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# Polytope of Type {6,20}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,20}*960e
if this polytope has a name.
Group : SmallGroup(960,10952)
Rank : 3
Schlafli Type : {6,20}
Number of vertices, edges, etc : 24, 240, 80
Order of s0s1s2 : 60
Order of s0s1s2s1 : 4
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{6,20,2} of size 1920
Vertex Figure Of :
{2,6,20} of size 1920
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {6,20}*480c
4-fold quotients : {6,20}*240a, {6,20}*240b
5-fold quotients : {6,4}*192b
8-fold quotients : {6,10}*120
10-fold quotients : {6,4}*96
12-fold quotients : {2,20}*80
20-fold quotients : {6,4}*48a, {3,4}*48, {6,4}*48b, {6,4}*48c
24-fold quotients : {2,10}*40
40-fold quotients : {3,4}*24, {6,2}*24
48-fold quotients : {2,5}*20
60-fold quotients : {2,4}*16
80-fold quotients : {3,2}*12
120-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
2-fold covers : {6,40}*1920b, {6,40}*1920d, {6,20}*1920b, {12,20}*1920b, {12,20}*1920c
Permutation Representation (GAP) :
```s0 := (  2,  3)(  6,  7)( 10, 11)( 14, 15)( 18, 19)( 21, 41)( 22, 43)( 23, 42)
( 24, 44)( 25, 45)( 26, 47)( 27, 46)( 28, 48)( 29, 49)( 30, 51)( 31, 50)
( 32, 52)( 33, 53)( 34, 55)( 35, 54)( 36, 56)( 37, 57)( 38, 59)( 39, 58)
( 40, 60)( 62, 63)( 66, 67)( 70, 71)( 74, 75)( 78, 79)( 81,101)( 82,103)
( 83,102)( 84,104)( 85,105)( 86,107)( 87,106)( 88,108)( 89,109)( 90,111)
( 91,110)( 92,112)( 93,113)( 94,115)( 95,114)( 96,116)( 97,117)( 98,119)
( 99,118)(100,120)(122,123)(126,127)(130,131)(134,135)(138,139)(141,161)
(142,163)(143,162)(144,164)(145,165)(146,167)(147,166)(148,168)(149,169)
(150,171)(151,170)(152,172)(153,173)(154,175)(155,174)(156,176)(157,177)
(158,179)(159,178)(160,180)(182,183)(186,187)(190,191)(194,195)(198,199)
(201,221)(202,223)(203,222)(204,224)(205,225)(206,227)(207,226)(208,228)
(209,229)(210,231)(211,230)(212,232)(213,233)(214,235)(215,234)(216,236)
(217,237)(218,239)(219,238)(220,240);;
s1 := (  1, 21)(  2, 22)(  3, 24)(  4, 23)(  5, 37)(  6, 38)(  7, 40)(  8, 39)
(  9, 33)( 10, 34)( 11, 36)( 12, 35)( 13, 29)( 14, 30)( 15, 32)( 16, 31)
( 17, 25)( 18, 26)( 19, 28)( 20, 27)( 43, 44)( 45, 57)( 46, 58)( 47, 60)
( 48, 59)( 49, 53)( 50, 54)( 51, 56)( 52, 55)( 61, 81)( 62, 82)( 63, 84)
( 64, 83)( 65, 97)( 66, 98)( 67,100)( 68, 99)( 69, 93)( 70, 94)( 71, 96)
( 72, 95)( 73, 89)( 74, 90)( 75, 92)( 76, 91)( 77, 85)( 78, 86)( 79, 88)
( 80, 87)(103,104)(105,117)(106,118)(107,120)(108,119)(109,113)(110,114)
(111,116)(112,115)(121,201)(122,202)(123,204)(124,203)(125,217)(126,218)
(127,220)(128,219)(129,213)(130,214)(131,216)(132,215)(133,209)(134,210)
(135,212)(136,211)(137,205)(138,206)(139,208)(140,207)(141,181)(142,182)
(143,184)(144,183)(145,197)(146,198)(147,200)(148,199)(149,193)(150,194)
(151,196)(152,195)(153,189)(154,190)(155,192)(156,191)(157,185)(158,186)
(159,188)(160,187)(161,221)(162,222)(163,224)(164,223)(165,237)(166,238)
(167,240)(168,239)(169,233)(170,234)(171,236)(172,235)(173,229)(174,230)
(175,232)(176,231)(177,225)(178,226)(179,228)(180,227);;
s2 := (  1,128)(  2,127)(  3,126)(  4,125)(  5,124)(  6,123)(  7,122)(  8,121)
(  9,140)( 10,139)( 11,138)( 12,137)( 13,136)( 14,135)( 15,134)( 16,133)
( 17,132)( 18,131)( 19,130)( 20,129)( 21,148)( 22,147)( 23,146)( 24,145)
( 25,144)( 26,143)( 27,142)( 28,141)( 29,160)( 30,159)( 31,158)( 32,157)
( 33,156)( 34,155)( 35,154)( 36,153)( 37,152)( 38,151)( 39,150)( 40,149)
( 41,168)( 42,167)( 43,166)( 44,165)( 45,164)( 46,163)( 47,162)( 48,161)
( 49,180)( 50,179)( 51,178)( 52,177)( 53,176)( 54,175)( 55,174)( 56,173)
( 57,172)( 58,171)( 59,170)( 60,169)( 61,188)( 62,187)( 63,186)( 64,185)
( 65,184)( 66,183)( 67,182)( 68,181)( 69,200)( 70,199)( 71,198)( 72,197)
( 73,196)( 74,195)( 75,194)( 76,193)( 77,192)( 78,191)( 79,190)( 80,189)
( 81,208)( 82,207)( 83,206)( 84,205)( 85,204)( 86,203)( 87,202)( 88,201)
( 89,220)( 90,219)( 91,218)( 92,217)( 93,216)( 94,215)( 95,214)( 96,213)
( 97,212)( 98,211)( 99,210)(100,209)(101,228)(102,227)(103,226)(104,225)
(105,224)(106,223)(107,222)(108,221)(109,240)(110,239)(111,238)(112,237)
(113,236)(114,235)(115,234)(116,233)(117,232)(118,231)(119,230)(120,229);;
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*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s2*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s2*s0*s1*s0*s1,
s0*s1*s2*s1*s0*s1*s2*s1*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*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 ];;
poly := F / rels;;

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

```
References : None.
to this polytope