Polytope of Type {12,12} with group SmallGroup(1944,806)

Overview

Group: SmallGroup(1944,806)
Rank: 3
Schläfli Type: {12,12}
Vertices, edges, …: 81, 486, 81
Order of s₀s₁s₂: 18
Order of s₀s₁s₂s₁: 9
Also known as: if this polytope has a name.

Special Properties

Compact Hyperbolic Quotient
Locally Spherical
Orientable
Self-Dual

Quotients maximal quotients in bold

27-fold

{4,4}*72

Covers minimal covers in bold

None in this atlas.

Irregular Quotients of which this is a minimal cover

Click an entry to reveal its facets and vertex figures.

P/N, where N=<(s1*s0*s1*s2)^3> of order 3

27 facets

27 of {12}*24

27 vertex figures

27 of {12}*24

Representations

Permutation Representation (GAP)

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

Permutation Representation (Magma)

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

References

None.

to this polytope.