# Polytope of Type {40,6}

Atlas Canonical Name : {40,6}*960d
if this polytope has a name.
Group : SmallGroup(960,10967)
Rank : 3
Schlafli Type : {40,6}
Number of vertices, edges, etc : 80, 240, 12
Order of s0s1s2 : 30
Order of s0s1s2s1 : 8
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Related Polytopes :
Facet
Vertex Figure
Dual
Petrial
Facet Of :
{40,6,2} of size 1920
Vertex Figure Of :
{2,40,6} of size 1920
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {20,6}*480c
4-fold quotients : {20,6}*240b
5-fold quotients : {8,6}*192c
8-fold quotients : {10,6}*120
10-fold quotients : {4,6}*96
20-fold quotients : {4,3}*48, {4,6}*48b, {4,6}*48c
24-fold quotients : {10,2}*40
40-fold quotients : {4,3}*24, {2,6}*24
48-fold quotients : {5,2}*20
80-fold quotients : {2,3}*12
120-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
2-fold covers : {40,12}*1920e, {40,6}*1920b, {40,12}*1920g
Permutation Representation (GAP) :
```s0 := (  1,125)(  2,126)(  3,128)(  4,127)(  5,122)(  6,121)(  7,123)(  8,124)
(  9,157)( 10,158)( 11,160)( 12,159)( 13,154)( 14,153)( 15,155)( 16,156)
( 17,149)( 18,150)( 19,152)( 20,151)( 21,146)( 22,145)( 23,147)( 24,148)
( 25,141)( 26,142)( 27,144)( 28,143)( 29,138)( 30,137)( 31,139)( 32,140)
( 33,133)( 34,134)( 35,136)( 36,135)( 37,130)( 38,129)( 39,131)( 40,132)
( 41,165)( 42,166)( 43,168)( 44,167)( 45,162)( 46,161)( 47,163)( 48,164)
( 49,197)( 50,198)( 51,200)( 52,199)( 53,194)( 54,193)( 55,195)( 56,196)
( 57,189)( 58,190)( 59,192)( 60,191)( 61,186)( 62,185)( 63,187)( 64,188)
( 65,181)( 66,182)( 67,184)( 68,183)( 69,178)( 70,177)( 71,179)( 72,180)
( 73,173)( 74,174)( 75,176)( 76,175)( 77,170)( 78,169)( 79,171)( 80,172)
( 81,205)( 82,206)( 83,208)( 84,207)( 85,202)( 86,201)( 87,203)( 88,204)
( 89,237)( 90,238)( 91,240)( 92,239)( 93,234)( 94,233)( 95,235)( 96,236)
( 97,229)( 98,230)( 99,232)(100,231)(101,226)(102,225)(103,227)(104,228)
(105,221)(106,222)(107,224)(108,223)(109,218)(110,217)(111,219)(112,220)
(113,213)(114,214)(115,216)(116,215)(117,210)(118,209)(119,211)(120,212);;
s1 := (  1,  9)(  2, 10)(  3, 14)(  4, 13)(  5, 12)(  6, 11)(  7, 16)(  8, 15)
( 17, 33)( 18, 34)( 19, 38)( 20, 37)( 21, 36)( 22, 35)( 23, 40)( 24, 39)
( 27, 30)( 28, 29)( 31, 32)( 41, 89)( 42, 90)( 43, 94)( 44, 93)( 45, 92)
( 46, 91)( 47, 96)( 48, 95)( 49, 81)( 50, 82)( 51, 86)( 52, 85)( 53, 84)
( 54, 83)( 55, 88)( 56, 87)( 57,113)( 58,114)( 59,118)( 60,117)( 61,116)
( 62,115)( 63,120)( 64,119)( 65,105)( 66,106)( 67,110)( 68,109)( 69,108)
( 70,107)( 71,112)( 72,111)( 73, 97)( 74, 98)( 75,102)( 76,101)( 77,100)
( 78, 99)( 79,104)( 80,103)(121,129)(122,130)(123,134)(124,133)(125,132)
(126,131)(127,136)(128,135)(137,153)(138,154)(139,158)(140,157)(141,156)
(142,155)(143,160)(144,159)(147,150)(148,149)(151,152)(161,209)(162,210)
(163,214)(164,213)(165,212)(166,211)(167,216)(168,215)(169,201)(170,202)
(171,206)(172,205)(173,204)(174,203)(175,208)(176,207)(177,233)(178,234)
(179,238)(180,237)(181,236)(182,235)(183,240)(184,239)(185,225)(186,226)
(187,230)(188,229)(189,228)(190,227)(191,232)(192,231)(193,217)(194,218)
(195,222)(196,221)(197,220)(198,219)(199,224)(200,223);;
s2 := (  1, 81)(  2, 82)(  3, 87)(  4, 88)(  5, 86)(  6, 85)(  7, 83)(  8, 84)
(  9, 89)( 10, 90)( 11, 95)( 12, 96)( 13, 94)( 14, 93)( 15, 91)( 16, 92)
( 17, 97)( 18, 98)( 19,103)( 20,104)( 21,102)( 22,101)( 23, 99)( 24,100)
( 25,105)( 26,106)( 27,111)( 28,112)( 29,110)( 30,109)( 31,107)( 32,108)
( 33,113)( 34,114)( 35,119)( 36,120)( 37,118)( 38,117)( 39,115)( 40,116)
( 43, 47)( 44, 48)( 45, 46)( 51, 55)( 52, 56)( 53, 54)( 59, 63)( 60, 64)
( 61, 62)( 67, 71)( 68, 72)( 69, 70)( 75, 79)( 76, 80)( 77, 78)(121,202)
(122,201)(123,208)(124,207)(125,205)(126,206)(127,204)(128,203)(129,210)
(130,209)(131,216)(132,215)(133,213)(134,214)(135,212)(136,211)(137,218)
(138,217)(139,224)(140,223)(141,221)(142,222)(143,220)(144,219)(145,226)
(146,225)(147,232)(148,231)(149,229)(150,230)(151,228)(152,227)(153,234)
(154,233)(155,240)(156,239)(157,237)(158,238)(159,236)(160,235)(161,162)
(163,168)(164,167)(169,170)(171,176)(172,175)(177,178)(179,184)(180,183)
(185,186)(187,192)(188,191)(193,194)(195,200)(196,199);;
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,
s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1,
s2*s0*s1*s2*s0*s1*s2*s1*s0*s1*s2*s0*s1*s2*s0*s1*s2*s1*s0*s1,
s1*s2*s0*s1*s0*s1*s0*s2*s1*s0*s2*s1*s0*s1*s0*s2*s1*s0*s1*s0*s1*s0*s1*s0 ];;
poly := F / rels;;

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

```
References : None.
to this polytope