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# Polytope of Type {4,6,30}

Atlas Canonical Name : {4,6,30}*1440a
if this polytope has a name.
Group : SmallGroup(1440,5360)
Rank : 4
Schlafli Type : {4,6,30}
Number of vertices, edges, etc : 4, 12, 90, 30
Order of s0s1s2s3 : 60
Order of s0s1s2s3s2s1 : 2
Special Properties :
Universal
Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
None in this Atlas
Vertex Figure Of :
None in this Atlas
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {2,6,30}*720a
3-fold quotients : {4,6,10}*480a
5-fold quotients : {4,6,6}*288b
6-fold quotients : {2,6,10}*240
9-fold quotients : {4,2,10}*160
10-fold quotients : {2,6,6}*144c
15-fold quotients : {4,6,2}*96a
18-fold quotients : {4,2,5}*80, {2,2,10}*80
20-fold quotients : {2,3,6}*72
30-fold quotients : {2,6,2}*48
36-fold quotients : {2,2,5}*40
45-fold quotients : {4,2,2}*32
60-fold quotients : {2,3,2}*24
90-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
```s0 := (  1, 91)(  2, 92)(  3, 93)(  4, 94)(  5, 95)(  6, 96)(  7, 97)(  8, 98)
(  9, 99)( 10,100)( 11,101)( 12,102)( 13,103)( 14,104)( 15,105)( 16,106)
( 17,107)( 18,108)( 19,109)( 20,110)( 21,111)( 22,112)( 23,113)( 24,114)
( 25,115)( 26,116)( 27,117)( 28,118)( 29,119)( 30,120)( 31,121)( 32,122)
( 33,123)( 34,124)( 35,125)( 36,126)( 37,127)( 38,128)( 39,129)( 40,130)
( 41,131)( 42,132)( 43,133)( 44,134)( 45,135)( 46,136)( 47,137)( 48,138)
( 49,139)( 50,140)( 51,141)( 52,142)( 53,143)( 54,144)( 55,145)( 56,146)
( 57,147)( 58,148)( 59,149)( 60,150)( 61,151)( 62,152)( 63,153)( 64,154)
( 65,155)( 66,156)( 67,157)( 68,158)( 69,159)( 70,160)( 71,161)( 72,162)
( 73,163)( 74,164)( 75,165)( 76,166)( 77,167)( 78,168)( 79,169)( 80,170)
( 81,171)( 82,172)( 83,173)( 84,174)( 85,175)( 86,176)( 87,177)( 88,178)
( 89,179)( 90,180);;
s1 := (  6, 11)(  7, 12)(  8, 13)(  9, 14)( 10, 15)( 16, 31)( 17, 32)( 18, 33)
( 19, 34)( 20, 35)( 21, 41)( 22, 42)( 23, 43)( 24, 44)( 25, 45)( 26, 36)
( 27, 37)( 28, 38)( 29, 39)( 30, 40)( 51, 56)( 52, 57)( 53, 58)( 54, 59)
( 55, 60)( 61, 76)( 62, 77)( 63, 78)( 64, 79)( 65, 80)( 66, 86)( 67, 87)
( 68, 88)( 69, 89)( 70, 90)( 71, 81)( 72, 82)( 73, 83)( 74, 84)( 75, 85)
( 91,136)( 92,137)( 93,138)( 94,139)( 95,140)( 96,146)( 97,147)( 98,148)
( 99,149)(100,150)(101,141)(102,142)(103,143)(104,144)(105,145)(106,166)
(107,167)(108,168)(109,169)(110,170)(111,176)(112,177)(113,178)(114,179)
(115,180)(116,171)(117,172)(118,173)(119,174)(120,175)(121,151)(122,152)
(123,153)(124,154)(125,155)(126,161)(127,162)(128,163)(129,164)(130,165)
(131,156)(132,157)(133,158)(134,159)(135,160);;
s2 := (  1, 21)(  2, 25)(  3, 24)(  4, 23)(  5, 22)(  6, 16)(  7, 20)(  8, 19)
(  9, 18)( 10, 17)( 11, 26)( 12, 30)( 13, 29)( 14, 28)( 15, 27)( 31, 36)
( 32, 40)( 33, 39)( 34, 38)( 35, 37)( 42, 45)( 43, 44)( 46, 66)( 47, 70)
( 48, 69)( 49, 68)( 50, 67)( 51, 61)( 52, 65)( 53, 64)( 54, 63)( 55, 62)
( 56, 71)( 57, 75)( 58, 74)( 59, 73)( 60, 72)( 76, 81)( 77, 85)( 78, 84)
( 79, 83)( 80, 82)( 87, 90)( 88, 89)( 91,111)( 92,115)( 93,114)( 94,113)
( 95,112)( 96,106)( 97,110)( 98,109)( 99,108)(100,107)(101,116)(102,120)
(103,119)(104,118)(105,117)(121,126)(122,130)(123,129)(124,128)(125,127)
(132,135)(133,134)(136,156)(137,160)(138,159)(139,158)(140,157)(141,151)
(142,155)(143,154)(144,153)(145,152)(146,161)(147,165)(148,164)(149,163)
(150,162)(166,171)(167,175)(168,174)(169,173)(170,172)(177,180)(178,179);;
s3 := (  1,  2)(  3,  5)(  6,  7)(  8, 10)( 11, 12)( 13, 15)( 16, 32)( 17, 31)
( 18, 35)( 19, 34)( 20, 33)( 21, 37)( 22, 36)( 23, 40)( 24, 39)( 25, 38)
( 26, 42)( 27, 41)( 28, 45)( 29, 44)( 30, 43)( 46, 47)( 48, 50)( 51, 52)
( 53, 55)( 56, 57)( 58, 60)( 61, 77)( 62, 76)( 63, 80)( 64, 79)( 65, 78)
( 66, 82)( 67, 81)( 68, 85)( 69, 84)( 70, 83)( 71, 87)( 72, 86)( 73, 90)
( 74, 89)( 75, 88)( 91, 92)( 93, 95)( 96, 97)( 98,100)(101,102)(103,105)
(106,122)(107,121)(108,125)(109,124)(110,123)(111,127)(112,126)(113,130)
(114,129)(115,128)(116,132)(117,131)(118,135)(119,134)(120,133)(136,137)
(138,140)(141,142)(143,145)(146,147)(148,150)(151,167)(152,166)(153,170)
(154,169)(155,168)(156,172)(157,171)(158,175)(159,174)(160,173)(161,177)
(162,176)(163,180)(164,179)(165,178);;
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, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s3*s1*s2*s3*s2*s1*s2*s3*s1*s2*s3*s2*s1*s2,
s3*s1*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s1*s2*s3*s2 ];;
poly := F / rels;;

```
Permutation Representation (Magma) :
```s0 := Sym(180)!(  1, 91)(  2, 92)(  3, 93)(  4, 94)(  5, 95)(  6, 96)(  7, 97)
(  8, 98)(  9, 99)( 10,100)( 11,101)( 12,102)( 13,103)( 14,104)( 15,105)
( 16,106)( 17,107)( 18,108)( 19,109)( 20,110)( 21,111)( 22,112)( 23,113)
( 24,114)( 25,115)( 26,116)( 27,117)( 28,118)( 29,119)( 30,120)( 31,121)
( 32,122)( 33,123)( 34,124)( 35,125)( 36,126)( 37,127)( 38,128)( 39,129)
( 40,130)( 41,131)( 42,132)( 43,133)( 44,134)( 45,135)( 46,136)( 47,137)
( 48,138)( 49,139)( 50,140)( 51,141)( 52,142)( 53,143)( 54,144)( 55,145)
( 56,146)( 57,147)( 58,148)( 59,149)( 60,150)( 61,151)( 62,152)( 63,153)
( 64,154)( 65,155)( 66,156)( 67,157)( 68,158)( 69,159)( 70,160)( 71,161)
( 72,162)( 73,163)( 74,164)( 75,165)( 76,166)( 77,167)( 78,168)( 79,169)
( 80,170)( 81,171)( 82,172)( 83,173)( 84,174)( 85,175)( 86,176)( 87,177)
( 88,178)( 89,179)( 90,180);
s1 := Sym(180)!(  6, 11)(  7, 12)(  8, 13)(  9, 14)( 10, 15)( 16, 31)( 17, 32)
( 18, 33)( 19, 34)( 20, 35)( 21, 41)( 22, 42)( 23, 43)( 24, 44)( 25, 45)
( 26, 36)( 27, 37)( 28, 38)( 29, 39)( 30, 40)( 51, 56)( 52, 57)( 53, 58)
( 54, 59)( 55, 60)( 61, 76)( 62, 77)( 63, 78)( 64, 79)( 65, 80)( 66, 86)
( 67, 87)( 68, 88)( 69, 89)( 70, 90)( 71, 81)( 72, 82)( 73, 83)( 74, 84)
( 75, 85)( 91,136)( 92,137)( 93,138)( 94,139)( 95,140)( 96,146)( 97,147)
( 98,148)( 99,149)(100,150)(101,141)(102,142)(103,143)(104,144)(105,145)
(106,166)(107,167)(108,168)(109,169)(110,170)(111,176)(112,177)(113,178)
(114,179)(115,180)(116,171)(117,172)(118,173)(119,174)(120,175)(121,151)
(122,152)(123,153)(124,154)(125,155)(126,161)(127,162)(128,163)(129,164)
(130,165)(131,156)(132,157)(133,158)(134,159)(135,160);
s2 := Sym(180)!(  1, 21)(  2, 25)(  3, 24)(  4, 23)(  5, 22)(  6, 16)(  7, 20)
(  8, 19)(  9, 18)( 10, 17)( 11, 26)( 12, 30)( 13, 29)( 14, 28)( 15, 27)
( 31, 36)( 32, 40)( 33, 39)( 34, 38)( 35, 37)( 42, 45)( 43, 44)( 46, 66)
( 47, 70)( 48, 69)( 49, 68)( 50, 67)( 51, 61)( 52, 65)( 53, 64)( 54, 63)
( 55, 62)( 56, 71)( 57, 75)( 58, 74)( 59, 73)( 60, 72)( 76, 81)( 77, 85)
( 78, 84)( 79, 83)( 80, 82)( 87, 90)( 88, 89)( 91,111)( 92,115)( 93,114)
( 94,113)( 95,112)( 96,106)( 97,110)( 98,109)( 99,108)(100,107)(101,116)
(102,120)(103,119)(104,118)(105,117)(121,126)(122,130)(123,129)(124,128)
(125,127)(132,135)(133,134)(136,156)(137,160)(138,159)(139,158)(140,157)
(141,151)(142,155)(143,154)(144,153)(145,152)(146,161)(147,165)(148,164)
(149,163)(150,162)(166,171)(167,175)(168,174)(169,173)(170,172)(177,180)
(178,179);
s3 := Sym(180)!(  1,  2)(  3,  5)(  6,  7)(  8, 10)( 11, 12)( 13, 15)( 16, 32)
( 17, 31)( 18, 35)( 19, 34)( 20, 33)( 21, 37)( 22, 36)( 23, 40)( 24, 39)
( 25, 38)( 26, 42)( 27, 41)( 28, 45)( 29, 44)( 30, 43)( 46, 47)( 48, 50)
( 51, 52)( 53, 55)( 56, 57)( 58, 60)( 61, 77)( 62, 76)( 63, 80)( 64, 79)
( 65, 78)( 66, 82)( 67, 81)( 68, 85)( 69, 84)( 70, 83)( 71, 87)( 72, 86)
( 73, 90)( 74, 89)( 75, 88)( 91, 92)( 93, 95)( 96, 97)( 98,100)(101,102)
(103,105)(106,122)(107,121)(108,125)(109,124)(110,123)(111,127)(112,126)
(113,130)(114,129)(115,128)(116,132)(117,131)(118,135)(119,134)(120,133)
(136,137)(138,140)(141,142)(143,145)(146,147)(148,150)(151,167)(152,166)
(153,170)(154,169)(155,168)(156,172)(157,171)(158,175)(159,174)(160,173)
(161,177)(162,176)(163,180)(164,179)(165,178);
poly := sub<Sym(180)|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,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s3*s1*s2*s3*s2*s1*s2*s3*s1*s2*s3*s2*s1*s2,
s3*s1*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s1*s2*s3*s2 >;

```
References : None.
to this polytope