Polytope of Type {84,6} with group SmallGroup(1512,827)

Overview

Group: SmallGroup(1512,827)
Rank: 3
Schläfli Type: {84,6}
Vertices, edges, …: 126, 378, 9
Order of s₀s₁s₂: 84
Order of s₀s₁s₂s₁: 6
Also known as: if this polytope has a name.

Special Properties

Compact Hyperbolic Quotient
Locally Spherical
Non-Orientable
Self-Petrie

Quotients maximal quotients in bold

3-fold

{28,6}*504

7-fold

{12,6}*216c

21-fold

{4,6}*72

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