Polytope of Type {30,6}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {30,6}*960
if this polytope has a name.
Group : SmallGroup(960,10967)
Rank : 3
Schlafli Type : {30,6}
Number of vertices, edges, etc : 80, 240, 16
Order of s0s1s2 : 40
Order of s0s1s2s1 : 12
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Related Polytopes :
Facet
Vertex Figure
Dual
Petrial
Facet Of :
{30,6,2} of size 1920
Vertex Figure Of :
{2,30,6} of size 1920
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {30,6}*480
4-fold quotients : {15,6}*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 : {10,2}*40
40-fold quotients : {3,3}*24
48-fold quotients : {5,2}*20
120-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
2-fold covers : {60,6}*1920, {30,12}*1920
Permutation Representation (GAP) :
s0 := ( 1,121)( 2,122)( 3,126)( 4,125)( 5,124)( 6,123)( 7,128)( 8,127)
( 9,153)( 10,154)( 11,158)( 12,157)( 13,156)( 14,155)( 15,160)( 16,159)
( 17,145)( 18,146)( 19,150)( 20,149)( 21,148)( 22,147)( 23,152)( 24,151)
( 25,137)( 26,138)( 27,142)( 28,141)( 29,140)( 30,139)( 31,144)( 32,143)
( 33,129)( 34,130)( 35,134)( 36,133)( 37,132)( 38,131)( 39,136)( 40,135)
( 41,201)( 42,202)( 43,206)( 44,205)( 45,204)( 46,203)( 47,208)( 48,207)
( 49,233)( 50,234)( 51,238)( 52,237)( 53,236)( 54,235)( 55,240)( 56,239)
( 57,225)( 58,226)( 59,230)( 60,229)( 61,228)( 62,227)( 63,232)( 64,231)
( 65,217)( 66,218)( 67,222)( 68,221)( 69,220)( 70,219)( 71,224)( 72,223)
( 73,209)( 74,210)( 75,214)( 76,213)( 77,212)( 78,211)( 79,216)( 80,215)
( 81,161)( 82,162)( 83,166)( 84,165)( 85,164)( 86,163)( 87,168)( 88,167)
( 89,193)( 90,194)( 91,198)( 92,197)( 93,196)( 94,195)( 95,200)( 96,199)
( 97,185)( 98,186)( 99,190)(100,189)(101,188)(102,187)(103,192)(104,191)
(105,177)(106,178)(107,182)(108,181)(109,180)(110,179)(111,184)(112,183)
(113,169)(114,170)(115,174)(116,173)(117,172)(118,171)(119,176)(120,175);;
s1 := ( 1, 49)( 2, 50)( 3, 52)( 4, 51)( 5, 55)( 6, 56)( 7, 53)( 8, 54)
( 9, 41)( 10, 42)( 11, 44)( 12, 43)( 13, 47)( 14, 48)( 15, 45)( 16, 46)
( 17, 73)( 18, 74)( 19, 76)( 20, 75)( 21, 79)( 22, 80)( 23, 77)( 24, 78)
( 25, 65)( 26, 66)( 27, 68)( 28, 67)( 29, 71)( 30, 72)( 31, 69)( 32, 70)
( 33, 57)( 34, 58)( 35, 60)( 36, 59)( 37, 63)( 38, 64)( 39, 61)( 40, 62)
( 81, 89)( 82, 90)( 83, 92)( 84, 91)( 85, 95)( 86, 96)( 87, 93)( 88, 94)
( 97,113)( 98,114)( 99,116)(100,115)(101,119)(102,120)(103,117)(104,118)
(107,108)(109,111)(110,112)(121,169)(122,170)(123,172)(124,171)(125,175)
(126,176)(127,173)(128,174)(129,161)(130,162)(131,164)(132,163)(133,167)
(134,168)(135,165)(136,166)(137,193)(138,194)(139,196)(140,195)(141,199)
(142,200)(143,197)(144,198)(145,185)(146,186)(147,188)(148,187)(149,191)
(150,192)(151,189)(152,190)(153,177)(154,178)(155,180)(156,179)(157,183)
(158,184)(159,181)(160,182)(201,209)(202,210)(203,212)(204,211)(205,215)
(206,216)(207,213)(208,214)(217,233)(218,234)(219,236)(220,235)(221,239)
(222,240)(223,237)(224,238)(227,228)(229,231)(230,232);;
s2 := ( 1, 7)( 2, 8)( 5, 6)( 9, 15)( 10, 16)( 13, 14)( 17, 23)( 18, 24)
( 21, 22)( 25, 31)( 26, 32)( 29, 30)( 33, 39)( 34, 40)( 37, 38)( 41, 87)
( 42, 88)( 43, 83)( 44, 84)( 45, 86)( 46, 85)( 47, 81)( 48, 82)( 49, 95)
( 50, 96)( 51, 91)( 52, 92)( 53, 94)( 54, 93)( 55, 89)( 56, 90)( 57,103)
( 58,104)( 59, 99)( 60,100)( 61,102)( 62,101)( 63, 97)( 64, 98)( 65,111)
( 66,112)( 67,107)( 68,108)( 69,110)( 70,109)( 71,105)( 72,106)( 73,119)
( 74,120)( 75,115)( 76,116)( 77,118)( 78,117)( 79,113)( 80,114)(121,128)
(122,127)(123,124)(129,136)(130,135)(131,132)(137,144)(138,143)(139,140)
(145,152)(146,151)(147,148)(153,160)(154,159)(155,156)(161,208)(162,207)
(163,204)(164,203)(165,205)(166,206)(167,202)(168,201)(169,216)(170,215)
(171,212)(172,211)(173,213)(174,214)(175,210)(176,209)(177,224)(178,223)
(179,220)(180,219)(181,221)(182,222)(183,218)(184,217)(185,232)(186,231)
(187,228)(188,227)(189,229)(190,230)(191,226)(192,225)(193,240)(194,239)
(195,236)(196,235)(197,237)(198,238)(199,234)(200,233);;
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,
s2*s0*s1*s2*s1*s0*s1*s0*s1*s2*s0*s1*s2*s1*s0*s1*s0*s1,
s2*s0*s1*s2*s1*s2*s0*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1,
s2*s0*s1*s2*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s0*s1,
s0*s2*s1*s0*s2*s1*s0*s1*s0*s1*s0*s1*s2*s0*s1*s0*s1*s0*s1*s0*s1*s2*s0*s1 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(240)!( 1,121)( 2,122)( 3,126)( 4,125)( 5,124)( 6,123)( 7,128)
( 8,127)( 9,153)( 10,154)( 11,158)( 12,157)( 13,156)( 14,155)( 15,160)
( 16,159)( 17,145)( 18,146)( 19,150)( 20,149)( 21,148)( 22,147)( 23,152)
( 24,151)( 25,137)( 26,138)( 27,142)( 28,141)( 29,140)( 30,139)( 31,144)
( 32,143)( 33,129)( 34,130)( 35,134)( 36,133)( 37,132)( 38,131)( 39,136)
( 40,135)( 41,201)( 42,202)( 43,206)( 44,205)( 45,204)( 46,203)( 47,208)
( 48,207)( 49,233)( 50,234)( 51,238)( 52,237)( 53,236)( 54,235)( 55,240)
( 56,239)( 57,225)( 58,226)( 59,230)( 60,229)( 61,228)( 62,227)( 63,232)
( 64,231)( 65,217)( 66,218)( 67,222)( 68,221)( 69,220)( 70,219)( 71,224)
( 72,223)( 73,209)( 74,210)( 75,214)( 76,213)( 77,212)( 78,211)( 79,216)
( 80,215)( 81,161)( 82,162)( 83,166)( 84,165)( 85,164)( 86,163)( 87,168)
( 88,167)( 89,193)( 90,194)( 91,198)( 92,197)( 93,196)( 94,195)( 95,200)
( 96,199)( 97,185)( 98,186)( 99,190)(100,189)(101,188)(102,187)(103,192)
(104,191)(105,177)(106,178)(107,182)(108,181)(109,180)(110,179)(111,184)
(112,183)(113,169)(114,170)(115,174)(116,173)(117,172)(118,171)(119,176)
(120,175);
s1 := Sym(240)!( 1, 49)( 2, 50)( 3, 52)( 4, 51)( 5, 55)( 6, 56)( 7, 53)
( 8, 54)( 9, 41)( 10, 42)( 11, 44)( 12, 43)( 13, 47)( 14, 48)( 15, 45)
( 16, 46)( 17, 73)( 18, 74)( 19, 76)( 20, 75)( 21, 79)( 22, 80)( 23, 77)
( 24, 78)( 25, 65)( 26, 66)( 27, 68)( 28, 67)( 29, 71)( 30, 72)( 31, 69)
( 32, 70)( 33, 57)( 34, 58)( 35, 60)( 36, 59)( 37, 63)( 38, 64)( 39, 61)
( 40, 62)( 81, 89)( 82, 90)( 83, 92)( 84, 91)( 85, 95)( 86, 96)( 87, 93)
( 88, 94)( 97,113)( 98,114)( 99,116)(100,115)(101,119)(102,120)(103,117)
(104,118)(107,108)(109,111)(110,112)(121,169)(122,170)(123,172)(124,171)
(125,175)(126,176)(127,173)(128,174)(129,161)(130,162)(131,164)(132,163)
(133,167)(134,168)(135,165)(136,166)(137,193)(138,194)(139,196)(140,195)
(141,199)(142,200)(143,197)(144,198)(145,185)(146,186)(147,188)(148,187)
(149,191)(150,192)(151,189)(152,190)(153,177)(154,178)(155,180)(156,179)
(157,183)(158,184)(159,181)(160,182)(201,209)(202,210)(203,212)(204,211)
(205,215)(206,216)(207,213)(208,214)(217,233)(218,234)(219,236)(220,235)
(221,239)(222,240)(223,237)(224,238)(227,228)(229,231)(230,232);
s2 := Sym(240)!( 1, 7)( 2, 8)( 5, 6)( 9, 15)( 10, 16)( 13, 14)( 17, 23)
( 18, 24)( 21, 22)( 25, 31)( 26, 32)( 29, 30)( 33, 39)( 34, 40)( 37, 38)
( 41, 87)( 42, 88)( 43, 83)( 44, 84)( 45, 86)( 46, 85)( 47, 81)( 48, 82)
( 49, 95)( 50, 96)( 51, 91)( 52, 92)( 53, 94)( 54, 93)( 55, 89)( 56, 90)
( 57,103)( 58,104)( 59, 99)( 60,100)( 61,102)( 62,101)( 63, 97)( 64, 98)
( 65,111)( 66,112)( 67,107)( 68,108)( 69,110)( 70,109)( 71,105)( 72,106)
( 73,119)( 74,120)( 75,115)( 76,116)( 77,118)( 78,117)( 79,113)( 80,114)
(121,128)(122,127)(123,124)(129,136)(130,135)(131,132)(137,144)(138,143)
(139,140)(145,152)(146,151)(147,148)(153,160)(154,159)(155,156)(161,208)
(162,207)(163,204)(164,203)(165,205)(166,206)(167,202)(168,201)(169,216)
(170,215)(171,212)(172,211)(173,213)(174,214)(175,210)(176,209)(177,224)
(178,223)(179,220)(180,219)(181,221)(182,222)(183,218)(184,217)(185,232)
(186,231)(187,228)(188,227)(189,229)(190,230)(191,226)(192,225)(193,240)
(194,239)(195,236)(196,235)(197,237)(198,238)(199,234)(200,233);
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, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s2*s0*s1*s2*s1*s0*s1*s0*s1*s2*s0*s1*s2*s1*s0*s1*s0*s1,
s2*s0*s1*s2*s1*s2*s0*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1,
s2*s0*s1*s2*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s0*s1,
s0*s2*s1*s0*s2*s1*s0*s1*s0*s1*s0*s1*s2*s0*s1*s0*s1*s0*s1*s0*s1*s2*s0*s1 >;
References : None.
to this polytope