Polytope of Type {108,6} with group SmallGroup(1296,826)

Overview

Group: SmallGroup(1296,826)
Rank: 3
Schläfli Type: {108,6}
Vertices, edges, …: 108, 324, 6
Order of s₀s₁s₂: 108
Order of s₀s₁s₂s₁: 2
Also known as: {108,6|2}. if this polytope has another name.

Special Properties

Compact Hyperbolic Quotient
Locally Spherical
Orientable
Flat

Quotients maximal quotients in bold

2-fold

{54,6}*648a

3-fold

6-fold

9-fold

12-fold

{27,2}*108

18-fold

27-fold

36-fold

{9,2}*36

54-fold

81-fold

{4,2}*16

108-fold

162-fold

{2,2}*8

Covers minimal covers in bold

None in this atlas.

Irregular Quotients of which this is a minimal cover

None.

Representations

Permutation Representation (GAP)

s0 := (  2,  3)(  4,  9)(  5,  8)(  6,  7)( 10, 27)( 11, 26)( 12, 25)( 13, 24)( 14, 23)( 15, 22)( 16, 21)( 17, 20)( 18, 19)( 29, 30)( 31, 36)( 32, 35)( 33, 34)( 37, 54)( 38, 53)( 39, 52)( 40, 51)( 41, 50)( 42, 49)( 43, 48)( 44, 47)( 45, 46)( 56, 57)( 58, 63)( 59, 62)( 60, 61)( 64, 81)( 65, 80)( 66, 79)( 67, 78)( 68, 77)( 69, 76)( 70, 75)( 71, 74)( 72, 73)( 83, 84)( 85, 90)( 86, 89)( 87, 88)( 91,108)( 92,107)( 93,106)( 94,105)( 95,104)( 96,103)( 97,102)( 98,101)( 99,100)(110,111)(112,117)(113,116)(114,115)(118,135)(119,134)(120,133)(121,132)(122,131)(123,130)(124,129)(125,128)(126,127)(137,138)(139,144)(140,143)(141,142)(145,162)(146,161)(147,160)(148,159)(149,158)(150,157)(151,156)(152,155)(153,154)(163,244)(164,246)(165,245)(166,252)(167,251)(168,250)(169,249)(170,248)(171,247)(172,270)(173,269)(174,268)(175,267)(176,266)(177,265)(178,264)(179,263)(180,262)(181,261)(182,260)(183,259)(184,258)(185,257)(186,256)(187,255)(188,254)(189,253)(190,271)(191,273)(192,272)(193,279)(194,278)(195,277)(196,276)(197,275)(198,274)(199,297)(200,296)(201,295)(202,294)(203,293)(204,292)(205,291)(206,290)(207,289)(208,288)(209,287)(210,286)(211,285)(212,284)(213,283)(214,282)(215,281)(216,280)(217,298)(218,300)(219,299)(220,306)(221,305)(222,304)(223,303)(224,302)(225,301)(226,324)(227,323)(228,322)(229,321)(230,320)(231,319)(232,318)(233,317)(234,316)(235,315)(236,314)(237,313)(238,312)(239,311)(240,310)(241,309)(242,308)(243,307);;
s1 := (  1,172)(  2,174)(  3,173)(  4,180)(  5,179)(  6,178)(  7,177)(  8,176)(  9,175)( 10,163)( 11,165)( 12,164)( 13,171)( 14,170)( 15,169)( 16,168)( 17,167)( 18,166)( 19,189)( 20,188)( 21,187)( 22,186)( 23,185)( 24,184)( 25,183)( 26,182)( 27,181)( 28,226)( 29,228)( 30,227)( 31,234)( 32,233)( 33,232)( 34,231)( 35,230)( 36,229)( 37,217)( 38,219)( 39,218)( 40,225)( 41,224)( 42,223)( 43,222)( 44,221)( 45,220)( 46,243)( 47,242)( 48,241)( 49,240)( 50,239)( 51,238)( 52,237)( 53,236)( 54,235)( 55,199)( 56,201)( 57,200)( 58,207)( 59,206)( 60,205)( 61,204)( 62,203)( 63,202)( 64,190)( 65,192)( 66,191)( 67,198)( 68,197)( 69,196)( 70,195)( 71,194)( 72,193)( 73,216)( 74,215)( 75,214)( 76,213)( 77,212)( 78,211)( 79,210)( 80,209)( 81,208)( 82,253)( 83,255)( 84,254)( 85,261)( 86,260)( 87,259)( 88,258)( 89,257)( 90,256)( 91,244)( 92,246)( 93,245)( 94,252)( 95,251)( 96,250)( 97,249)( 98,248)( 99,247)(100,270)(101,269)(102,268)(103,267)(104,266)(105,265)(106,264)(107,263)(108,262)(109,307)(110,309)(111,308)(112,315)(113,314)(114,313)(115,312)(116,311)(117,310)(118,298)(119,300)(120,299)(121,306)(122,305)(123,304)(124,303)(125,302)(126,301)(127,324)(128,323)(129,322)(130,321)(131,320)(132,319)(133,318)(134,317)(135,316)(136,280)(137,282)(138,281)(139,288)(140,287)(141,286)(142,285)(143,284)(144,283)(145,271)(146,273)(147,272)(148,279)(149,278)(150,277)(151,276)(152,275)(153,274)(154,297)(155,296)(156,295)(157,294)(158,293)(159,292)(160,291)(161,290)(162,289);;
s2 := (  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)(244,271)(245,272)(246,273)(247,274)(248,275)(249,276)(250,277)(251,278)(252,279)(253,280)(254,281)(255,282)(256,283)(257,284)(258,285)(259,286)(260,287)(261,288)(262,289)(263,290)(264,291)(265,292)(266,293)(267,294)(268,295)(269,296)(270,297);;
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*s2*s1*s0*s1*s2*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
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*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*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*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(324)!(  2,  3)(  4,  9)(  5,  8)(  6,  7)( 10, 27)( 11, 26)( 12, 25)( 13, 24)( 14, 23)( 15, 22)( 16, 21)( 17, 20)( 18, 19)( 29, 30)( 31, 36)( 32, 35)( 33, 34)( 37, 54)( 38, 53)( 39, 52)( 40, 51)( 41, 50)( 42, 49)( 43, 48)( 44, 47)( 45, 46)( 56, 57)( 58, 63)( 59, 62)( 60, 61)( 64, 81)( 65, 80)( 66, 79)( 67, 78)( 68, 77)( 69, 76)( 70, 75)( 71, 74)( 72, 73)( 83, 84)( 85, 90)( 86, 89)( 87, 88)( 91,108)( 92,107)( 93,106)( 94,105)( 95,104)( 96,103)( 97,102)( 98,101)( 99,100)(110,111)(112,117)(113,116)(114,115)(118,135)(119,134)(120,133)(121,132)(122,131)(123,130)(124,129)(125,128)(126,127)(137,138)(139,144)(140,143)(141,142)(145,162)(146,161)(147,160)(148,159)(149,158)(150,157)(151,156)(152,155)(153,154)(163,244)(164,246)(165,245)(166,252)(167,251)(168,250)(169,249)(170,248)(171,247)(172,270)(173,269)(174,268)(175,267)(176,266)(177,265)(178,264)(179,263)(180,262)(181,261)(182,260)(183,259)(184,258)(185,257)(186,256)(187,255)(188,254)(189,253)(190,271)(191,273)(192,272)(193,279)(194,278)(195,277)(196,276)(197,275)(198,274)(199,297)(200,296)(201,295)(202,294)(203,293)(204,292)(205,291)(206,290)(207,289)(208,288)(209,287)(210,286)(211,285)(212,284)(213,283)(214,282)(215,281)(216,280)(217,298)(218,300)(219,299)(220,306)(221,305)(222,304)(223,303)(224,302)(225,301)(226,324)(227,323)(228,322)(229,321)(230,320)(231,319)(232,318)(233,317)(234,316)(235,315)(236,314)(237,313)(238,312)(239,311)(240,310)(241,309)(242,308)(243,307);
s1 := Sym(324)!(  1,172)(  2,174)(  3,173)(  4,180)(  5,179)(  6,178)(  7,177)(  8,176)(  9,175)( 10,163)( 11,165)( 12,164)( 13,171)( 14,170)( 15,169)( 16,168)( 17,167)( 18,166)( 19,189)( 20,188)( 21,187)( 22,186)( 23,185)( 24,184)( 25,183)( 26,182)( 27,181)( 28,226)( 29,228)( 30,227)( 31,234)( 32,233)( 33,232)( 34,231)( 35,230)( 36,229)( 37,217)( 38,219)( 39,218)( 40,225)( 41,224)( 42,223)( 43,222)( 44,221)( 45,220)( 46,243)( 47,242)( 48,241)( 49,240)( 50,239)( 51,238)( 52,237)( 53,236)( 54,235)( 55,199)( 56,201)( 57,200)( 58,207)( 59,206)( 60,205)( 61,204)( 62,203)( 63,202)( 64,190)( 65,192)( 66,191)( 67,198)( 68,197)( 69,196)( 70,195)( 71,194)( 72,193)( 73,216)( 74,215)( 75,214)( 76,213)( 77,212)( 78,211)( 79,210)( 80,209)( 81,208)( 82,253)( 83,255)( 84,254)( 85,261)( 86,260)( 87,259)( 88,258)( 89,257)( 90,256)( 91,244)( 92,246)( 93,245)( 94,252)( 95,251)( 96,250)( 97,249)( 98,248)( 99,247)(100,270)(101,269)(102,268)(103,267)(104,266)(105,265)(106,264)(107,263)(108,262)(109,307)(110,309)(111,308)(112,315)(113,314)(114,313)(115,312)(116,311)(117,310)(118,298)(119,300)(120,299)(121,306)(122,305)(123,304)(124,303)(125,302)(126,301)(127,324)(128,323)(129,322)(130,321)(131,320)(132,319)(133,318)(134,317)(135,316)(136,280)(137,282)(138,281)(139,288)(140,287)(141,286)(142,285)(143,284)(144,283)(145,271)(146,273)(147,272)(148,279)(149,278)(150,277)(151,276)(152,275)(153,274)(154,297)(155,296)(156,295)(157,294)(158,293)(159,292)(160,291)(161,290)(162,289);
s2 := Sym(324)!(  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)(244,271)(245,272)(246,273)(247,274)(248,275)(249,276)(250,277)(251,278)(252,279)(253,280)(254,281)(255,282)(256,283)(257,284)(258,285)(259,286)(260,287)(261,288)(262,289)(263,290)(264,291)(265,292)(266,293)(267,294)(268,295)(269,296)(270,297);
poly := sub<Sym(324)|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*s2*s1*s0*s1*s2*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
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*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*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*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