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# Polytope of Type {40,12}

Atlas Canonical Name : {40,12}*960b
if this polytope has a name.
Group : SmallGroup(960,2444)
Rank : 3
Schlafli Type : {40,12}
Number of vertices, edges, etc : 40, 240, 12
Order of s0s1s2 : 120
Order of s0s1s2s1 : 4
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 : {20,12}*480
3-fold quotients : {40,4}*320b
4-fold quotients : {10,12}*240, {20,6}*240a
5-fold quotients : {8,12}*192b
6-fold quotients : {20,4}*160
8-fold quotients : {10,6}*120
10-fold quotients : {4,12}*96a
12-fold quotients : {20,2}*80, {10,4}*80
15-fold quotients : {8,4}*64b
20-fold quotients : {2,12}*48, {4,6}*48a
24-fold quotients : {10,2}*40
30-fold quotients : {4,4}*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}*1920b, {40,24}*1920d
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)( 31, 46)( 32, 50)( 33, 49)( 34, 48)
( 35, 47)( 36, 51)( 37, 55)( 38, 54)( 39, 53)( 40, 52)( 41, 56)( 42, 60)
( 43, 59)( 44, 58)( 45, 57)( 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)( 92, 95)( 93, 94)( 97,100)( 98, 99)(102,105)(103,104)
(107,110)(108,109)(112,115)(113,114)(117,120)(118,119)(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)(151,226)(152,230)(153,229)(154,228)
(155,227)(156,231)(157,235)(158,234)(159,233)(160,232)(161,236)(162,240)
(163,239)(164,238)(165,237)(166,211)(167,215)(168,214)(169,213)(170,212)
(171,216)(172,220)(173,219)(174,218)(175,217)(176,221)(177,225)(178,224)
(179,223)(180,222);;
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, 81)( 62, 82)( 63, 83)( 64, 84)
( 65, 85)( 66, 76)( 67, 77)( 68, 78)( 69, 79)( 70, 80)( 71, 86)( 72, 87)
( 73, 88)( 74, 89)( 75, 90)( 91,111)( 92,112)( 93,113)( 94,114)( 95,115)
( 96,106)( 97,107)( 98,108)( 99,109)(100,110)(101,116)(102,117)(103,118)
(104,119)(105,120)(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)
(181,231)(182,232)(183,233)(184,234)(185,235)(186,226)(187,227)(188,228)
(189,229)(190,230)(191,236)(192,237)(193,238)(194,239)(195,240)(196,216)
(197,217)(198,218)(199,219)(200,220)(201,211)(202,212)(203,213)(204,214)
(205,215)(206,221)(207,222)(208,223)(209,224)(210,225);;
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*s2*s1*s0*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,
s0*s1*s0*s1*s0*s1*s0*s2*s1*s0*s1*s0*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*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)( 31, 46)( 32, 50)( 33, 49)
( 34, 48)( 35, 47)( 36, 51)( 37, 55)( 38, 54)( 39, 53)( 40, 52)( 41, 56)
( 42, 60)( 43, 59)( 44, 58)( 45, 57)( 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)( 92, 95)( 93, 94)( 97,100)( 98, 99)(102,105)
(103,104)(107,110)(108,109)(112,115)(113,114)(117,120)(118,119)(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)(151,226)(152,230)(153,229)
(154,228)(155,227)(156,231)(157,235)(158,234)(159,233)(160,232)(161,236)
(162,240)(163,239)(164,238)(165,237)(166,211)(167,215)(168,214)(169,213)
(170,212)(171,216)(172,220)(173,219)(174,218)(175,217)(176,221)(177,225)
(178,224)(179,223)(180,222);
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, 81)( 62, 82)( 63, 83)
( 64, 84)( 65, 85)( 66, 76)( 67, 77)( 68, 78)( 69, 79)( 70, 80)( 71, 86)
( 72, 87)( 73, 88)( 74, 89)( 75, 90)( 91,111)( 92,112)( 93,113)( 94,114)
( 95,115)( 96,106)( 97,107)( 98,108)( 99,109)(100,110)(101,116)(102,117)
(103,118)(104,119)(105,120)(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)(181,231)(182,232)(183,233)(184,234)(185,235)(186,226)(187,227)
(188,228)(189,229)(190,230)(191,236)(192,237)(193,238)(194,239)(195,240)
(196,216)(197,217)(198,218)(199,219)(200,220)(201,211)(202,212)(203,213)
(204,214)(205,215)(206,221)(207,222)(208,223)(209,224)(210,225);
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, s2*s0*s1*s2*s1*s0*s1*s2*s0*s1*s2*s1*s0*s1,
s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s2*s1*s0*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,
s0*s1*s0*s1*s0*s1*s0*s2*s1*s0*s1*s0*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*s0*s1*s0*s1 >;

```
References : None.
to this polytope