Polytope of Type {6,8} with group SmallGroup(768,1086649)

Overview

Group: SmallGroup(768,1086649)
Rank: 3
Schläfli Type: {6,8}
Vertices, edges, …: 48, 192, 64
Order of s₀s₁s₂: 24
Order of s₀s₁s₂s₁: 8
Also known as: if this polytope has a name.

Special Properties

Compact Hyperbolic Quotient
Locally Spherical
Orientable

Quotients maximal quotients in bold

2-fold

4-fold

8-fold

16-fold

24-fold

{2,8}*32

32-fold

48-fold

{2,4}*16

64-fold

{3,2}*12

96-fold

{2,2}*8

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=<(s0*s1)^3> of order 2

40 facets

16 of {3}*6
24 of {6}*12

24 vertex figures

24 of {8}*16

P/N, where N=<(s0*s1)^2> of order 3

32 facets

16 of {2}*4
16 of {6}*12

16 vertex figures

16 of {8}*16

Representations

Permutation Representation (GAP)

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

Permutation Representation (Magma)

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

References

None.

to this polytope.