# Polytope of Type {20,6,4}

Atlas Canonical Name : {20,6,4}*960b
if this polytope has a name.
Group : SmallGroup(960,10952)
Rank : 4
Schlafli Type : {20,6,4}
Number of vertices, edges, etc : 20, 60, 12, 4
Order of s0s1s2s3 : 60
Order of s0s1s2s3s2s1 : 2
Special Properties :
Universal
Non-Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{20,6,4,2} of size 1920
Vertex Figure Of :
{2,20,6,4} of size 1920
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {10,6,4}*480b
5-fold quotients : {4,6,4}*192b
10-fold quotients : {2,6,4}*96c
20-fold quotients : {2,3,4}*48
Covers (Minimal Covers in Boldface) :
2-fold covers : {40,6,4}*1920b, {20,12,4}*1920b, {20,12,4}*1920c, {20,6,4}*1920a
Permutation Representation (GAP) :
```s0 := (  5, 17)(  6, 18)(  7, 19)(  8, 20)(  9, 13)( 10, 14)( 11, 15)( 12, 16)
( 25, 37)( 26, 38)( 27, 39)( 28, 40)( 29, 33)( 30, 34)( 31, 35)( 32, 36)
( 45, 57)( 46, 58)( 47, 59)( 48, 60)( 49, 53)( 50, 54)( 51, 55)( 52, 56)
( 65, 77)( 66, 78)( 67, 79)( 68, 80)( 69, 73)( 70, 74)( 71, 75)( 72, 76)
( 85, 97)( 86, 98)( 87, 99)( 88,100)( 89, 93)( 90, 94)( 91, 95)( 92, 96)
(105,117)(106,118)(107,119)(108,120)(109,113)(110,114)(111,115)(112,116)
(121,181)(122,182)(123,183)(124,184)(125,197)(126,198)(127,199)(128,200)
(129,193)(130,194)(131,195)(132,196)(133,189)(134,190)(135,191)(136,192)
(137,185)(138,186)(139,187)(140,188)(141,201)(142,202)(143,203)(144,204)
(145,217)(146,218)(147,219)(148,220)(149,213)(150,214)(151,215)(152,216)
(153,209)(154,210)(155,211)(156,212)(157,205)(158,206)(159,207)(160,208)
(161,221)(162,222)(163,223)(164,224)(165,237)(166,238)(167,239)(168,240)
(169,233)(170,234)(171,235)(172,236)(173,229)(174,230)(175,231)(176,232)
(177,225)(178,226)(179,227)(180,228);;
s1 := (  1,125)(  2,127)(  3,126)(  4,128)(  5,121)(  6,123)(  7,122)(  8,124)
(  9,137)( 10,139)( 11,138)( 12,140)( 13,133)( 14,135)( 15,134)( 16,136)
( 17,129)( 18,131)( 19,130)( 20,132)( 21,165)( 22,167)( 23,166)( 24,168)
( 25,161)( 26,163)( 27,162)( 28,164)( 29,177)( 30,179)( 31,178)( 32,180)
( 33,173)( 34,175)( 35,174)( 36,176)( 37,169)( 38,171)( 39,170)( 40,172)
( 41,145)( 42,147)( 43,146)( 44,148)( 45,141)( 46,143)( 47,142)( 48,144)
( 49,157)( 50,159)( 51,158)( 52,160)( 53,153)( 54,155)( 55,154)( 56,156)
( 57,149)( 58,151)( 59,150)( 60,152)( 61,185)( 62,187)( 63,186)( 64,188)
( 65,181)( 66,183)( 67,182)( 68,184)( 69,197)( 70,199)( 71,198)( 72,200)
( 73,193)( 74,195)( 75,194)( 76,196)( 77,189)( 78,191)( 79,190)( 80,192)
( 81,225)( 82,227)( 83,226)( 84,228)( 85,221)( 86,223)( 87,222)( 88,224)
( 89,237)( 90,239)( 91,238)( 92,240)( 93,233)( 94,235)( 95,234)( 96,236)
( 97,229)( 98,231)( 99,230)(100,232)(101,205)(102,207)(103,206)(104,208)
(105,201)(106,203)(107,202)(108,204)(109,217)(110,219)(111,218)(112,220)
(113,213)(114,215)(115,214)(116,216)(117,209)(118,211)(119,210)(120,212);;
s2 := (  1, 21)(  2, 22)(  3, 24)(  4, 23)(  5, 25)(  6, 26)(  7, 28)(  8, 27)
(  9, 29)( 10, 30)( 11, 32)( 12, 31)( 13, 33)( 14, 34)( 15, 36)( 16, 35)
( 17, 37)( 18, 38)( 19, 40)( 20, 39)( 43, 44)( 47, 48)( 51, 52)( 55, 56)
( 59, 60)( 61, 81)( 62, 82)( 63, 84)( 64, 83)( 65, 85)( 66, 86)( 67, 88)
( 68, 87)( 69, 89)( 70, 90)( 71, 92)( 72, 91)( 73, 93)( 74, 94)( 75, 96)
( 76, 95)( 77, 97)( 78, 98)( 79,100)( 80, 99)(103,104)(107,108)(111,112)
(115,116)(119,120)(121,141)(122,142)(123,144)(124,143)(125,145)(126,146)
(127,148)(128,147)(129,149)(130,150)(131,152)(132,151)(133,153)(134,154)
(135,156)(136,155)(137,157)(138,158)(139,160)(140,159)(163,164)(167,168)
(171,172)(175,176)(179,180)(181,201)(182,202)(183,204)(184,203)(185,205)
(186,206)(187,208)(188,207)(189,209)(190,210)(191,212)(192,211)(193,213)
(194,214)(195,216)(196,215)(197,217)(198,218)(199,220)(200,219)(223,224)
(227,228)(231,232)(235,236)(239,240);;
s3 := (  1,  4)(  2,  3)(  5,  8)(  6,  7)(  9, 12)( 10, 11)( 13, 16)( 14, 15)
( 17, 20)( 18, 19)( 21, 24)( 22, 23)( 25, 28)( 26, 27)( 29, 32)( 30, 31)
( 33, 36)( 34, 35)( 37, 40)( 38, 39)( 41, 44)( 42, 43)( 45, 48)( 46, 47)
( 49, 52)( 50, 51)( 53, 56)( 54, 55)( 57, 60)( 58, 59)( 61, 64)( 62, 63)
( 65, 68)( 66, 67)( 69, 72)( 70, 71)( 73, 76)( 74, 75)( 77, 80)( 78, 79)
( 81, 84)( 82, 83)( 85, 88)( 86, 87)( 89, 92)( 90, 91)( 93, 96)( 94, 95)
( 97,100)( 98, 99)(101,104)(102,103)(105,108)(106,107)(109,112)(110,111)
(113,116)(114,115)(117,120)(118,119)(121,124)(122,123)(125,128)(126,127)
(129,132)(130,131)(133,136)(134,135)(137,140)(138,139)(141,144)(142,143)
(145,148)(146,147)(149,152)(150,151)(153,156)(154,155)(157,160)(158,159)
(161,164)(162,163)(165,168)(166,167)(169,172)(170,171)(173,176)(174,175)
(177,180)(178,179)(181,184)(182,183)(185,188)(186,187)(189,192)(190,191)
(193,196)(194,195)(197,200)(198,199)(201,204)(202,203)(205,208)(206,207)
(209,212)(210,211)(213,216)(214,215)(217,220)(218,219)(221,224)(222,223)
(225,228)(226,227)(229,232)(230,231)(233,236)(234,235)(237,240)(238,239);;
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, s0*s1*s2*s1*s0*s1*s2*s1,
s2*s3*s2*s3*s2*s3*s2*s3, s3*s2*s1*s3*s2*s3*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 ];;
poly := F / rels;;

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

```
References : None.
to this polytope