Polytope of Type {40,12}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {40,12}*960a
Also Known As : {40,12|2}. if this polytope has another name.
Group : SmallGroup(960,2316)
Rank : 3
Schlafli Type : {40,12}
Number of vertices, edges, etc : 40, 240, 12
Order of s0s1s2 : 120
Order of s0s1s2s1 : 2
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{40,12,2} of size 1920
Vertex Figure Of :
{2,40,12} of size 1920
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {40,6}*480, {20,12}*480
3-fold quotients : {40,4}*320a
4-fold quotients : {10,12}*240, {20,6}*240a
5-fold quotients : {8,12}*192a
6-fold quotients : {20,4}*160, {40,2}*160
8-fold quotients : {10,6}*120
10-fold quotients : {4,12}*96a, {8,6}*96
12-fold quotients : {20,2}*80, {10,4}*80
15-fold quotients : {8,4}*64a
20-fold quotients : {2,12}*48, {4,6}*48a
24-fold quotients : {10,2}*40
30-fold quotients : {4,4}*32, {8,2}*32
40-fold quotients : {2,6}*24
48-fold quotients : {5,2}*20
60-fold quotients : {2,4}*16, {4,2}*16
80-fold quotients : {2,3}*12
120-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
2-fold covers : {40,12}*1920a, {40,24}*1920a, {40,24}*1920c, {80,12}*1920a, {80,12}*1920b
Permutation Representation (GAP) :
s0 := ( 2, 5)( 3, 4)( 7, 10)( 8, 9)( 12, 15)( 13, 14)( 17, 20)( 18, 19)
( 22, 25)( 23, 24)( 27, 30)( 28, 29)( 32, 35)( 33, 34)( 37, 40)( 38, 39)
( 42, 45)( 43, 44)( 47, 50)( 48, 49)( 52, 55)( 53, 54)( 57, 60)( 58, 59)
( 61, 76)( 62, 80)( 63, 79)( 64, 78)( 65, 77)( 66, 81)( 67, 85)( 68, 84)
( 69, 83)( 70, 82)( 71, 86)( 72, 90)( 73, 89)( 74, 88)( 75, 87)( 91,106)
( 92,110)( 93,109)( 94,108)( 95,107)( 96,111)( 97,115)( 98,114)( 99,113)
(100,112)(101,116)(102,120)(103,119)(104,118)(105,117)(121,181)(122,185)
(123,184)(124,183)(125,182)(126,186)(127,190)(128,189)(129,188)(130,187)
(131,191)(132,195)(133,194)(134,193)(135,192)(136,196)(137,200)(138,199)
(139,198)(140,197)(141,201)(142,205)(143,204)(144,203)(145,202)(146,206)
(147,210)(148,209)(149,208)(150,207)(151,211)(152,215)(153,214)(154,213)
(155,212)(156,216)(157,220)(158,219)(159,218)(160,217)(161,221)(162,225)
(163,224)(164,223)(165,222)(166,226)(167,230)(168,229)(169,228)(170,227)
(171,231)(172,235)(173,234)(174,233)(175,232)(176,236)(177,240)(178,239)
(179,238)(180,237);;
s1 := ( 1,122)( 2,121)( 3,125)( 4,124)( 5,123)( 6,132)( 7,131)( 8,135)
( 9,134)( 10,133)( 11,127)( 12,126)( 13,130)( 14,129)( 15,128)( 16,137)
( 17,136)( 18,140)( 19,139)( 20,138)( 21,147)( 22,146)( 23,150)( 24,149)
( 25,148)( 26,142)( 27,141)( 28,145)( 29,144)( 30,143)( 31,152)( 32,151)
( 33,155)( 34,154)( 35,153)( 36,162)( 37,161)( 38,165)( 39,164)( 40,163)
( 41,157)( 42,156)( 43,160)( 44,159)( 45,158)( 46,167)( 47,166)( 48,170)
( 49,169)( 50,168)( 51,177)( 52,176)( 53,180)( 54,179)( 55,178)( 56,172)
( 57,171)( 58,175)( 59,174)( 60,173)( 61,197)( 62,196)( 63,200)( 64,199)
( 65,198)( 66,207)( 67,206)( 68,210)( 69,209)( 70,208)( 71,202)( 72,201)
( 73,205)( 74,204)( 75,203)( 76,182)( 77,181)( 78,185)( 79,184)( 80,183)
( 81,192)( 82,191)( 83,195)( 84,194)( 85,193)( 86,187)( 87,186)( 88,190)
( 89,189)( 90,188)( 91,227)( 92,226)( 93,230)( 94,229)( 95,228)( 96,237)
( 97,236)( 98,240)( 99,239)(100,238)(101,232)(102,231)(103,235)(104,234)
(105,233)(106,212)(107,211)(108,215)(109,214)(110,213)(111,222)(112,221)
(113,225)(114,224)(115,223)(116,217)(117,216)(118,220)(119,219)(120,218);;
s2 := ( 1, 6)( 2, 7)( 3, 8)( 4, 9)( 5, 10)( 16, 21)( 17, 22)( 18, 23)
( 19, 24)( 20, 25)( 31, 36)( 32, 37)( 33, 38)( 34, 39)( 35, 40)( 46, 51)
( 47, 52)( 48, 53)( 49, 54)( 50, 55)( 61, 66)( 62, 67)( 63, 68)( 64, 69)
( 65, 70)( 76, 81)( 77, 82)( 78, 83)( 79, 84)( 80, 85)( 91, 96)( 92, 97)
( 93, 98)( 94, 99)( 95,100)(106,111)(107,112)(108,113)(109,114)(110,115)
(121,156)(122,157)(123,158)(124,159)(125,160)(126,151)(127,152)(128,153)
(129,154)(130,155)(131,161)(132,162)(133,163)(134,164)(135,165)(136,171)
(137,172)(138,173)(139,174)(140,175)(141,166)(142,167)(143,168)(144,169)
(145,170)(146,176)(147,177)(148,178)(149,179)(150,180)(181,216)(182,217)
(183,218)(184,219)(185,220)(186,211)(187,212)(188,213)(189,214)(190,215)
(191,221)(192,222)(193,223)(194,224)(195,225)(196,231)(197,232)(198,233)
(199,234)(200,235)(201,226)(202,227)(203,228)(204,229)(205,230)(206,236)
(207,237)(208,238)(209,239)(210,240);;
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*s2*s1*s0*s1*s2*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,
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*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 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(240)!( 2, 5)( 3, 4)( 7, 10)( 8, 9)( 12, 15)( 13, 14)( 17, 20)
( 18, 19)( 22, 25)( 23, 24)( 27, 30)( 28, 29)( 32, 35)( 33, 34)( 37, 40)
( 38, 39)( 42, 45)( 43, 44)( 47, 50)( 48, 49)( 52, 55)( 53, 54)( 57, 60)
( 58, 59)( 61, 76)( 62, 80)( 63, 79)( 64, 78)( 65, 77)( 66, 81)( 67, 85)
( 68, 84)( 69, 83)( 70, 82)( 71, 86)( 72, 90)( 73, 89)( 74, 88)( 75, 87)
( 91,106)( 92,110)( 93,109)( 94,108)( 95,107)( 96,111)( 97,115)( 98,114)
( 99,113)(100,112)(101,116)(102,120)(103,119)(104,118)(105,117)(121,181)
(122,185)(123,184)(124,183)(125,182)(126,186)(127,190)(128,189)(129,188)
(130,187)(131,191)(132,195)(133,194)(134,193)(135,192)(136,196)(137,200)
(138,199)(139,198)(140,197)(141,201)(142,205)(143,204)(144,203)(145,202)
(146,206)(147,210)(148,209)(149,208)(150,207)(151,211)(152,215)(153,214)
(154,213)(155,212)(156,216)(157,220)(158,219)(159,218)(160,217)(161,221)
(162,225)(163,224)(164,223)(165,222)(166,226)(167,230)(168,229)(169,228)
(170,227)(171,231)(172,235)(173,234)(174,233)(175,232)(176,236)(177,240)
(178,239)(179,238)(180,237);
s1 := Sym(240)!( 1,122)( 2,121)( 3,125)( 4,124)( 5,123)( 6,132)( 7,131)
( 8,135)( 9,134)( 10,133)( 11,127)( 12,126)( 13,130)( 14,129)( 15,128)
( 16,137)( 17,136)( 18,140)( 19,139)( 20,138)( 21,147)( 22,146)( 23,150)
( 24,149)( 25,148)( 26,142)( 27,141)( 28,145)( 29,144)( 30,143)( 31,152)
( 32,151)( 33,155)( 34,154)( 35,153)( 36,162)( 37,161)( 38,165)( 39,164)
( 40,163)( 41,157)( 42,156)( 43,160)( 44,159)( 45,158)( 46,167)( 47,166)
( 48,170)( 49,169)( 50,168)( 51,177)( 52,176)( 53,180)( 54,179)( 55,178)
( 56,172)( 57,171)( 58,175)( 59,174)( 60,173)( 61,197)( 62,196)( 63,200)
( 64,199)( 65,198)( 66,207)( 67,206)( 68,210)( 69,209)( 70,208)( 71,202)
( 72,201)( 73,205)( 74,204)( 75,203)( 76,182)( 77,181)( 78,185)( 79,184)
( 80,183)( 81,192)( 82,191)( 83,195)( 84,194)( 85,193)( 86,187)( 87,186)
( 88,190)( 89,189)( 90,188)( 91,227)( 92,226)( 93,230)( 94,229)( 95,228)
( 96,237)( 97,236)( 98,240)( 99,239)(100,238)(101,232)(102,231)(103,235)
(104,234)(105,233)(106,212)(107,211)(108,215)(109,214)(110,213)(111,222)
(112,221)(113,225)(114,224)(115,223)(116,217)(117,216)(118,220)(119,219)
(120,218);
s2 := Sym(240)!( 1, 6)( 2, 7)( 3, 8)( 4, 9)( 5, 10)( 16, 21)( 17, 22)
( 18, 23)( 19, 24)( 20, 25)( 31, 36)( 32, 37)( 33, 38)( 34, 39)( 35, 40)
( 46, 51)( 47, 52)( 48, 53)( 49, 54)( 50, 55)( 61, 66)( 62, 67)( 63, 68)
( 64, 69)( 65, 70)( 76, 81)( 77, 82)( 78, 83)( 79, 84)( 80, 85)( 91, 96)
( 92, 97)( 93, 98)( 94, 99)( 95,100)(106,111)(107,112)(108,113)(109,114)
(110,115)(121,156)(122,157)(123,158)(124,159)(125,160)(126,151)(127,152)
(128,153)(129,154)(130,155)(131,161)(132,162)(133,163)(134,164)(135,165)
(136,171)(137,172)(138,173)(139,174)(140,175)(141,166)(142,167)(143,168)
(144,169)(145,170)(146,176)(147,177)(148,178)(149,179)(150,180)(181,216)
(182,217)(183,218)(184,219)(185,220)(186,211)(187,212)(188,213)(189,214)
(190,215)(191,221)(192,222)(193,223)(194,224)(195,225)(196,231)(197,232)
(198,233)(199,234)(200,235)(201,226)(202,227)(203,228)(204,229)(205,230)
(206,236)(207,237)(208,238)(209,239)(210,240);
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*s2*s1*s0*s1*s2*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,
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*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 >;
References : None.
to this polytope