Polytope of Type {6,72} with group SmallGroup(864,770)

Overview

Group: SmallGroup(864,770)
Rank: 3
Schläfli Type: {6,72}
Vertices, edges, …: 6, 216, 72
Order of s₀s₁s₂: 72
Order of s₀s₁s₂s₁: 2
Also known as: {6,72|2}. if this polytope has another name.

Special Properties

Compact Hyperbolic Quotient
Locally Spherical
Orientable
Flat

Quotients maximal quotients in bold

2-fold

{6,36}*432a

3-fold

4-fold

{6,18}*216a

6-fold

9-fold

12-fold

18-fold

24-fold

{2,9}*36

27-fold

{2,8}*32

36-fold

54-fold

{2,4}*16

72-fold

108-fold

{2,2}*8

Covers minimal covers in bold

2-fold

Irregular Quotients of which this is a minimal cover

None.

Representations

Permutation Representation (GAP)

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

Permutation Representation (Magma)

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

References