Polytope of Type {42,18}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {42,18}*1512a
if this polytope has a name.
Group : SmallGroup(1512,485)
Rank : 3
Schlafli Type : {42,18}
Number of vertices, edges, etc : 42, 378, 18
Order of s₀s₁s₂ : 126
Order of s₀s₁s₂s₁ : 6
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {14,18}*504, {42,6}*504a
   7-fold quotients : {6,18}*216b
   9-fold quotients : {14,6}*168
   14-fold quotients : {6,9}*108
   21-fold quotients : {2,18}*72, {6,6}*72b
   27-fold quotients : {14,2}*56
   42-fold quotients : {2,9}*36, {6,3}*36
   54-fold quotients : {7,2}*28
   63-fold quotients : {2,6}*24
   126-fold quotients : {2,3}*12
   189-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

References : None.
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