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# Polytope of Type {2,12,30}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,12,30}*1440c
if this polytope has a name.
Group : SmallGroup(1440,5712)
Rank : 4
Schlafli Type : {2,12,30}
Number of vertices, edges, etc : 2, 12, 180, 30
Order of s0s1s2s3 : 60
Order of s0s1s2s3s2s1 : 2
Special Properties :
Degenerate
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}*720c
3-fold quotients : {2,4,30}*480a
4-fold quotients : {2,6,15}*360
5-fold quotients : {2,12,6}*288c
6-fold quotients : {2,2,30}*240
9-fold quotients : {2,4,10}*160
10-fold quotients : {2,6,6}*144b
12-fold quotients : {2,2,15}*120
15-fold quotients : {2,4,6}*96a
18-fold quotients : {2,2,10}*80
20-fold quotients : {2,6,3}*72
30-fold quotients : {2,2,6}*48
36-fold quotients : {2,2,5}*40
45-fold quotients : {2,4,2}*32
60-fold quotients : {2,2,3}*24
90-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
```s0 := (1,2);;
s1 := ( 18, 33)( 19, 34)( 20, 35)( 21, 36)( 22, 37)( 23, 38)( 24, 39)( 25, 40)
( 26, 41)( 27, 42)( 28, 43)( 29, 44)( 30, 45)( 31, 46)( 32, 47)( 63, 78)
( 64, 79)( 65, 80)( 66, 81)( 67, 82)( 68, 83)( 69, 84)( 70, 85)( 71, 86)
( 72, 87)( 73, 88)( 74, 89)( 75, 90)( 76, 91)( 77, 92)( 93,138)( 94,139)
( 95,140)( 96,141)( 97,142)( 98,143)( 99,144)(100,145)(101,146)(102,147)
(103,148)(104,149)(105,150)(106,151)(107,152)(108,168)(109,169)(110,170)
(111,171)(112,172)(113,173)(114,174)(115,175)(116,176)(117,177)(118,178)
(119,179)(120,180)(121,181)(122,182)(123,153)(124,154)(125,155)(126,156)
(127,157)(128,158)(129,159)(130,160)(131,161)(132,162)(133,163)(134,164)
(135,165)(136,166)(137,167);;
s2 := (  3,108)(  4,112)(  5,111)(  6,110)(  7,109)(  8,118)(  9,122)( 10,121)
( 11,120)( 12,119)( 13,113)( 14,117)( 15,116)( 16,115)( 17,114)( 18, 93)
( 19, 97)( 20, 96)( 21, 95)( 22, 94)( 23,103)( 24,107)( 25,106)( 26,105)
( 27,104)( 28, 98)( 29,102)( 30,101)( 31,100)( 32, 99)( 33,123)( 34,127)
( 35,126)( 36,125)( 37,124)( 38,133)( 39,137)( 40,136)( 41,135)( 42,134)
( 43,128)( 44,132)( 45,131)( 46,130)( 47,129)( 48,153)( 49,157)( 50,156)
( 51,155)( 52,154)( 53,163)( 54,167)( 55,166)( 56,165)( 57,164)( 58,158)
( 59,162)( 60,161)( 61,160)( 62,159)( 63,138)( 64,142)( 65,141)( 66,140)
( 67,139)( 68,148)( 69,152)( 70,151)( 71,150)( 72,149)( 73,143)( 74,147)
( 75,146)( 76,145)( 77,144)( 78,168)( 79,172)( 80,171)( 81,170)( 82,169)
( 83,178)( 84,182)( 85,181)( 86,180)( 87,179)( 88,173)( 89,177)( 90,176)
( 91,175)( 92,174);;
s3 := (  3,  9)(  4,  8)(  5, 12)(  6, 11)(  7, 10)( 13, 14)( 15, 17)( 18, 39)
( 19, 38)( 20, 42)( 21, 41)( 22, 40)( 23, 34)( 24, 33)( 25, 37)( 26, 36)
( 27, 35)( 28, 44)( 29, 43)( 30, 47)( 31, 46)( 32, 45)( 48, 54)( 49, 53)
( 50, 57)( 51, 56)( 52, 55)( 58, 59)( 60, 62)( 63, 84)( 64, 83)( 65, 87)
( 66, 86)( 67, 85)( 68, 79)( 69, 78)( 70, 82)( 71, 81)( 72, 80)( 73, 89)
( 74, 88)( 75, 92)( 76, 91)( 77, 90)( 93, 99)( 94, 98)( 95,102)( 96,101)
( 97,100)(103,104)(105,107)(108,129)(109,128)(110,132)(111,131)(112,130)
(113,124)(114,123)(115,127)(116,126)(117,125)(118,134)(119,133)(120,137)
(121,136)(122,135)(138,144)(139,143)(140,147)(141,146)(142,145)(148,149)
(150,152)(153,174)(154,173)(155,177)(156,176)(157,175)(158,169)(159,168)
(160,172)(161,171)(162,170)(163,179)(164,178)(165,182)(166,181)(167,180);;
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*s1*s0*s1,
s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3,
s3*s1*s2*s1*s2*s3*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,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 ];;
poly := F / rels;;

```
Permutation Representation (Magma) :
```s0 := Sym(182)!(1,2);
s1 := Sym(182)!( 18, 33)( 19, 34)( 20, 35)( 21, 36)( 22, 37)( 23, 38)( 24, 39)
( 25, 40)( 26, 41)( 27, 42)( 28, 43)( 29, 44)( 30, 45)( 31, 46)( 32, 47)
( 63, 78)( 64, 79)( 65, 80)( 66, 81)( 67, 82)( 68, 83)( 69, 84)( 70, 85)
( 71, 86)( 72, 87)( 73, 88)( 74, 89)( 75, 90)( 76, 91)( 77, 92)( 93,138)
( 94,139)( 95,140)( 96,141)( 97,142)( 98,143)( 99,144)(100,145)(101,146)
(102,147)(103,148)(104,149)(105,150)(106,151)(107,152)(108,168)(109,169)
(110,170)(111,171)(112,172)(113,173)(114,174)(115,175)(116,176)(117,177)
(118,178)(119,179)(120,180)(121,181)(122,182)(123,153)(124,154)(125,155)
(126,156)(127,157)(128,158)(129,159)(130,160)(131,161)(132,162)(133,163)
(134,164)(135,165)(136,166)(137,167);
s2 := Sym(182)!(  3,108)(  4,112)(  5,111)(  6,110)(  7,109)(  8,118)(  9,122)
( 10,121)( 11,120)( 12,119)( 13,113)( 14,117)( 15,116)( 16,115)( 17,114)
( 18, 93)( 19, 97)( 20, 96)( 21, 95)( 22, 94)( 23,103)( 24,107)( 25,106)
( 26,105)( 27,104)( 28, 98)( 29,102)( 30,101)( 31,100)( 32, 99)( 33,123)
( 34,127)( 35,126)( 36,125)( 37,124)( 38,133)( 39,137)( 40,136)( 41,135)
( 42,134)( 43,128)( 44,132)( 45,131)( 46,130)( 47,129)( 48,153)( 49,157)
( 50,156)( 51,155)( 52,154)( 53,163)( 54,167)( 55,166)( 56,165)( 57,164)
( 58,158)( 59,162)( 60,161)( 61,160)( 62,159)( 63,138)( 64,142)( 65,141)
( 66,140)( 67,139)( 68,148)( 69,152)( 70,151)( 71,150)( 72,149)( 73,143)
( 74,147)( 75,146)( 76,145)( 77,144)( 78,168)( 79,172)( 80,171)( 81,170)
( 82,169)( 83,178)( 84,182)( 85,181)( 86,180)( 87,179)( 88,173)( 89,177)
( 90,176)( 91,175)( 92,174);
s3 := Sym(182)!(  3,  9)(  4,  8)(  5, 12)(  6, 11)(  7, 10)( 13, 14)( 15, 17)
( 18, 39)( 19, 38)( 20, 42)( 21, 41)( 22, 40)( 23, 34)( 24, 33)( 25, 37)
( 26, 36)( 27, 35)( 28, 44)( 29, 43)( 30, 47)( 31, 46)( 32, 45)( 48, 54)
( 49, 53)( 50, 57)( 51, 56)( 52, 55)( 58, 59)( 60, 62)( 63, 84)( 64, 83)
( 65, 87)( 66, 86)( 67, 85)( 68, 79)( 69, 78)( 70, 82)( 71, 81)( 72, 80)
( 73, 89)( 74, 88)( 75, 92)( 76, 91)( 77, 90)( 93, 99)( 94, 98)( 95,102)
( 96,101)( 97,100)(103,104)(105,107)(108,129)(109,128)(110,132)(111,131)
(112,130)(113,124)(114,123)(115,127)(116,126)(117,125)(118,134)(119,133)
(120,137)(121,136)(122,135)(138,144)(139,143)(140,147)(141,146)(142,145)
(148,149)(150,152)(153,174)(154,173)(155,177)(156,176)(157,175)(158,169)
(159,168)(160,172)(161,171)(162,170)(163,179)(164,178)(165,182)(166,181)
(167,180);
poly := sub<Sym(182)|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*s1*s0*s1, s0*s2*s0*s2, s0*s3*s0*s3,
s1*s3*s1*s3, s3*s1*s2*s1*s2*s3*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,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 >;

```

to this polytope