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# Polytope of Type {30,12}

Atlas Canonical Name : {30,12}*720b
Also Known As : {30,12|2}. if this polytope has another name.
Group : SmallGroup(720,672)
Rank : 3
Schlafli Type : {30,12}
Number of vertices, edges, etc : 30, 180, 12
Order of s0s1s2 : 60
Order of s0s1s2s1 : 2
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{30,12,2} of size 1440
Vertex Figure Of :
{2,30,12} of size 1440
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {30,6}*360b
3-fold quotients : {10,12}*240, {30,4}*240a
5-fold quotients : {6,12}*144a
6-fold quotients : {10,6}*120, {30,2}*120
9-fold quotients : {10,4}*80
10-fold quotients : {6,6}*72a
12-fold quotients : {15,2}*60
15-fold quotients : {2,12}*48, {6,4}*48a
18-fold quotients : {10,2}*40
30-fold quotients : {2,6}*24, {6,2}*24
36-fold quotients : {5,2}*20
45-fold quotients : {2,4}*16
60-fold quotients : {2,3}*12, {3,2}*12
90-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
2-fold covers : {30,24}*1440b, {60,12}*1440b
Permutation Representation (GAP) :
```s0 := (  2,  5)(  3,  4)(  6, 11)(  7, 15)(  8, 14)(  9, 13)( 10, 12)( 17, 20)
( 18, 19)( 21, 26)( 22, 30)( 23, 29)( 24, 28)( 25, 27)( 32, 35)( 33, 34)
( 36, 41)( 37, 45)( 38, 44)( 39, 43)( 40, 42)( 47, 50)( 48, 49)( 51, 56)
( 52, 60)( 53, 59)( 54, 58)( 55, 57)( 62, 65)( 63, 64)( 66, 71)( 67, 75)
( 68, 74)( 69, 73)( 70, 72)( 77, 80)( 78, 79)( 81, 86)( 82, 90)( 83, 89)
( 84, 88)( 85, 87)( 92, 95)( 93, 94)( 96,101)( 97,105)( 98,104)( 99,103)
(100,102)(107,110)(108,109)(111,116)(112,120)(113,119)(114,118)(115,117)
(122,125)(123,124)(126,131)(127,135)(128,134)(129,133)(130,132)(137,140)
(138,139)(141,146)(142,150)(143,149)(144,148)(145,147)(152,155)(153,154)
(156,161)(157,165)(158,164)(159,163)(160,162)(167,170)(168,169)(171,176)
(172,180)(173,179)(174,178)(175,177);;
s1 := (  1,  7)(  2,  6)(  3, 10)(  4,  9)(  5,  8)( 11, 12)( 13, 15)( 16, 37)
( 17, 36)( 18, 40)( 19, 39)( 20, 38)( 21, 32)( 22, 31)( 23, 35)( 24, 34)
( 25, 33)( 26, 42)( 27, 41)( 28, 45)( 29, 44)( 30, 43)( 46, 52)( 47, 51)
( 48, 55)( 49, 54)( 50, 53)( 56, 57)( 58, 60)( 61, 82)( 62, 81)( 63, 85)
( 64, 84)( 65, 83)( 66, 77)( 67, 76)( 68, 80)( 69, 79)( 70, 78)( 71, 87)
( 72, 86)( 73, 90)( 74, 89)( 75, 88)( 91,142)( 92,141)( 93,145)( 94,144)
( 95,143)( 96,137)( 97,136)( 98,140)( 99,139)(100,138)(101,147)(102,146)
(103,150)(104,149)(105,148)(106,172)(107,171)(108,175)(109,174)(110,173)
(111,167)(112,166)(113,170)(114,169)(115,168)(116,177)(117,176)(118,180)
(119,179)(120,178)(121,157)(122,156)(123,160)(124,159)(125,158)(126,152)
(127,151)(128,155)(129,154)(130,153)(131,162)(132,161)(133,165)(134,164)
(135,163);;
s2 := (  1,106)(  2,107)(  3,108)(  4,109)(  5,110)(  6,111)(  7,112)(  8,113)
(  9,114)( 10,115)( 11,116)( 12,117)( 13,118)( 14,119)( 15,120)( 16, 91)
( 17, 92)( 18, 93)( 19, 94)( 20, 95)( 21, 96)( 22, 97)( 23, 98)( 24, 99)
( 25,100)( 26,101)( 27,102)( 28,103)( 29,104)( 30,105)( 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)( 46,151)( 47,152)( 48,153)
( 49,154)( 50,155)( 51,156)( 52,157)( 53,158)( 54,159)( 55,160)( 56,161)
( 57,162)( 58,163)( 59,164)( 60,165)( 61,136)( 62,137)( 63,138)( 64,139)
( 65,140)( 66,141)( 67,142)( 68,143)( 69,144)( 70,145)( 71,146)( 72,147)
( 73,148)( 74,149)( 75,150)( 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);;
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*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,
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*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(180)!(  2,  5)(  3,  4)(  6, 11)(  7, 15)(  8, 14)(  9, 13)( 10, 12)
( 17, 20)( 18, 19)( 21, 26)( 22, 30)( 23, 29)( 24, 28)( 25, 27)( 32, 35)
( 33, 34)( 36, 41)( 37, 45)( 38, 44)( 39, 43)( 40, 42)( 47, 50)( 48, 49)
( 51, 56)( 52, 60)( 53, 59)( 54, 58)( 55, 57)( 62, 65)( 63, 64)( 66, 71)
( 67, 75)( 68, 74)( 69, 73)( 70, 72)( 77, 80)( 78, 79)( 81, 86)( 82, 90)
( 83, 89)( 84, 88)( 85, 87)( 92, 95)( 93, 94)( 96,101)( 97,105)( 98,104)
( 99,103)(100,102)(107,110)(108,109)(111,116)(112,120)(113,119)(114,118)
(115,117)(122,125)(123,124)(126,131)(127,135)(128,134)(129,133)(130,132)
(137,140)(138,139)(141,146)(142,150)(143,149)(144,148)(145,147)(152,155)
(153,154)(156,161)(157,165)(158,164)(159,163)(160,162)(167,170)(168,169)
(171,176)(172,180)(173,179)(174,178)(175,177);
s1 := Sym(180)!(  1,  7)(  2,  6)(  3, 10)(  4,  9)(  5,  8)( 11, 12)( 13, 15)
( 16, 37)( 17, 36)( 18, 40)( 19, 39)( 20, 38)( 21, 32)( 22, 31)( 23, 35)
( 24, 34)( 25, 33)( 26, 42)( 27, 41)( 28, 45)( 29, 44)( 30, 43)( 46, 52)
( 47, 51)( 48, 55)( 49, 54)( 50, 53)( 56, 57)( 58, 60)( 61, 82)( 62, 81)
( 63, 85)( 64, 84)( 65, 83)( 66, 77)( 67, 76)( 68, 80)( 69, 79)( 70, 78)
( 71, 87)( 72, 86)( 73, 90)( 74, 89)( 75, 88)( 91,142)( 92,141)( 93,145)
( 94,144)( 95,143)( 96,137)( 97,136)( 98,140)( 99,139)(100,138)(101,147)
(102,146)(103,150)(104,149)(105,148)(106,172)(107,171)(108,175)(109,174)
(110,173)(111,167)(112,166)(113,170)(114,169)(115,168)(116,177)(117,176)
(118,180)(119,179)(120,178)(121,157)(122,156)(123,160)(124,159)(125,158)
(126,152)(127,151)(128,155)(129,154)(130,153)(131,162)(132,161)(133,165)
(134,164)(135,163);
s2 := Sym(180)!(  1,106)(  2,107)(  3,108)(  4,109)(  5,110)(  6,111)(  7,112)
(  8,113)(  9,114)( 10,115)( 11,116)( 12,117)( 13,118)( 14,119)( 15,120)
( 16, 91)( 17, 92)( 18, 93)( 19, 94)( 20, 95)( 21, 96)( 22, 97)( 23, 98)
( 24, 99)( 25,100)( 26,101)( 27,102)( 28,103)( 29,104)( 30,105)( 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)( 46,151)( 47,152)
( 48,153)( 49,154)( 50,155)( 51,156)( 52,157)( 53,158)( 54,159)( 55,160)
( 56,161)( 57,162)( 58,163)( 59,164)( 60,165)( 61,136)( 62,137)( 63,138)
( 64,139)( 65,140)( 66,141)( 67,142)( 68,143)( 69,144)( 70,145)( 71,146)
( 72,147)( 73,148)( 74,149)( 75,150)( 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);
poly := sub<Sym(180)|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*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,
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*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