Polytope of Type {6,12,10}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,12,10}*1920a
if this polytope has a name.
Group : SmallGroup(1920,240151)
Rank : 4
Schlafli Type : {6,12,10}
Number of vertices, edges, etc : 8, 48, 80, 10
Order of s₀s₁s₂s₃ : 20
Order of s₀s₁s₂s₃s₂s₁ : 2
Special Properties :
   Universal
   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) :
   2-fold quotients : {6,6,10}*960
   4-fold quotients : {3,6,10}*480
   5-fold quotients : {6,12,2}*384a
   10-fold quotients : {6,6,2}*192
   12-fold quotients : {2,4,10}*160
   20-fold quotients : {3,6,2}*96, {6,3,2}*96
   24-fold quotients : {2,2,10}*80
   40-fold quotients : {3,3,2}*48
   48-fold quotients : {2,2,5}*40
   60-fold quotients : {2,4,2}*32
   120-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

s0 := (  3,  4)(  7,  8)( 11, 12)( 15, 16)( 19, 20)( 21, 41)( 22, 42)( 23, 44)
( 24, 43)( 25, 45)( 26, 46)( 27, 48)( 28, 47)( 29, 49)( 30, 50)( 31, 52)
( 32, 51)( 33, 53)( 34, 54)( 35, 56)( 36, 55)( 37, 57)( 38, 58)( 39, 60)
( 40, 59)( 63, 64)( 67, 68)( 71, 72)( 75, 76)( 79, 80)( 81,101)( 82,102)
( 83,104)( 84,103)( 85,105)( 86,106)( 87,108)( 88,107)( 89,109)( 90,110)
( 91,112)( 92,111)( 93,113)( 94,114)( 95,116)( 96,115)( 97,117)( 98,118)
( 99,120)(100,119)(123,124)(127,128)(131,132)(135,136)(139,140)(141,161)
(142,162)(143,164)(144,163)(145,165)(146,166)(147,168)(148,167)(149,169)
(150,170)(151,172)(152,171)(153,173)(154,174)(155,176)(156,175)(157,177)
(158,178)(159,180)(160,179)(183,184)(187,188)(191,192)(195,196)(199,200)
(201,221)(202,222)(203,224)(204,223)(205,225)(206,226)(207,228)(208,227)
(209,229)(210,230)(211,232)(212,231)(213,233)(214,234)(215,236)(216,235)
(217,237)(218,238)(219,240)(220,239);;
s1 := (  1, 21)(  2, 24)(  3, 23)(  4, 22)(  5, 25)(  6, 28)(  7, 27)(  8, 26)
(  9, 29)( 10, 32)( 11, 31)( 12, 30)( 13, 33)( 14, 36)( 15, 35)( 16, 34)
( 17, 37)( 18, 40)( 19, 39)( 20, 38)( 42, 44)( 46, 48)( 50, 52)( 54, 56)
( 58, 60)( 61, 81)( 62, 84)( 63, 83)( 64, 82)( 65, 85)( 66, 88)( 67, 87)
( 68, 86)( 69, 89)( 70, 92)( 71, 91)( 72, 90)( 73, 93)( 74, 96)( 75, 95)
( 76, 94)( 77, 97)( 78,100)( 79, 99)( 80, 98)(102,104)(106,108)(110,112)
(114,116)(118,120)(121,201)(122,204)(123,203)(124,202)(125,205)(126,208)
(127,207)(128,206)(129,209)(130,212)(131,211)(132,210)(133,213)(134,216)
(135,215)(136,214)(137,217)(138,220)(139,219)(140,218)(141,181)(142,184)
(143,183)(144,182)(145,185)(146,188)(147,187)(148,186)(149,189)(150,192)
(151,191)(152,190)(153,193)(154,196)(155,195)(156,194)(157,197)(158,200)
(159,199)(160,198)(161,221)(162,224)(163,223)(164,222)(165,225)(166,228)
(167,227)(168,226)(169,229)(170,232)(171,231)(172,230)(173,233)(174,236)
(175,235)(176,234)(177,237)(178,240)(179,239)(180,238);;
s2 := (  1,122)(  2,121)(  3,123)(  4,124)(  5,138)(  6,137)(  7,139)(  8,140)
(  9,134)( 10,133)( 11,135)( 12,136)( 13,130)( 14,129)( 15,131)( 16,132)
( 17,126)( 18,125)( 19,127)( 20,128)( 21,162)( 22,161)( 23,163)( 24,164)
( 25,178)( 26,177)( 27,179)( 28,180)( 29,174)( 30,173)( 31,175)( 32,176)
( 33,170)( 34,169)( 35,171)( 36,172)( 37,166)( 38,165)( 39,167)( 40,168)
( 41,142)( 42,141)( 43,143)( 44,144)( 45,158)( 46,157)( 47,159)( 48,160)
( 49,154)( 50,153)( 51,155)( 52,156)( 53,150)( 54,149)( 55,151)( 56,152)
( 57,146)( 58,145)( 59,147)( 60,148)( 61,182)( 62,181)( 63,183)( 64,184)
( 65,198)( 66,197)( 67,199)( 68,200)( 69,194)( 70,193)( 71,195)( 72,196)
( 73,190)( 74,189)( 75,191)( 76,192)( 77,186)( 78,185)( 79,187)( 80,188)
( 81,222)( 82,221)( 83,223)( 84,224)( 85,238)( 86,237)( 87,239)( 88,240)
( 89,234)( 90,233)( 91,235)( 92,236)( 93,230)( 94,229)( 95,231)( 96,232)
( 97,226)( 98,225)( 99,227)(100,228)(101,202)(102,201)(103,203)(104,204)
(105,218)(106,217)(107,219)(108,220)(109,214)(110,213)(111,215)(112,216)
(113,210)(114,209)(115,211)(116,212)(117,206)(118,205)(119,207)(120,208);;
s3 := (  1,  5)(  2,  6)(  3,  7)(  4,  8)(  9, 17)( 10, 18)( 11, 19)( 12, 20)
( 21, 25)( 22, 26)( 23, 27)( 24, 28)( 29, 37)( 30, 38)( 31, 39)( 32, 40)
( 41, 45)( 42, 46)( 43, 47)( 44, 48)( 49, 57)( 50, 58)( 51, 59)( 52, 60)
( 61, 65)( 62, 66)( 63, 67)( 64, 68)( 69, 77)( 70, 78)( 71, 79)( 72, 80)
( 81, 85)( 82, 86)( 83, 87)( 84, 88)( 89, 97)( 90, 98)( 91, 99)( 92,100)
(101,105)(102,106)(103,107)(104,108)(109,117)(110,118)(111,119)(112,120)
(121,125)(122,126)(123,127)(124,128)(129,137)(130,138)(131,139)(132,140)
(141,145)(142,146)(143,147)(144,148)(149,157)(150,158)(151,159)(152,160)
(161,165)(162,166)(163,167)(164,168)(169,177)(170,178)(171,179)(172,180)
(181,185)(182,186)(183,187)(184,188)(189,197)(190,198)(191,199)(192,200)
(201,205)(202,206)(203,207)(204,208)(209,217)(210,218)(211,219)(212,220)
(221,225)(222,226)(223,227)(224,228)(229,237)(230,238)(231,239)(232,240);;
poly := Group([s0,s1,s2,s3]);;

Finitely Presented Group Representation (GAP) :

F := FreeGroup("s0","s1","s2","s3");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;  
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s0*s2*s0*s2, 
s0*s3*s0*s3, s1*s3*s1*s3, s1*s2*s3*s2*s1*s2*s3*s2, 
s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(240)!(  3,  4)(  7,  8)( 11, 12)( 15, 16)( 19, 20)( 21, 41)( 22, 42)
( 23, 44)( 24, 43)( 25, 45)( 26, 46)( 27, 48)( 28, 47)( 29, 49)( 30, 50)
( 31, 52)( 32, 51)( 33, 53)( 34, 54)( 35, 56)( 36, 55)( 37, 57)( 38, 58)
( 39, 60)( 40, 59)( 63, 64)( 67, 68)( 71, 72)( 75, 76)( 79, 80)( 81,101)
( 82,102)( 83,104)( 84,103)( 85,105)( 86,106)( 87,108)( 88,107)( 89,109)
( 90,110)( 91,112)( 92,111)( 93,113)( 94,114)( 95,116)( 96,115)( 97,117)
( 98,118)( 99,120)(100,119)(123,124)(127,128)(131,132)(135,136)(139,140)
(141,161)(142,162)(143,164)(144,163)(145,165)(146,166)(147,168)(148,167)
(149,169)(150,170)(151,172)(152,171)(153,173)(154,174)(155,176)(156,175)
(157,177)(158,178)(159,180)(160,179)(183,184)(187,188)(191,192)(195,196)
(199,200)(201,221)(202,222)(203,224)(204,223)(205,225)(206,226)(207,228)
(208,227)(209,229)(210,230)(211,232)(212,231)(213,233)(214,234)(215,236)
(216,235)(217,237)(218,238)(219,240)(220,239);
s1 := Sym(240)!(  1, 21)(  2, 24)(  3, 23)(  4, 22)(  5, 25)(  6, 28)(  7, 27)
(  8, 26)(  9, 29)( 10, 32)( 11, 31)( 12, 30)( 13, 33)( 14, 36)( 15, 35)
( 16, 34)( 17, 37)( 18, 40)( 19, 39)( 20, 38)( 42, 44)( 46, 48)( 50, 52)
( 54, 56)( 58, 60)( 61, 81)( 62, 84)( 63, 83)( 64, 82)( 65, 85)( 66, 88)
( 67, 87)( 68, 86)( 69, 89)( 70, 92)( 71, 91)( 72, 90)( 73, 93)( 74, 96)
( 75, 95)( 76, 94)( 77, 97)( 78,100)( 79, 99)( 80, 98)(102,104)(106,108)
(110,112)(114,116)(118,120)(121,201)(122,204)(123,203)(124,202)(125,205)
(126,208)(127,207)(128,206)(129,209)(130,212)(131,211)(132,210)(133,213)
(134,216)(135,215)(136,214)(137,217)(138,220)(139,219)(140,218)(141,181)
(142,184)(143,183)(144,182)(145,185)(146,188)(147,187)(148,186)(149,189)
(150,192)(151,191)(152,190)(153,193)(154,196)(155,195)(156,194)(157,197)
(158,200)(159,199)(160,198)(161,221)(162,224)(163,223)(164,222)(165,225)
(166,228)(167,227)(168,226)(169,229)(170,232)(171,231)(172,230)(173,233)
(174,236)(175,235)(176,234)(177,237)(178,240)(179,239)(180,238);
s2 := Sym(240)!(  1,122)(  2,121)(  3,123)(  4,124)(  5,138)(  6,137)(  7,139)
(  8,140)(  9,134)( 10,133)( 11,135)( 12,136)( 13,130)( 14,129)( 15,131)
( 16,132)( 17,126)( 18,125)( 19,127)( 20,128)( 21,162)( 22,161)( 23,163)
( 24,164)( 25,178)( 26,177)( 27,179)( 28,180)( 29,174)( 30,173)( 31,175)
( 32,176)( 33,170)( 34,169)( 35,171)( 36,172)( 37,166)( 38,165)( 39,167)
( 40,168)( 41,142)( 42,141)( 43,143)( 44,144)( 45,158)( 46,157)( 47,159)
( 48,160)( 49,154)( 50,153)( 51,155)( 52,156)( 53,150)( 54,149)( 55,151)
( 56,152)( 57,146)( 58,145)( 59,147)( 60,148)( 61,182)( 62,181)( 63,183)
( 64,184)( 65,198)( 66,197)( 67,199)( 68,200)( 69,194)( 70,193)( 71,195)
( 72,196)( 73,190)( 74,189)( 75,191)( 76,192)( 77,186)( 78,185)( 79,187)
( 80,188)( 81,222)( 82,221)( 83,223)( 84,224)( 85,238)( 86,237)( 87,239)
( 88,240)( 89,234)( 90,233)( 91,235)( 92,236)( 93,230)( 94,229)( 95,231)
( 96,232)( 97,226)( 98,225)( 99,227)(100,228)(101,202)(102,201)(103,203)
(104,204)(105,218)(106,217)(107,219)(108,220)(109,214)(110,213)(111,215)
(112,216)(113,210)(114,209)(115,211)(116,212)(117,206)(118,205)(119,207)
(120,208);
s3 := Sym(240)!(  1,  5)(  2,  6)(  3,  7)(  4,  8)(  9, 17)( 10, 18)( 11, 19)
( 12, 20)( 21, 25)( 22, 26)( 23, 27)( 24, 28)( 29, 37)( 30, 38)( 31, 39)
( 32, 40)( 41, 45)( 42, 46)( 43, 47)( 44, 48)( 49, 57)( 50, 58)( 51, 59)
( 52, 60)( 61, 65)( 62, 66)( 63, 67)( 64, 68)( 69, 77)( 70, 78)( 71, 79)
( 72, 80)( 81, 85)( 82, 86)( 83, 87)( 84, 88)( 89, 97)( 90, 98)( 91, 99)
( 92,100)(101,105)(102,106)(103,107)(104,108)(109,117)(110,118)(111,119)
(112,120)(121,125)(122,126)(123,127)(124,128)(129,137)(130,138)(131,139)
(132,140)(141,145)(142,146)(143,147)(144,148)(149,157)(150,158)(151,159)
(152,160)(161,165)(162,166)(163,167)(164,168)(169,177)(170,178)(171,179)
(172,180)(181,185)(182,186)(183,187)(184,188)(189,197)(190,198)(191,199)
(192,200)(201,205)(202,206)(203,207)(204,208)(209,217)(210,218)(211,219)
(212,220)(221,225)(222,226)(223,227)(224,228)(229,237)(230,238)(231,239)
(232,240);
poly := sub<Sym(240)|s0,s1,s2,s3>;

Finitely Presented Group Representation (Magma) :

poly<s0,s1,s2,s3> := Group< s0,s1,s2,s3 | s0*s0, s1*s1, s2*s2, 
s3*s3, s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s1*s2*s3*s2*s1*s2*s3*s2, s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 >;

References : None.
to this polytope