(104,210)(105,209)(106,214)(107,216)(108,215);
s2 := Sym(216)!(  1, 10)(  2, 12)(  3, 11)(  4, 14)(  5, 13)(  6, 15)(  7, 18)(  8, 17)(  9, 16)( 20, 21)( 22, 23)( 25, 27)( 28, 37)( 29, 39)( 30, 38)( 31, 41)( 32, 40)( 33, 42)( 34, 45)( 35, 44)( 36, 43)( 47, 48)( 49, 50)( 52, 54)( 55, 91)( 56, 93)( 57, 92)( 58, 95)( 59, 94)( 60, 96)( 61, 99)( 62, 98)( 63, 97)( 64, 82)( 65, 84)( 66, 83)( 67, 86)( 68, 85)( 69, 87)( 70, 90)( 71, 89)( 72, 88)( 73,100)( 74,102)( 75,101)( 76,104)( 77,103)( 78,105)( 79,108)( 80,107)( 81,106)(109,118)(110,120)(111,119)(112,122)(113,121)(114,123)(115,126)(116,125)(117,124)(128,129)(130,131)(133,135)(136,145)(137,147)(138,146)(139,149)(140,148)(141,150)(142,153)(143,152)(144,151)(155,156)(157,158)(160,162)(163,199)(164,201)(165,200)(166,203)(167,202)(168,204)(169,207)(170,206)(171,205)(172,190)(173,192)(174,191)(175,194)(176,193)(177,195)(178,198)(179,197)(180,196)(181,208)(182,210)(183,209)(184,212)(185,211)(186,213)(187,216)(188,215)(189,214);
s3 := Sym(216)!(  1, 55)(  2, 57)(  3, 56)(  4, 58)(  5, 60)(  6, 59)(  7, 61)(  8, 63)(  9, 62)( 10, 73)( 11, 75)( 12, 74)( 13, 76)( 14, 78)( 15, 77)( 16, 79)( 17, 81)( 18, 80)( 19, 64)( 20, 66)( 21, 65)( 22, 67)( 23, 69)( 24, 68)( 25, 70)( 26, 72)( 27, 71)( 28, 82)( 29, 84)( 30, 83)( 31, 85)( 32, 87)( 33, 86)( 34, 88)( 35, 90)( 36, 89)( 37,100)( 38,102)( 39,101)( 40,103)( 41,105)( 42,104)( 43,106)( 44,108)( 45,107)( 46, 91)( 47, 93)( 48, 92)( 49, 94)( 50, 96)( 51, 95)( 52, 97)( 53, 99)( 54, 98)(109,163)(110,165)(111,164)(112,166)(113,168)(114,167)(115,169)(116,171)(117,170)(118,181)(119,183)(120,182)(121,184)(122,186)(123,185)(124,187)(125,189)(126,188)(127,172)(128,174)(129,173)(130,175)(131,177)(132,176)(133,178)(134,180)(135,179)(136,190)(137,192)(138,191)(139,193)(140,195)(141,194)(142,196)(143,198)(144,197)(145,208)(146,210)(147,209)(148,211)(149,213)(150,212)(151,214)(152,216)(153,215)(154,199)(155,201)(156,200)(157,202)(158,204)(159,203)(160,205)(161,207)(162,206);
poly := sub<Sym(216)|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, 
s2*s0*s1*s2*s1*s2*s0*s1*s2*s1, s1*s2*s3*s1*s2*s1*s2*s3*s1*s2, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2*s1, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 >;

References