Polytope of Type {6,30}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,30}*960
if this polytope has a name.
Group : SmallGroup(960,10967)
Rank : 3
Schlafli Type : {6,30}
Number of vertices, edges, etc : 16, 240, 80
Order of s₀s₁s₂ : 40
Order of s₀s₁s₂s₁ : 12
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {6,30,2} of size 1920
Vertex Figure Of :
   {2,6,30} of size 1920
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {6,30}*480
   4-fold quotients : {6,15}*240
   5-fold quotients : {6,6}*192b
   10-fold quotients : {6,6}*96
   20-fold quotients : {3,6}*48, {6,3}*48
   24-fold quotients : {2,10}*40
   40-fold quotients : {3,3}*24
   48-fold quotients : {2,5}*20
   120-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   2-fold covers : {6,60}*1920, {12,30}*1920
Permutation Representation (GAP) :

s0 := (  3,  6)(  4,  5)(  7,  8)( 11, 14)( 12, 13)( 15, 16)( 19, 22)( 20, 21)
( 23, 24)( 27, 30)( 28, 29)( 31, 32)( 35, 38)( 36, 37)( 39, 40)( 41, 81)
( 42, 82)( 43, 86)( 44, 85)( 45, 84)( 46, 83)( 47, 88)( 48, 87)( 49, 89)
( 50, 90)( 51, 94)( 52, 93)( 53, 92)( 54, 91)( 55, 96)( 56, 95)( 57, 97)
( 58, 98)( 59,102)( 60,101)( 61,100)( 62, 99)( 63,104)( 64,103)( 65,105)
( 66,106)( 67,110)( 68,109)( 69,108)( 70,107)( 71,112)( 72,111)( 73,113)
( 74,114)( 75,118)( 76,117)( 77,116)( 78,115)( 79,120)( 80,119)(121,122)
(123,125)(124,126)(129,130)(131,133)(132,134)(137,138)(139,141)(140,142)
(145,146)(147,149)(148,150)(153,154)(155,157)(156,158)(161,202)(162,201)
(163,205)(164,206)(165,203)(166,204)(167,207)(168,208)(169,210)(170,209)
(171,213)(172,214)(173,211)(174,212)(175,215)(176,216)(177,218)(178,217)
(179,221)(180,222)(181,219)(182,220)(183,223)(184,224)(185,226)(186,225)
(187,229)(188,230)(189,227)(190,228)(191,231)(192,232)(193,234)(194,233)
(195,237)(196,238)(197,235)(198,236)(199,239)(200,240);;
s1 := (  1, 41)(  2, 42)(  3, 44)(  4, 43)(  5, 47)(  6, 48)(  7, 45)(  8, 46)
(  9, 73)( 10, 74)( 11, 76)( 12, 75)( 13, 79)( 14, 80)( 15, 77)( 16, 78)
( 17, 65)( 18, 66)( 19, 68)( 20, 67)( 21, 71)( 22, 72)( 23, 69)( 24, 70)
( 25, 57)( 26, 58)( 27, 60)( 28, 59)( 29, 63)( 30, 64)( 31, 61)( 32, 62)
( 33, 49)( 34, 50)( 35, 52)( 36, 51)( 37, 55)( 38, 56)( 39, 53)( 40, 54)
( 83, 84)( 85, 87)( 86, 88)( 89,113)( 90,114)( 91,116)( 92,115)( 93,119)
( 94,120)( 95,117)( 96,118)( 97,105)( 98,106)( 99,108)(100,107)(101,111)
(102,112)(103,109)(104,110)(121,161)(122,162)(123,164)(124,163)(125,167)
(126,168)(127,165)(128,166)(129,193)(130,194)(131,196)(132,195)(133,199)
(134,200)(135,197)(136,198)(137,185)(138,186)(139,188)(140,187)(141,191)
(142,192)(143,189)(144,190)(145,177)(146,178)(147,180)(148,179)(149,183)
(150,184)(151,181)(152,182)(153,169)(154,170)(155,172)(156,171)(157,175)
(158,176)(159,173)(160,174)(203,204)(205,207)(206,208)(209,233)(210,234)
(211,236)(212,235)(213,239)(214,240)(215,237)(216,238)(217,225)(218,226)
(219,228)(220,227)(221,231)(222,232)(223,229)(224,230);;
s2 := (  1,135)(  2,136)(  3,131)(  4,132)(  5,134)(  6,133)(  7,129)(  8,130)
(  9,127)( 10,128)( 11,123)( 12,124)( 13,126)( 14,125)( 15,121)( 16,122)
( 17,159)( 18,160)( 19,155)( 20,156)( 21,158)( 22,157)( 23,153)( 24,154)
( 25,151)( 26,152)( 27,147)( 28,148)( 29,150)( 30,149)( 31,145)( 32,146)
( 33,143)( 34,144)( 35,139)( 36,140)( 37,142)( 38,141)( 39,137)( 40,138)
( 41,215)( 42,216)( 43,211)( 44,212)( 45,214)( 46,213)( 47,209)( 48,210)
( 49,207)( 50,208)( 51,203)( 52,204)( 53,206)( 54,205)( 55,201)( 56,202)
( 57,239)( 58,240)( 59,235)( 60,236)( 61,238)( 62,237)( 63,233)( 64,234)
( 65,231)( 66,232)( 67,227)( 68,228)( 69,230)( 70,229)( 71,225)( 72,226)
( 73,223)( 74,224)( 75,219)( 76,220)( 77,222)( 78,221)( 79,217)( 80,218)
( 81,175)( 82,176)( 83,171)( 84,172)( 85,174)( 86,173)( 87,169)( 88,170)
( 89,167)( 90,168)( 91,163)( 92,164)( 93,166)( 94,165)( 95,161)( 96,162)
( 97,199)( 98,200)( 99,195)(100,196)(101,198)(102,197)(103,193)(104,194)
(105,191)(106,192)(107,187)(108,188)(109,190)(110,189)(111,185)(112,186)
(113,183)(114,184)(115,179)(116,180)(117,182)(118,181)(119,177)(120,178);;
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, 
s2*s0*s1*s2*s1*s2*s1*s0*s1*s2*s0*s1*s2*s1*s2*s1*s0*s1, 
s2*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s0*s2*s1*s0*s1*s2*s1*s0*s1*s0*s1, 
s1*s2*s1*s0*s2*s1*s0*s2*s1*s2*s0*s1*s2*s1*s2*s1*s2*s1*s2*s0*s1*s2*s1*s2 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(240)!(  3,  6)(  4,  5)(  7,  8)( 11, 14)( 12, 13)( 15, 16)( 19, 22)
( 20, 21)( 23, 24)( 27, 30)( 28, 29)( 31, 32)( 35, 38)( 36, 37)( 39, 40)
( 41, 81)( 42, 82)( 43, 86)( 44, 85)( 45, 84)( 46, 83)( 47, 88)( 48, 87)
( 49, 89)( 50, 90)( 51, 94)( 52, 93)( 53, 92)( 54, 91)( 55, 96)( 56, 95)
( 57, 97)( 58, 98)( 59,102)( 60,101)( 61,100)( 62, 99)( 63,104)( 64,103)
( 65,105)( 66,106)( 67,110)( 68,109)( 69,108)( 70,107)( 71,112)( 72,111)
( 73,113)( 74,114)( 75,118)( 76,117)( 77,116)( 78,115)( 79,120)( 80,119)
(121,122)(123,125)(124,126)(129,130)(131,133)(132,134)(137,138)(139,141)
(140,142)(145,146)(147,149)(148,150)(153,154)(155,157)(156,158)(161,202)
(162,201)(163,205)(164,206)(165,203)(166,204)(167,207)(168,208)(169,210)
(170,209)(171,213)(172,214)(173,211)(174,212)(175,215)(176,216)(177,218)
(178,217)(179,221)(180,222)(181,219)(182,220)(183,223)(184,224)(185,226)
(186,225)(187,229)(188,230)(189,227)(190,228)(191,231)(192,232)(193,234)
(194,233)(195,237)(196,238)(197,235)(198,236)(199,239)(200,240);
s1 := Sym(240)!(  1, 41)(  2, 42)(  3, 44)(  4, 43)(  5, 47)(  6, 48)(  7, 45)
(  8, 46)(  9, 73)( 10, 74)( 11, 76)( 12, 75)( 13, 79)( 14, 80)( 15, 77)
( 16, 78)( 17, 65)( 18, 66)( 19, 68)( 20, 67)( 21, 71)( 22, 72)( 23, 69)
( 24, 70)( 25, 57)( 26, 58)( 27, 60)( 28, 59)( 29, 63)( 30, 64)( 31, 61)
( 32, 62)( 33, 49)( 34, 50)( 35, 52)( 36, 51)( 37, 55)( 38, 56)( 39, 53)
( 40, 54)( 83, 84)( 85, 87)( 86, 88)( 89,113)( 90,114)( 91,116)( 92,115)
( 93,119)( 94,120)( 95,117)( 96,118)( 97,105)( 98,106)( 99,108)(100,107)
(101,111)(102,112)(103,109)(104,110)(121,161)(122,162)(123,164)(124,163)
(125,167)(126,168)(127,165)(128,166)(129,193)(130,194)(131,196)(132,195)
(133,199)(134,200)(135,197)(136,198)(137,185)(138,186)(139,188)(140,187)
(141,191)(142,192)(143,189)(144,190)(145,177)(146,178)(147,180)(148,179)
(149,183)(150,184)(151,181)(152,182)(153,169)(154,170)(155,172)(156,171)
(157,175)(158,176)(159,173)(160,174)(203,204)(205,207)(206,208)(209,233)
(210,234)(211,236)(212,235)(213,239)(214,240)(215,237)(216,238)(217,225)
(218,226)(219,228)(220,227)(221,231)(222,232)(223,229)(224,230);
s2 := Sym(240)!(  1,135)(  2,136)(  3,131)(  4,132)(  5,134)(  6,133)(  7,129)
(  8,130)(  9,127)( 10,128)( 11,123)( 12,124)( 13,126)( 14,125)( 15,121)
( 16,122)( 17,159)( 18,160)( 19,155)( 20,156)( 21,158)( 22,157)( 23,153)
( 24,154)( 25,151)( 26,152)( 27,147)( 28,148)( 29,150)( 30,149)( 31,145)
( 32,146)( 33,143)( 34,144)( 35,139)( 36,140)( 37,142)( 38,141)( 39,137)
( 40,138)( 41,215)( 42,216)( 43,211)( 44,212)( 45,214)( 46,213)( 47,209)
( 48,210)( 49,207)( 50,208)( 51,203)( 52,204)( 53,206)( 54,205)( 55,201)
( 56,202)( 57,239)( 58,240)( 59,235)( 60,236)( 61,238)( 62,237)( 63,233)
( 64,234)( 65,231)( 66,232)( 67,227)( 68,228)( 69,230)( 70,229)( 71,225)
( 72,226)( 73,223)( 74,224)( 75,219)( 76,220)( 77,222)( 78,221)( 79,217)
( 80,218)( 81,175)( 82,176)( 83,171)( 84,172)( 85,174)( 86,173)( 87,169)
( 88,170)( 89,167)( 90,168)( 91,163)( 92,164)( 93,166)( 94,165)( 95,161)
( 96,162)( 97,199)( 98,200)( 99,195)(100,196)(101,198)(102,197)(103,193)
(104,194)(105,191)(106,192)(107,187)(108,188)(109,190)(110,189)(111,185)
(112,186)(113,183)(114,184)(115,179)(116,180)(117,182)(118,181)(119,177)
(120,178);
poly := sub<Sym(240)|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, 
s2*s0*s1*s2*s1*s2*s1*s0*s1*s2*s0*s1*s2*s1*s2*s1*s0*s1, 
s2*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s0*s2*s1*s0*s1*s2*s1*s0*s1*s0*s1, 
s1*s2*s1*s0*s2*s1*s0*s2*s1*s2*s0*s1*s2*s1*s2*s1*s2*s1*s2*s0*s1*s2*s1*s2 >;

References : None.
to this polytope