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# Polytope of Type {4,30,4}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,30,4}*960b
if this polytope has a name.
Group : SmallGroup(960,11092)
Rank : 4
Schlafli Type : {4,30,4}
Number of vertices, edges, etc : 4, 60, 60, 4
Order of s0s1s2s3 : 60
Order of s0s1s2s3s2s1 : 2
Special Properties :
Universal
Non-Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{4,30,4,2} of size 1920
Vertex Figure Of :
{2,4,30,4} of size 1920
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {2,30,4}*480b
4-fold quotients : {2,15,4}*240
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 : {4,60,4}*1920b, {4,60,4}*1920c, {8,30,4}*1920b, {4,30,4}*1920a
Permutation Representation (GAP) :
```s0 := (121,181)(122,182)(123,183)(124,184)(125,185)(126,186)(127,187)(128,188)
(129,189)(130,190)(131,191)(132,192)(133,193)(134,194)(135,195)(136,196)
(137,197)(138,198)(139,199)(140,200)(141,201)(142,202)(143,203)(144,204)
(145,205)(146,206)(147,207)(148,208)(149,209)(150,210)(151,211)(152,212)
(153,213)(154,214)(155,215)(156,216)(157,217)(158,218)(159,219)(160,220)
(161,221)(162,222)(163,223)(164,224)(165,225)(166,226)(167,227)(168,228)
(169,229)(170,230)(171,231)(172,232)(173,233)(174,234)(175,235)(176,236)
(177,237)(178,238)(179,239)(180,240);;
s1 := (  1,121)(  2,123)(  3,122)(  4,124)(  5,137)(  6,139)(  7,138)(  8,140)
(  9,133)( 10,135)( 11,134)( 12,136)( 13,129)( 14,131)( 15,130)( 16,132)
( 17,125)( 18,127)( 19,126)( 20,128)( 21,161)( 22,163)( 23,162)( 24,164)
( 25,177)( 26,179)( 27,178)( 28,180)( 29,173)( 30,175)( 31,174)( 32,176)
( 33,169)( 34,171)( 35,170)( 36,172)( 37,165)( 38,167)( 39,166)( 40,168)
( 41,141)( 42,143)( 43,142)( 44,144)( 45,157)( 46,159)( 47,158)( 48,160)
( 49,153)( 50,155)( 51,154)( 52,156)( 53,149)( 54,151)( 55,150)( 56,152)
( 57,145)( 58,147)( 59,146)( 60,148)( 61,181)( 62,183)( 63,182)( 64,184)
( 65,197)( 66,199)( 67,198)( 68,200)( 69,193)( 70,195)( 71,194)( 72,196)
( 73,189)( 74,191)( 75,190)( 76,192)( 77,185)( 78,187)( 79,186)( 80,188)
( 81,221)( 82,223)( 83,222)( 84,224)( 85,237)( 86,239)( 87,238)( 88,240)
( 89,233)( 90,235)( 91,234)( 92,236)( 93,229)( 94,231)( 95,230)( 96,232)
( 97,225)( 98,227)( 99,226)(100,228)(101,201)(102,203)(103,202)(104,204)
(105,217)(106,219)(107,218)(108,220)(109,213)(110,215)(111,214)(112,216)
(113,209)(114,211)(115,210)(116,212)(117,205)(118,207)(119,206)(120,208);;
s2 := (  1, 25)(  2, 26)(  3, 28)(  4, 27)(  5, 21)(  6, 22)(  7, 24)(  8, 23)
(  9, 37)( 10, 38)( 11, 40)( 12, 39)( 13, 33)( 14, 34)( 15, 36)( 16, 35)
( 17, 29)( 18, 30)( 19, 32)( 20, 31)( 41, 45)( 42, 46)( 43, 48)( 44, 47)
( 49, 57)( 50, 58)( 51, 60)( 52, 59)( 55, 56)( 61, 85)( 62, 86)( 63, 88)
( 64, 87)( 65, 81)( 66, 82)( 67, 84)( 68, 83)( 69, 97)( 70, 98)( 71,100)
( 72, 99)( 73, 93)( 74, 94)( 75, 96)( 76, 95)( 77, 89)( 78, 90)( 79, 92)
( 80, 91)(101,105)(102,106)(103,108)(104,107)(109,117)(110,118)(111,120)
(112,119)(115,116)(121,145)(122,146)(123,148)(124,147)(125,141)(126,142)
(127,144)(128,143)(129,157)(130,158)(131,160)(132,159)(133,153)(134,154)
(135,156)(136,155)(137,149)(138,150)(139,152)(140,151)(161,165)(162,166)
(163,168)(164,167)(169,177)(170,178)(171,180)(172,179)(175,176)(181,205)
(182,206)(183,208)(184,207)(185,201)(186,202)(187,204)(188,203)(189,217)
(190,218)(191,220)(192,219)(193,213)(194,214)(195,216)(196,215)(197,209)
(198,210)(199,212)(200,211)(221,225)(222,226)(223,228)(224,227)(229,237)
(230,238)(231,240)(232,239)(235,236);;
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*s0*s1*s0*s1*s0*s1,
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*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*s1*s2*s1*s2*s1*s2*s1*s2 ];;
poly := F / rels;;

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

```
References : None.
to this polytope